diff --git a/parse/train/0WASBV4xkhy/0WASBV4xkhy_content_list.json b/parse/train/0WASBV4xkhy/0WASBV4xkhy_content_list.json new file mode 100644 index 0000000000000000000000000000000000000000..e6556c6d14e2e55c5b3f24aa91c5cfb1322bebe6 --- /dev/null +++ b/parse/train/0WASBV4xkhy/0WASBV4xkhy_content_list.json @@ -0,0 +1,1490 @@ +[ + { + "type": "text", + "text": "Towards a Theoretical Framework of Out-of-Distribution Generalization ", + "text_level": 1, + "bbox": [ + 272, + 122, + 725, + 172 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Chuanlong XieHuawei Noah’s Ark Labxie.chuanlong@huawei.com", + "bbox": [ + 377, + 215, + 584, + 257 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Haotian Ye \nPeking University \nPazhou Lab \nhaotianye@pku.edu.cn ", + "bbox": [ + 184, + 215, + 359, + 271 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Tianle Cai Peking University caitianle1998@pku.edu.cn ", + "bbox": [ + 604, + 215, + 812, + 257 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Ruichen Li ", + "text_level": 1, + "bbox": [ + 218, + 292, + 297, + 305 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Zhenguo Li ", + "text_level": 1, + "bbox": [ + 400, + 292, + 483, + 306 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Liwei Wang ", + "text_level": 1, + "bbox": [ + 637, + 292, + 725, + 306 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Peking University xk-lrc@pku.edu.cn ", + "bbox": [ + 184, + 306, + 331, + 333 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Huawei Noah’s Ark Lab Li.Zhenguo@huawei.com ", + "bbox": [ + 352, + 306, + 532, + 333 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Key Laboratory of Machine Perception, MOE, School of EECS, Institute for Artificial Intelligence, Peking University wanglw@cis.pku.edu.cn ", + "bbox": [ + 552, + 305, + 812, + 376 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Abstract ", + "text_level": 1, + "bbox": [ + 462, + 411, + 535, + 426 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Generalization to out-of-distribution (OOD) data is one of the central problems in modern machine learning. Recently, there is a surge of attempts to propose algorithms that mainly build upon the idea of extracting invariant features. Although intuitively reasonable, theoretical understanding of what kind of invariance can guarantee OOD generalization is still limited, and generalization to arbitrary out-of-distribution is clearly impossible. In this work, we take the first step towards rigorous and quantitative definitions of 1) what is OOD; and 2) what does it mean by saying an OOD problem is learnable. We also introduce a new concept of expansion function, which characterizes to what extent the variance is amplified in the test domains over the training domains, and therefore give a quantitative meaning of invariant features. Based on these, we prove OOD generalization error bounds. It turns out that OOD generalization largely depends on the expansion function. As recently pointed out by [21], any OOD learning algorithm without a model selection module is incomplete. Our theory naturally induces a model selection criterion. Extensive experiments on benchmark OOD datasets demonstrate that our model selection criterion has a significant advantage over baselines. ", + "bbox": [ + 233, + 441, + 766, + 662 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "1 Introduction ", + "text_level": 1, + "bbox": [ + 174, + 686, + 310, + 704 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "One of the most fundamental assumptions of classic supervised learning is the “i.i.d. assumption”, which states that the training and the test data are independent and identically distributed. However, this assumption can be easily violated in a reality [8, 10, 11, 17, 38, 48, 56] where the test data usually have a different distribution than the training data. This motivates the research on the out-ofdistribution (OOD) generalization, or domain generalization problem, which assumes access only to data drawn from a set $\\mathcal { E } _ { a v a i l }$ of available domains during training, and the goal is to generalize to a larger domain set ${ \\mathcal { E } } _ { a l l }$ including unseen domains. ", + "bbox": [ + 174, + 718, + 825, + 815 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "To generalize to OOD data, most existing algorithms attempt to learn features that are invariant to a certain extent across training domains in the hope that such invariance also holds in unseen domains. For example, distributional matching-based methods [20, 35, 55] seek to learn features that have the same distribution across different domains; IRM [5] and its variants [1, 32, 33] learn feature representations such that the optimal linear classifier on top of the representation matches across domains. ", + "bbox": [ + 174, + 821, + 825, + 904 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "35th Conference on Neural Information Processing Systems (NeurIPS 2021). ", + "bbox": [ + 174, + 922, + 629, + 936 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Though the idea of learning invariant features is intuitively reasonable, there is only limited theoretical understanding of what kind of invariance can guarantee OOD generalization. Clearly, generalization to an arbitrary out-of-distribution domain is impossible and in practice, the features can hardly be absolutely invariant from $\\mathcal { E } _ { a v a i l }$ to ${ \\mathcal { E } } _ { a l l }$ unless all the domains are identical. So it is necessary to first formulate what OOD data can be generalized to, or, what is the relation between the available training domain set $\\mathcal { E } _ { a v a i l }$ and the entire domain set ${ \\mathcal { E } } _ { a l l }$ . Meanwhile, to what extent the invariance of features on $\\mathcal { E } _ { a v a i l }$ can be preserved in ${ \\mathcal { E } } _ { a l l }$ should be rigorously characterized. ", + "bbox": [ + 174, + 92, + 825, + 188 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "In this paper, we take the first step towards a general OOD framework by quantitatively formalizing the relationship between $\\mathcal { E } _ { a v a i l }$ and ${ \\mathcal { E } } _ { a l l }$ in terms of the distributions of features and provide OOD generalization guarantees based on our quantification of the difficulty of OOD generalization problem. Specifically, we first rigorously formulate the intuition of invariant features used in previous works by introducing the “variation” and “informativeness” (Definition 3.1 and 3.2) of each feature. Our theoretical insight can then be informally stated as: for learnable OOD problems, if a feature is informative for the classification task as well as invariant over $\\mathcal { E } _ { a v a i l }$ , then it is still invariant over ${ \\mathcal { E } } _ { a l l }$ . In other words, invariance of informative features in $\\mathcal { E } _ { a v a i l }$ can be preserved in ${ \\mathcal { E } } _ { a l l }$ . We further introduce a class of functions, dubbed expansion function (Definition 3.3), to quantitatively characterize to what extent the variance of features on $\\mathcal { E } _ { a v a i l }$ is amplified on ${ \\mathcal { E } } _ { a l l }$ . ", + "bbox": [ + 174, + 194, + 825, + 333 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "Based on our formulation, we derive theoretical guarantees on the OOD generalization error, i.e., the gap of largest error between the domain in $\\mathcal { E } _ { a v a i l }$ and domain in ${ \\mathcal { E } } _ { a l l }$ . Specifically, we prove the upper and lower bound of OOD generalization error in terms of the expansion function and the variation of learned features over $\\mathcal { E } _ { a v a i l }$ . Our results theoretically confirm that 1) the expansion function can reflect the difficulty of OOD generalization problem, i.e., problems with more rapidly increasing expansion functions are harder and have worse generalization guarantees; 2) the generalization error gap can tend to zero when the variation of learned features tend to zero, so minimizing the variation in $\\mathcal { E } _ { a v a i l }$ can reduce the generalization error. ", + "bbox": [ + 174, + 339, + 825, + 449 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "As pointed out by Gulrajani and Lopez-Paz [21], any OOD algorithm without a specified model selection criterion is not complete. Since ${ \\mathcal { E } } _ { a l l }$ is unseen, hyper-parameters can only be chosen according to $\\mathcal { E } _ { a v a i l }$ . Previous selection methods mainly focus on validation accuracy over $\\mathcal { E } _ { a v a i l }$ which is only a biased metric of OOD performance. On the contrary, a promising model selection method should instead be predictive of OOD performance. Inspired by our bounds, we propose a model selection method to select models with high validation accuracy and low variation, which corresponds to the upper bound of OOD error. The introduction of a model’s variation relieves the problem of classic selection methods, in which models that overfit $\\mathcal { E } _ { a v a i l }$ tend to be selected. Experimental results show that our method can outperform baselines and select models with higher OOD accuracy. ", + "bbox": [ + 174, + 457, + 825, + 594 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "Contributions. We summarize our major contributions here: ", + "bbox": [ + 174, + 622, + 583, + 636 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "• We introduce a quantitative and rigorous formulation of OOD generalization problem that characterizes the relation of invariance over the training domain set $\\mathcal { E } _ { a v a i l }$ and test domain set ${ \\mathcal { E } } _ { a l l }$ . The core quantity in our characterization, the expansion function, determines the difficulty of an OOD generalization problem. ", + "bbox": [ + 173, + 654, + 826, + 709 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "• We prove novel OOD generalization error bounds based on our formulation. The upper and lower bounds together indicate that the expansion function well characterizes the OOD generalization ability of features with different levels of variation. ", + "bbox": [ + 174, + 724, + 823, + 766 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "• We design a model selection criterion that is inspired by our generalization bounds. Our criterion takes both the performance on training domains and the variation of models into consideration and is predictive of OOD performance according to our bounds. Experimental results demonstrate our selection criterion can choose models with higher OOD accuracy. ", + "bbox": [ + 173, + 782, + 825, + 838 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "The rest of the paper is organized as follows: Section 2 is our preliminary. In Section 3, we give our theoretical formulation. Section 4 gives our generalization bound. We propose our model selection method in Section 5. In Section 6 we conduct experiments on expansion function and model selection. We review more related works in Section 7 and conclude our work in Section 8. ", + "bbox": [ + 176, + 856, + 825, + 911 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "2 Preliminary ", + "text_level": 1, + "bbox": [ + 173, + 88, + 305, + 107 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Throughout the paper, we consider a multi-class classification task $\\mathcal { X } \\to \\mathcal { Y } = \\{ 1 , . . . , K \\}$ .1 Let ${ \\mathcal { E } } _ { a l l }$ be the domain set we want to generalize to, and $\\mathcal { E } _ { a v a i l } \\subseteq \\mathcal { E } _ { a l l }$ be the available domain set, i.e., all domains we have during the training procedure. We denote $( X ^ { e } , Y ^ { e } )$ to be the input-label pair drawn from the data distribution of domain $e$ . The OOD generalization goal is to find a classifier $f ^ { * }$ that minimizes the worst-domain loss on ${ \\mathcal { E } } _ { a l l }$ : ", + "bbox": [ + 174, + 119, + 825, + 190 + ], + "page_idx": 2 + }, + { + "type": "equation", + "img_path": "images/92ece3d80a18d8b09d86d490b1aa6271dee39a3b130e884626dc355a67e41bdf.jpg", + "text": "$$\nf ^ { * } = \\underset { f \\in \\mathcal { F } } { \\mathrm { a r g m i n } } \\mathcal { L } ( \\mathcal { E } _ { a l l } , f ) , \\mathcal { L } ( \\mathcal { E } , f ) \\triangleq \\underset { e \\in \\mathcal { E } } { \\mathrm { m a x } } \\mathbb { E } \\big [ \\ell \\big ( f ( X ^ { e } ) , Y ^ { e } \\big ) \\big ]\n$$", + "text_format": "latex", + "bbox": [ + 300, + 195, + 696, + 224 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "where $\\mathcal { F } : \\mathcal { X } \\xrightarrow { } \\mathbb { R } ^ { K }$ is the the hypothetical space and $\\ell ( \\cdot , \\cdot )$ is a loss function. Similar to previous works [5, 16, 27, 33], we assume that $f$ can be decomposed into $g \\circ h$ , where $g \\in \\mathcal { G } : \\mathbb { R } ^ { d } \\mathbb { R } ^ { K }$ is the top classifier and $h : \\mathcal { X } \\mathbb { R } ^ { d }$ is a $d$ -dimensional feature extractor, i.e., ", + "bbox": [ + 173, + 232, + 825, + 276 + ], + "page_idx": 2 + }, + { + "type": "equation", + "img_path": "images/ee0a57e21bd4e230b25e434833fe9ced52367d0fd15ac040eda84f95a58b070d.jpg", + "text": "$$\nh ( x ) = ( \\phi _ { 1 } ( x ) , \\phi _ { 2 } ( x ) , \\ldots , \\phi _ { d } ( x ) ) ^ { \\top } , \\quad \\phi _ { i } \\in \\Phi .\n$$", + "text_format": "latex", + "bbox": [ + 323, + 284, + 640, + 303 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Here $\\Phi$ is the set of scalar feature maps which map $\\mathcal { X }$ to $\\mathbb { R }$ and $d$ is fixed. We will call each $\\phi \\in \\Phi$ a feature for simplicity. Given a domain $e$ , we denote the $d$ -dimensional random vector $h ( X ^ { e } )$ as $h ^ { e }$ , one-dimensional feature $\\phi ( X ^ { e } )$ as $\\phi ^ { e }$ , and the conditional distribution of $h ^ { e } , \\phi ^ { e }$ given $Y ^ { e } = y$ as $\\mathbb { P } ( h ^ { e } | y ) , \\mathbb { P } ( \\phi ^ { e } | y )$ . For simplicity, we assume the data distribution is balanced in every domain, i.e., $P ( Y ^ { e } = y ) = \\frac { 1 } { K } , \\forall y \\in \\mathcal { Y } , e \\in \\mathcal { E } _ { a l l }$ . Our framework can be easily extended to the case where the balanced assumption is removed, with an additional term corresponding to the imbalance adding to the generalization bounds. ", + "bbox": [ + 173, + 309, + 826, + 407 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "3 Framework of OOD Generalization Problem ", + "text_level": 1, + "bbox": [ + 173, + 426, + 580, + 444 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "The main challenge of formalizing the OOD generalization problem is to mathematically describe the connection between $\\mathcal { E } _ { a v a i l }$ and ${ \\mathcal { E } } _ { a l l }$ and how generalization depends on this relation. Towards this goal, we introduce several quantities to characterize the relation of feature distributions over different domains and bridge $\\mathcal { E } _ { a v a i l }$ and ${ \\mathcal { E } } _ { a l l }$ by expansion function (Definition 3.3) over the quantities we have introduced. Our framework is motivated by the understanding that, in an OOD generalization task, certain “property” of “good” features in $\\mathcal { E } _ { a v a i l }$ should be “preserved” in ${ \\mathcal { E } } _ { a l l }$ (the reason is described in Section 1). In Section 3.1, we will go into details on what we mean by “property” (variation, Definition 3.1), “good” (informativeness, Definition 3.2), and “preserved” (measured by expansion function). In Section 6.2, we further illustrate the key concepts in our framework by a real-world OOD problem. ", + "bbox": [ + 173, + 457, + 826, + 597 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "3.1 Formalizing OOD Problem by Quantifying Feature Distribution ", + "text_level": 1, + "bbox": [ + 174, + 612, + 658, + 628 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "We first introduce the concepts “variation\" and “informativeness\" of a feature $\\phi$ . The first one is what we expect to be preserved in ${ \\mathcal { E } } _ { a l l }$ and the second one characterizes what features will be considered. Specifically, let $\\rho ( \\mathbb { P } , \\mathbb { Q } )$ be a symmetric “distance” of two distributions. Note that $\\rho$ can have many choices, like $L _ { 2 }$ Distance, Total Variation and symmetric KL-divergence, etc. The variation and informativeness are defined as follows: ", + "bbox": [ + 173, + 638, + 825, + 708 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Definition 3.1 (Variation). The variation of feature $\\phi ( \\cdot )$ across a domain set $\\mathcal { E }$ is ", + "bbox": [ + 173, + 712, + 705, + 728 + ], + "page_idx": 2 + }, + { + "type": "equation", + "img_path": "images/b5e6d90d4f6e707f43d0a5fd09cd50b46b59f7364145ae7e58a3a5a8d9599fbf.jpg", + "text": "$$\n\\mathcal { V } _ { \\rho } ( \\phi , \\mathcal { E } ) = \\operatorname* { m a x } _ { y \\in \\mathcal { V } } \\operatorname* { s u p } _ { e , e ^ { \\prime } \\in \\mathcal { E } } \\rho \\big ( \\mathbb { P } ( \\phi ^ { e } | y ) , \\mathbb { P } ( \\phi ^ { e ^ { \\prime } } | y ) \\big ) .\n$$", + "text_format": "latex", + "bbox": [ + 348, + 736, + 648, + 765 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "A feature $\\phi ( \\cdot )$ is $\\varepsilon$ -invariant across $\\mathcal { E }$ , $i f \\varepsilon \\geq \\mathcal { V } ( \\phi , \\mathcal { E } )$ (We omit the subscript $\\rho$ in case of no ambiguity). Definition 3.2 (Informativeness). The informativeness of feature $\\phi ( \\cdot )$ across a domain set $\\mathcal { E }$ is ", + "bbox": [ + 171, + 772, + 828, + 808 + ], + "page_idx": 2 + }, + { + "type": "equation", + "img_path": "images/b5d2d9f29351ffd114b5530974aa3d8ba7d34f3c8275c029e6a978717a1c0731.jpg", + "text": "$$\n\\mathcal { T } _ { \\rho } ( \\phi , \\mathcal { E } ) = \\frac { 1 } { K ( K - 1 ) } \\sum _ { \\stackrel { y \\neq y ^ { \\prime } } { y , y ^ { \\prime } \\in y } } \\operatorname* { m i n } _ { e \\in \\mathcal { E } } \\rho \\big ( \\mathbb { P } ( \\phi ^ { e } | y ) , \\mathbb { P } ( \\phi ^ { e } | y ^ { \\prime } ) \\big ) .\n$$", + "text_format": "latex", + "bbox": [ + 292, + 813, + 671, + 864 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "A feature $\\phi ( \\cdot )$ is $\\delta$ -informative across $\\varepsilon$ , if $\\quad \\delta \\leq { \\mathcal { I } } ( \\phi , { \\mathcal { E } } )$ . ", + "bbox": [ + 171, + 871, + 539, + 887 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "The variation $\\mathcal { V } ( \\phi , \\mathcal { E } )$ measures the stability of $\\phi ( \\cdot )$ over the domains in $\\mathcal { E }$ and the informativeness ${ \\mathcal { T } } ( \\phi , { \\mathcal { E } } )$ captures the ability of $\\phi ( \\cdot )$ to distinguish different labels. We would like to highlight that the variation and informativeness are defined on each one-dimensional feature $\\phi ( \\cdot )$ . Unlike previous distance between distributions defined in $d$ -dimensional space, our definitions are more reasonable and practical, since it can be easily calculated and analyzed. ", + "bbox": [ + 174, + 90, + 825, + 161 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "We are now ready to introduce the core quantity for connecting $\\mathcal { E } _ { a v a i l }$ and ${ \\mathcal { E } } _ { a l l }$ . Our motivation, as elaborated in the introduction section, is that, if a feature is informative for the classification task and invariant over $\\mathcal { E } _ { a v a i l }$ , then to enable OOD generalization from $\\mathcal { E } _ { a v a i l }$ to ${ \\mathcal { E } } _ { a l l }$ , it should be still invariant over ${ \\mathcal { E } } _ { a l l }$ . So the relation between $\\mathcal { V } ( \\phi , \\mathcal { E } _ { a v a i l } )$ and $\\mathinner { \\gamma \\mathopen { \\left( \\phi , \\mathcal { E } _ { a l l } \\right) } }$ of an informative feature captures the feasibility and difficulty of OOD generalization. To quantitatively measure this relation, we define the following function class: ", + "bbox": [ + 174, + 166, + 825, + 251 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Definition 3.3 (Expansion Function). We say a function $s : \\mathbb { R } ^ { + } \\cup \\{ 0 \\} \\mathbb { R } ^ { + } \\cup \\{ 0 , + \\infty \\}$ is an expansion function, iff the following properties hold: 1) $s ( \\cdot )$ is monotonically increasing and $s ( x ) \\geq x , \\forall x \\geq 0 ; 2 ,$ ) $\\begin{array} { r } { \\operatorname* { l i m } _ { x 0 ^ { + } } s ( x ) = s ( 0 ) = 0 } \\end{array}$ . ", + "bbox": [ + 174, + 255, + 825, + 297 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "This function class gives a full characterization of how the variation between $\\mathcal { E } _ { a v a i l }$ and ${ \\mathcal { E } } _ { a l l }$ is related. Based on this function class, we can introduce our formulation of the learnability of OOD generalization as follows: ", + "bbox": [ + 174, + 309, + 825, + 351 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Definition 3.4 (Learnability). Let $\\Phi$ be the feature space and $\\rho$ be a distribution distance. We say an OOD generalization problem from $\\mathcal { E } _ { a v a i l }$ to ${ \\mathcal { E } } _ { a l l }$ is learnable if there exists an expansion function $s ( \\cdot )$ and $\\delta \\geq 0$ , such that: for all $\\phi \\in \\Phi$ satisfying ${ \\mathcal { T } } _ { \\rho } ( \\phi , { \\mathcal { E } } _ { a v a i l } ) \\geq \\delta$ , we have $s ( \\mathcal { V } _ { \\rho } ( \\phi , \\mathcal { E } _ { a v a i l } ) ) \\geq$ $\\mathcal { V } _ { \\rho } \\big ( \\phi , \\mathcal { E } _ { a l l } \\big )$ . If such $s ( \\cdot )$ and $\\delta$ exist, we further call this problem $( s ( \\cdot ) , \\delta )$ -learnable. If an $o o D$ generalization problem is not learnable, we call $i t$ unlearnable. ", + "bbox": [ + 174, + 356, + 825, + 425 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "To understand the intuition and rationality of our formulation, several discussions are in order. ", + "bbox": [ + 171, + 436, + 787, + 452 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Properties of the expansion function. In Definition 3.3, we highlight two properties of the expansion function. The first property comes naturally from the monotonicity properties of variation: any $\\varepsilon _ { 1 }$ -invariant feature is also $\\varepsilon _ { 2 }$ -invariant for $\\varepsilon _ { 2 } \\geq \\varepsilon _ { 1 }$ ; and $\\mathcal { V } ( \\phi , \\mathcal { E } _ { 1 } ) \\leq \\mathcal { V } ( \\phi , \\mathcal { E } _ { 2 } )$ for any ${ \\mathcal { E } } _ { 1 } \\subseteq { \\mathcal { E } } _ { 2 }$ . The monotonicity also implies that larger ${ \\mathcal { E } } _ { a l l }$ will induce larger $s ( \\cdot ) ^ { 2 }$ and it is also harder to be generalized to. From this view, we can see that the scale of $s ( \\cdot )$ can reflect the difficulty of OOD generalization. The second property is more crucial since it formulates the intuition that if an informative feature is almost invariant in $\\mathcal { E } _ { a v a i l }$ , it should remain invariant in ${ \\mathcal { E } } _ { a l l }$ . Without this assumption, OOD generalization can never be guaranteed because we cannot predict whether an invariant and informative feature in $\\mathcal { E } _ { a v a i l }$ will vary severely in unseen ${ \\mathcal { E } } _ { a l l }$ . ", + "bbox": [ + 173, + 467, + 826, + 592 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Necessity of informativeness. We include a seemingly redundant quantity informativeness in the definition of learnability. However, this term is necessary because only informative features are responsible for the performance of classification. Non-informative but invariant features over $\\mathcal { E } _ { a v a i l }$ may only capture some noise that is irrelevant to the classification problem, and we shall not expect the noise to be invariant over ${ \\mathcal { E } } _ { a l l }$ . Moreover, we show in Figure 1 that in practice, many invariant but useless features in $\\mathcal { E } _ { a v a i l }$ vary a lot in ${ \\mathcal { E } } _ { a l l }$ , and adding the constraint of informativeness makes the expansion function reasonable. In addition, there are multiple choices of $( s ( \\cdot ) , \\delta )$ to make an OOD generalization problem learnable: larger $\\delta$ will filter out more features, and so $\\dot { s } ( \\cdot )$ can be smaller (flatter). This multiplicity will result in a trade-off between $s ( \\cdot )$ and $\\delta$ , which will be discussed in Section 6.2. ", + "bbox": [ + 174, + 607, + 825, + 746 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Two extreme cases: i.i.d. & unlearnable. To better understand the concept of learnability, we consider two extreme cases. (1) The first example is when all data from different $\\textit { e } \\in \\mathcal { E } _ { a l l }$ are identically distributed, i.e., the classic supervised learning setting. This problem is $( s ( \\cdot ) , 0 )$ -learnable with $s ( x ) = x$ , implying no extra difficulty in OOD generalization. (2) As an example of unlearnable, consider the following case (modified from Colored MNIST [5]): For $e \\in \\mathcal { E } _ { a v a i l }$ , images with label 0 always has a red background while images with label 1 has a blue background. For $e \\in \\mathcal { E } _ { a l l } \\ \\backslash \\ \\mathcal { E } _ { a v a i l }$ , this relationship is entirely inverse. Since data from different $e \\in \\mathcal { E } _ { a v a i l }$ are identically distributed but different from other $e \\in \\mathcal { E } _ { a l l }$ , no expansion function can make it learnable, i.e., it is OOD-unlearnable. ", + "bbox": [ + 173, + 761, + 825, + 873 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "The unlearnability of this case also coincides with our intuition: Without prior knowledge, it is not clear from merely the training data, whether the task is to distinguish digit 0 from 1, or to distinguish color red from blue. As a result, generalization to ${ \\mathcal { E } } _ { a l l }$ cannot be guaranteed. ", + "bbox": [ + 173, + 90, + 825, + 133 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "4 Generalization Bound ", + "text_level": 1, + "bbox": [ + 174, + 151, + 390, + 170 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "In this section, we consider an OOD generalization problem from $\\mathcal { E } _ { a v a i l }$ to ${ \\mathcal { E } } _ { a l l }$ , and our goal is to analyze the OOD generalization error of classifier $f = g \\circ h$ defined by ", + "bbox": [ + 173, + 183, + 823, + 212 + ], + "page_idx": 4 + }, + { + "type": "equation", + "img_path": "images/f61601e575911b7b38e2ff41bc91c80f1ca3bdfad17110177fbe71a235a390ce.jpg", + "text": "$$\n\\mathrm { e r r } ( f ) = \\mathcal { L } ( \\mathcal { E } _ { a l l } , f ) - \\mathcal { L } ( \\mathcal { E } _ { a v a i l } , f ) ,\n$$", + "text_format": "latex", + "bbox": [ + 362, + 217, + 599, + 234 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "where we assume the loss function $l ( \\cdot , \\cdot )$ is bounded by $[ 0 , C ]$ . We prove two upper bounds (4.1, 4.2) as well as a lower bound (4.3) for $\\operatorname { e r r } ( f )$ based on our formulation. Our bounds together provide a complete characterization of the difficulty of OOD generalization. Since we expect that an invariant classifier can generalize to unseen domains, we hope to bound $\\operatorname { e r r } ( f )$ in terms of the certain variation of $f$ . To this end, we define the variation and informativeness of $f$ in terms of its features, i.e., ", + "bbox": [ + 173, + 238, + 825, + 309 + ], + "page_idx": 4 + }, + { + "type": "equation", + "img_path": "images/37ed4f7a0dc7d2319bdc5086f2478da0e7cc992b3f71f92313ccf9703cd17289.jpg", + "text": "$$\n\\begin{array} { r l r } { \\mathcal { V } ^ { \\operatorname* { s u p } } ( h , \\mathcal { E } _ { a v a i l } ) } & { \\triangleq } & { \\underset { \\beta \\in S ^ { d - 1 } } { \\operatorname* { s u p } } \\mathcal { V } ( \\beta ^ { \\top } h , \\mathcal { E } _ { a v a i l } ) , } \\\\ { \\mathcal { T } ^ { \\operatorname* { i n f } } ( h , \\mathcal { E } _ { a v a i l } ) } & { \\triangleq } & { \\underset { \\beta \\in S ^ { d - 1 } } { \\operatorname* { i n f } } \\mathcal { T } ( \\beta ^ { \\top } h , \\mathcal { E } _ { a v a i l } ) , } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 348, + 313, + 647, + 372 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "where $( \\beta ^ { \\top } h ) ( x ) = \\beta ^ { \\top } h ( x )$ is a feature and $S ^ { d - 1 } = \\{ \\beta \\in \\mathbb { R } ^ { d } : \\| \\beta \\| _ { 2 } = 1 \\}$ is the unit $( d - 1 )$ -sphere. ", + "bbox": [ + 171, + 376, + 825, + 395 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Necessity of using supremum over linear combination. One seemingly plausible definition of the variation of a classifier $f$ can be the supremum over all $\\mathcal { V } ( \\phi _ { i } , \\mathcal { E } _ { a v a i l } ) , i \\in [ d ]$ . However, as is shown in Appendix 1, it is possible that two high-dimensional joint distributions have close marginal distribution in each dimension, while they do not overlap. In other words, there exist cases where $\\mathcal { V } ( \\phi _ { i } , \\mathcal { E } _ { a l l } ) = 0 , \\forall i \\in [ d ]$ but after applying the top model $g$ over $\\phi _ { i }$ ’s, the distribution varies a lot in $\\mathcal { E } _ { a v a i l }$ . Our definition comes from the simple idea that the class of top model $\\mathcal { G }$ is at least a linear space, so we should at least consider the variation of every (normalized) linear combination of $h ( \\cdot )$ With this, we can guarantee the joint distribution distance is still small. ", + "bbox": [ + 173, + 406, + 825, + 518 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Theorem 4.1 (Main Theorem). Suppose we have learned a classifier $f ( x ) = g ( h ( x ) )$ such that $\\forall e \\in \\mathcal { E } _ { a l l }$ and $\\forall y \\in \\mathcal { V } ;$ $, p _ { h ^ { e } | Y ^ { e } } ( h | y ) \\in L ^ { 2 } ( \\mathbb R ^ { d } )$ . Denote the characteristic function of random variable $h ^ { e } | Y ^ { e }$ as $\\hat { p } _ { h ^ { e } | Y ^ { e } } ( t | y ) = \\mathbb { E } [ \\exp \\{ i \\langle t , h ^ { e } \\rangle \\} | Y ^ { e } = y ]$ . Assume the hypothetical space $\\mathcal { F }$ satisfies the following regularity conditions that $\\exists \\alpha , M _ { 1 } , M _ { 2 } > 0 , \\forall f \\in \\mathcal { F } , \\forall e \\in \\mathcal { E } _ { a l l } , y \\in \\mathcal { V }$ , ", + "bbox": [ + 173, + 521, + 825, + 582 + ], + "page_idx": 4 + }, + { + "type": "equation", + "img_path": "images/ee4b4c076d84f9c894a2e146552ae306c55f38d76c671d508d171889b7b12f9f.jpg", + "text": "$$\n\\int _ { h \\in \\mathbb { R } ^ { d } } p _ { h ^ { e } | Y ^ { e } } ( h | y ) | h | ^ { \\alpha } \\mathrm { d } h \\leq M _ { 1 } \\quad a n d \\quad \\int _ { t \\in \\mathbb { R } ^ { d } } | \\hat { p } _ { h ^ { e } | Y ^ { e } } ( t | y ) | | t | ^ { \\alpha } \\mathrm { d } t \\leq M _ { 2 } .\n$$", + "text_format": "latex", + "bbox": [ + 251, + 585, + 745, + 619 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "$I f ( \\mathcal { E } _ { a v a i l } , \\mathcal { E } _ { a l l } )$ is $\\left( s ( \\cdot ) , \\mathcal { T } ^ { i n f } ( h , \\mathcal { E } _ { a v a i l } ) \\right)$ -learnable under $\\Phi$ with Total Variation $\\rho ^ { 3 }$ , then we have ", + "bbox": [ + 174, + 625, + 802, + 641 + ], + "page_idx": 4 + }, + { + "type": "equation", + "img_path": "images/47d72154cd89cf0c6271de3f68baa90e658deea6245d64bec4f750c758b38256.jpg", + "text": "$$\n\\operatorname { e r r } ( f ) \\leq O \\left( s \\left( \\mathcal { V } _ { \\rho } ^ { s u p } ( h , \\mathcal { E } _ { a v a i l } ) \\right) ^ { \\frac { \\alpha ^ { 2 } } { ( \\alpha + d ) ^ { 2 } } } \\right) .\n$$", + "text_format": "latex", + "bbox": [ + 349, + 647, + 616, + 676 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Here $\\rho$ is total variation distance, and $O ( \\cdot )$ depends on $d , C , \\alpha , M _ { 1 } , M _ { 2 }$ . ", + "bbox": [ + 176, + 681, + 648, + 696 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "The above theorem holds for a general classifier learned by any algorithms. Due to its generality, we need to introduce some technical regularity conditions on the density function. The assumption (4) assume the decay rate of density and its characteristic function, which is common in the literature, e.g. [14]. This theorem demonstrates that, the generalization error can be bounded by a function of the variation of $h$ , and it converges to 0 as the variation approaches to 0. Under some special but typical case where the top model $g$ is linear, we can further show that even without the regularity conditions in Theorem 4.1, we have a much better (linear) convergence rate. ", + "bbox": [ + 173, + 705, + 826, + 804 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Theorem 4.2 (Linear Top Model). Consider any loss satisfying $\\begin{array} { r } { \\ell ( \\hat { y } , y ) = \\sum _ { k = 1 } ^ { K } \\ell _ { 0 } ( \\hat { y } _ { k } , y _ { k } ) } \\end{array}$ .4 For any classifier with linear top model $g$ , i.e., ", + "bbox": [ + 173, + 809, + 823, + 839 + ], + "page_idx": 4 + }, + { + "type": "equation", + "img_path": "images/fd483c8cfb69d41c51abea8bfcbef01bb15ff1bdd62fa9a0e20893ef0103bc4c.jpg", + "text": "$$\nf ( x ) = A h ( x ) + b w i t h A \\in \\mathbb { R } ^ { K \\times d } , b \\in \\mathbb { R } ^ { K } ,\n$$", + "text_format": "latex", + "bbox": [ + 318, + 843, + 647, + 862 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "$i f ( \\mathcal { E } _ { a v a i l } , \\mathcal { E } _ { a l l } )$ is $\\left( s ( \\cdot ) , \\mathcal { T } ^ { i n f } ( h , \\mathcal { E } _ { a v a i l } ) \\right)$ -learnable under $\\Phi$ with Total Variation $\\rho$ , then we have ", + "bbox": [ + 171, + 90, + 797, + 108 + ], + "page_idx": 5 + }, + { + "type": "equation", + "img_path": "images/bff65b7e956cd09791c5ed2c8933cda9684cbbae3c4dc554a897bdb02f4ed4da.jpg", + "text": "$$\n\\mathrm { e r r } ( f ) \\leq O \\Bigl ( s \\bigl ( \\mathcal { V } ^ { s u p } ( h , \\mathcal { E } _ { a v a i l } ) \\bigr ) \\Bigr ) .\n$$", + "text_format": "latex", + "bbox": [ + 369, + 113, + 594, + 140 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Here $O ( \\cdot )$ depends only on d and $C$ . ", + "bbox": [ + 174, + 145, + 413, + 160 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Discussion. Theorem 4.1 shows that, for any model, the generalization gap depends largely on the model’s variation captured by $\\mathcal { V } ^ { \\mathrm { s u p } } ( h , \\mathcal { E } _ { a v a i l } )$ . The result is irrelevant to the algorithm and provides a guarantee for the generalization gap from $\\mathcal { E } _ { a v a i l }$ to ${ \\mathcal { E } } _ { a l l }$ , so long as the learned model $f$ is invariant, i.e. $\\mathcal { V } ^ { \\mathrm { s u p } } ( h , \\mathcal { E } _ { a v a i l } )$ is small. When $s ( \\cdot )$ is fixed, a model with smaller $\\mathcal { V } ^ { \\mathrm { s u p } } ( h , \\mathcal { E } _ { a v a i l } )$ results in a smaller gap, which matches our understanding that invariant features in $\\mathcal { E } _ { a v a i l }$ are somehow invariant in ${ \\mathcal { E } } _ { a l l }$ . When $\\mathcal { V } ^ { \\mathrm { s u p } } ( h , \\mathcal { E } _ { a v a i l } )$ is fixed, more difficult generalization will generate a larger expansion function, which leads to a larger gap. For the Gaussian class with bounded mean and variance, $\\alpha \\gg d$ and the convergent rate is almost linear. ", + "bbox": [ + 173, + 174, + 826, + 286 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "However, without any constraint to $g$ , the convergent rate might be small. Theorem 4.2 then offers a generalization bound with a linear convergent rate under mild assumptions when $g$ is linear, which is common in reality. It relaxes the concentration assumption (Formula 4) and asks only for the integrability of the density. The convergent rate is identical to the convergent rate of the expansion function, showing that $s ( \\cdot )$ captures the generalization quite well. ", + "bbox": [ + 173, + 291, + 825, + 362 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Proof Sketch of Theorem 4.1. The proof of the main result, Theorem 4.1, is decomposed into the following steps. First, we transform $\\operatorname { e r r } ( f )$ into the Total Variation of joint distributions of features in different domains (Step 1). To bound the Total Variation, it is sufficient to bound the distance of the corresponding Fourier transform, and further, it is equivalent to bound the Radon transform of joint distributions (Step 2). Eventually, we show that $\\mathcal { V } ^ { \\mathrm { s u p } } ( \\beta ^ { \\top } h , \\mathcal { E } _ { a v a i l } )$ can be used to bound the Radon transform, which finishes the proof (Step 3). ", + "bbox": [ + 173, + 375, + 825, + 460 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Step 1. The OOD generalization error can be bounded as: ", + "bbox": [ + 173, + 464, + 553, + 479 + ], + "page_idx": 5 + }, + { + "type": "equation", + "img_path": "images/9c22961cf79642649ac7052d3371c49427c7a1ab896566ae2ddd688515f2d0dd.jpg", + "text": "$$\n\\mathrm { e r r } ( f ) \\le \\operatorname* { s u p } _ { ( e , e ^ { \\prime } ) \\in ( \\mathcal { E } _ { a v a i l } , \\mathcal { E } _ { a l l } ) } \\frac { C } { K } \\sum _ { y \\in \\mathcal { Y } } \\int _ { h \\in \\mathbb { R } ^ { d } } \\big | p _ { h ^ { e } | Y ^ { e } } ( h | y ) - p _ { h ^ { e ^ { \\prime } } | Y ^ { e ^ { \\prime } } } ( h | y ) ) \\big | \\mathrm { d } h .\n$$", + "text_format": "latex", + "bbox": [ + 232, + 484, + 733, + 525 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Step 2. According to the assumption (4), the dominant term in (7) is ", + "bbox": [ + 173, + 536, + 622, + 551 + ], + "page_idx": 5 + }, + { + "type": "equation", + "img_path": "images/8d84abc42db5ac0bce20ba33a361ed49a07b3c2d298320a7da3642158de97201.jpg", + "text": "$$\n\\int _ { | h | \\leq r _ { 1 } } \\Big | \\int _ { | t | \\leq r _ { 2 } } e ^ { - i \\langle h , t \\rangle } \\big ( \\hat { p } _ { h ^ { e } | Y ^ { e } } ( t | y ) - \\hat { p } _ { h ^ { e ^ { \\prime } } | Y ^ { e ^ { \\prime } } } ( t | y ) \\big ) \\big ) \\mathrm { d } t \\Big | \\mathrm { d } t ,\n$$", + "text_format": "latex", + "bbox": [ + 281, + 555, + 681, + 592 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "where $r _ { 1 }$ and $r _ { 2 }$ are well-selected scalars that depend on $s \\big ( \\mathcal { V } _ { \\rho } ^ { \\mathrm { s u p } } ( h , \\mathcal { E } _ { a v a i l } ) \\big )$ . By the Projection Theorem [31, 42] and the Fourier Inversion Formula, (8) is bounded above by ", + "bbox": [ + 173, + 597, + 826, + 627 + ], + "page_idx": 5 + }, + { + "type": "equation", + "img_path": "images/49950396e3d8096d0baf981fde2f0be38a8c185cda445644133fe0a23d73c6a7.jpg", + "text": "$$\nO ( r _ { 1 } ^ { d } r _ { 2 } ^ { d } ) \\times \\int _ { u \\in \\mathbb { R } } \\big | \\mathscr { R } _ { e ^ { \\prime } } ( \\beta , u ) - \\mathscr { R } _ { e } ( \\beta , u ) \\big | \\mathrm { d } u ,\n$$", + "text_format": "latex", + "bbox": [ + 333, + 632, + 630, + 666 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "where $\\mathcal { R } _ { e } ( \\beta , u )$ is the Radon transform of $p _ { h ^ { e } | Y ^ { e } } ( t | y )$ . ", + "bbox": [ + 174, + 671, + 534, + 688 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Step 3. The right-hand side of Formula 8 can be bounded by $O \\big ( r _ { 1 } ^ { d } r _ { 2 } ^ { d } s \\big ( \\mathcal { V } _ { \\rho } ^ { \\mathrm { s u p } } ( h , \\mathcal { E } _ { a v a i l } ) \\big ) \\big )$ . We finish the proof by selecting appropriate $r _ { 1 }$ and $r _ { 2 }$ to balance the rate of the dominant term and other minor terms. For more details, please see Appendix 2 for the complete proofs. ", + "bbox": [ + 173, + 694, + 825, + 737 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Now we turn to the lower bound of $\\operatorname { e r r } ( f )$ ", + "bbox": [ + 174, + 742, + 452, + 757 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Theorem 4.3 (Lower Bound). Consider 0-1 loss: $\\ell ( \\hat { y } , y ) = \\mathbb { I } ( \\hat { y } \\neq y )$ . For any $\\delta > 0$ and any exps.t. $k x \\leq s ( x ) < + \\infty , x \\in [ 0 , t ]$ $\\begin{array} { r } { s _ { + } ^ { \\prime } ( 0 ) \\triangleq \\operatorname* { l i m } _ { x \\to 0 ^ { + } } \\frac { s ( x ) - s ( 0 ) } { x } \\in ( 1 , + \\infty ) } \\end{array}$ s(x)−s(0) ∈ (1, +∞); 2) exists k > 1, t > 0, $C _ { 0 }$ $O O D$ generaliz $( \\mathcal { E } _ { a v a i l } , \\mathcal { E } _ { a l l } )$ that is $( s ( \\cdot ) , \\delta )$ -learnable under linear feature space $\\Phi$ w.r.t symmetric $K L$ -divergence $\\rho$ , s.t. $\\forall \\varepsilon \\ \\in \\ [ 0 , \\frac { t } { 2 } ] .$ , the optimal classifier $f$ satisfying $\\mathcal { V } ^ { s u p } ( h , \\mathcal { E } _ { a v a i l } ) = \\varepsilon$ will have the OOD generalization error lower bounded by ", + "bbox": [ + 173, + 761, + 825, + 848 + ], + "page_idx": 5 + }, + { + "type": "equation", + "img_path": "images/b71427ee15ff225ca02625e3250ad817c12b27937efe9a03a49e7ec1eee529f3.jpg", + "text": "$$\n\\mathrm { e r r } ( f ) \\geq C _ { 0 } \\cdot s ( \\mathcal { V } ^ { s u p } ( h , \\mathcal { E } _ { a v a i l } ) ) .\n$$", + "text_format": "latex", + "bbox": [ + 372, + 853, + 593, + 871 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Theorem 4.3 shows that $\\operatorname { e r r } ( f )$ of optimal classifier $f$ is lower bounded by its variation. Here “optimal” means the classifier that minimize $\\mathcal { L } ( f , \\mathcal { E } _ { a v a i l } )$ . Altogether, the above three theorems offer a bidirectional control of OOD generalization error, showing that our formulation can offer a fine-grained description of most OOD generalization problem in a theoretical way. To pursue a good OOD performance, OOD algorithm should focus on improving predictive performance on $\\mathcal { E } _ { a v a i l }$ and controlling the variation $\\mathcal { V } ^ { \\mathrm { s u p } } ( h , \\mathcal { E } _ { a v a i l } )$ simultaneously. Note that this bound starts from population error, and we call for future works to combine our generalization bound and traditional bound from data samples to population error, giving a more complete characterization of the problem. ", + "bbox": [ + 171, + 882, + 823, + 912 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "", + "bbox": [ + 173, + 90, + 825, + 175 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "5 Variation as a Factor of Model Selection Criterion ", + "text_level": 1, + "bbox": [ + 173, + 194, + 622, + 212 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "As is pointed out in [21], model selection has a significant effect on domain generalization, and any OOD algorithm without a model selection criterion is not complete. [21] trained more than 45,900 models with different algorithms, and results show that when traditional selection methods are applied, none of OOD algorithms can outperform ERM [58] by a significant margin. This result is not strange, since traditional selection methods focus mainly on (validation) accuracy, which is biased in OOD generalization [21, 63]. A very typical example is Colored MNIST [5], where the image is colored according to the label, but the relationship varies across domains. As explained in [5], ERM principle will only capture this spurious feature (color) and performs badly in ${ \\mathcal { E } } _ { a l l }$ . Since ERM is exactly minimizing loss in $\\mathcal { E } _ { a v a i l }$ , any model selection method using validation accuracy alone is likely to choose ERM rather than any other OOD algorithm [63]. Thus no algorithm will have a significant improvement compared to ERM. ", + "bbox": [ + 173, + 227, + 825, + 380 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "A natural question arises: what else can we use, in addition to accuracy? Theorem 4.1 points out that, learning feature with small variation across $\\mathcal { E } _ { a v a i l }$ is important for decreasing OOD generalization error. Once a model $f$ achieves a small $\\mathcal { V } ^ { \\mathrm { s u p } } ( h , \\mathcal { E } _ { a v a i l } )$ , then $\\operatorname { e r r } ( f )$ will be small. If the validation accuracy is also high, we shall know that the OOD accuracy will remain high. To this end, we propose our heuristic selection criterion (Algorithm 1). Instead of considering validation accuracy alone, we combine it with feature variation and select the model with high validation accuracy as well as low variation. ", + "bbox": [ + 173, + 386, + 825, + 482 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "Algorithm 1: Model Selection ", + "text_level": 1, + "bbox": [ + 174, + 501, + 377, + 515 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "Input: available dataset $\\mathcal { X } _ { a v a i l } = ( \\mathcal { X } _ { t r a i n } , \\mathcal { X } _ { v a l } )$ , candidate models set $\\mathcal { M }$ , var_acc_rate $r _ { 0 }$ . \nfor $f = g \\circ h$ in $\\mathcal { M }$ do for $i$ in $[ d ]$ do $\\hat { \\mathcal { V } } _ { i } \\gets \\operatorname* { m a x } _ { y \\in \\mathcal { V } , \\mathcal { X } ^ { e } \\neq \\mathcal { X } ^ { e ^ { \\prime } } \\in \\mathcal { X } _ { a v a i l } }$ Total Variation $( \\mathbb { P } ( \\phi _ { i } ^ { e } | y ) , \\mathbb { P } ( \\phi _ { i } ^ { e ^ { \\prime } } | y ) )$ ; .Use GPU KDE end $\\mathcal { V } _ { f } \\gets \\mathrm { m e a n } _ { i \\in [ d ] } \\hat { \\mathcal { V } } _ { i }$ $\\operatorname { A c c } _ { f } $ compute validation accuracy of $f$ using $\\mathcal { X } _ { v a l }$ \nend \nRetur ${ \\mathfrak { n } } \\operatorname { a r g m a x } _ { f \\in { \\mathcal { M } } } ( \\operatorname { A c c } _ { f } - r _ { 0 } \\mathcal { V } _ { f } )$ ", + "bbox": [ + 173, + 517, + 810, + 657 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "We briefly explain Algorithm 1 here. For each candidate model, we calculate its variation using the average of each feature’s variation, i.e., $\\begin{array} { r } { \\frac { 1 } { d } \\sum _ { i \\in [ d ] } \\mathcal { V } ( \\phi _ { i } , \\mathcal { X } _ { a v a i l } ) } \\end{array}$ . When deriving the bounds, we use $\\mathcal { V } ^ { \\mathrm { s u p } }$ instead of their average because we need to consider the worst case, i.e., the worst top model. In practice, we find out that the average of $\\mathcal { V } ( \\phi _ { i } , \\mathcal { X } _ { a v a i l } )$ is enough to improve selection. ", + "bbox": [ + 174, + 679, + 825, + 738 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "Our criterion of model selection is ", + "bbox": [ + 174, + 744, + 400, + 758 + ], + "page_idx": 6 + }, + { + "type": "equation", + "img_path": "images/f4a8d3348d12ad167259139dedde77e64f96549c3f99a91438c1b14e5a3f0957.jpg", + "text": "$$\n\\operatorname { A c c } _ { f } { - r _ { 0 } \\gamma _ { f } } ,\n$$", + "text_format": "latex", + "bbox": [ + 434, + 767, + 531, + 784 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "i.e., we select a model with high validation accuracy and low variation simultaneously. Here $r _ { 0 }$ is a hyper-parameter representing the concrete relationship between $\\operatorname { e r r } ( f )$ and $\\mathcal { V } _ { f }$ . Although we have already used one hyper-parameter to help select multiple hyper-parameter combinations, it is natural to ask whether we can further get rid of the selection of $r _ { 0 }$ . Since $r _ { 0 }$ represents the relationship between variation and accuracy, which is actually determined by the unknown expansion function, explicitly calculating $r _ { 0 }$ is not possible. However, we can empirically estimate $r _ { 0 }$ using $\\begin{array} { r } { r _ { 0 } = \\frac { \\mathrm { S t d } _ { f \\in \\hat { \\mathcal { M } } } \\mathrm { A c c } _ { f } } { \\mathrm { S t d } _ { f \\in \\hat { \\mathcal { M } } } \\mathcal { V } _ { f } } } \\end{array}$ where $\\hat { \\mathcal { M } } \\subset \\mathcal { M }$ is the model with not bad validation accuracy. We do not use the whole set $\\mathcal { M }$ because some OOD algorithms will perform extremely bad when the penalty is huge, and these models will influence our estimation of the ratio. Since high validation means large informativeness in learned features, the use of $\\hat { \\mathcal { M } }$ is an implicit application of informative assumption. ", + "bbox": [ + 173, + 791, + 825, + 912 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "", + "bbox": [ + 171, + 90, + 825, + 122 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "As shown in Section 6.1, our method can select models with higher OOD accuracy in various OOD datasets. We also explain in Appendix 3 why our method can outperform the traditional method in Color MNIST, where the dataset is hand-make and simple enough to calculate the expansion function. ", + "bbox": [ + 174, + 127, + 825, + 170 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "6 Experiments ", + "text_level": 1, + "bbox": [ + 174, + 188, + 312, + 205 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "In this section, we conduct experiments to compare our model selection criterion (Section 5) with the baseline method5 [21]. Since both the variation and informativeness in Definition 3.1 are based on one-dimensional features, we can directly estimate these quantities feature-by-feature and design model selection method based on them. To verify the existence of the expansion function and to see what it’s like in a real-world dataset, we plot nearly 2 million features trained in a common-used OOD dataset and compute their variation and informativeness. We then draw the expansion function for this problem. ", + "bbox": [ + 174, + 218, + 825, + 315 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "6.1 Experiments on Model Selection ", + "text_level": 1, + "bbox": [ + 174, + 330, + 437, + 345 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "In this section, we conduct experiments to compare the performance of models selected by our method and by validation accuracy. We train models on different datasets, different $\\mathcal { E } _ { a v a i l }$ , and select models according to a different selection criteria. We then compare the OOD accuracy of selected models. ", + "bbox": [ + 174, + 356, + 825, + 411 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "Settings We train our model on three benchmark OOD datasets (PACS [34], OfficeHome [59], VLCS [57]) and consider all possible selections of $( \\mathcal { E } _ { a v a i l } , \\mathcal { E } _ { a l l } )$ . We choose ResNet–50 as our network architecture. We use ERM [58] and four common-used OOD algorithms (CORAL [55], Inter-domain Mixup [62], Group DRO [51], and IRM [5]). For each environment setup, we train 200 models using different algorithms, penalties, learning rates, and epoch. After training, we employ different selection methods and compare the OOD accuracy of the selected models. As stated in Section 5, we use the standard deviation of $\\nu$ and validation accuracy in $\\hat { \\mathcal { M } }$ to estimate $r _ { 0 }$ , where ${ \\hat { \\mathcal { M } } } = \\{ f \\in { \\mathcal { M } } : \\operatorname { A c c } _ { f } \\geq \\operatorname* { m a x } _ { \\hat { f } } \\operatorname { A c c } _ { \\hat { f } } - 0 . 1 \\}$ . Note that calculating $\\mathcal { V } ( \\phi _ { i } , \\mathcal { X } _ { a v a i l } )$ takes calculus many times, so we design a parallel GPU kernel density estimation to speed up the whole process a hundred times and manage to finish one model in seconds. For more details about the experiments, see Appendix 4. ", + "bbox": [ + 173, + 425, + 826, + 584 + ], + "page_idx": 7 + }, + { + "type": "table", + "img_path": "images/cb2470f47e4fb512f2684a9e17c4e49890f16f65e01308e130f28a69279cde8a.jpg", + "table_caption": [ + "Table 1: Model Selection Result. “Env” denotes the unseen domain during training. “Val” denotes the OOD accuracy of model selected by validation accuracy. " + ], + "table_footnote": [], + "table_body": "
PACSEnvACPSavgacc inc
Val85.20%80.42%96.17%77.86%84.91%1
Ours88.72%81.74%96.83%79.00%86.57%1.66%↑
OfficeHomeEnvACPRavgacc inc
Val61.85%55.56%74.72%76.25%67.09%-
Ours65.76%55.07%75.20%76.31%68.09%1.00%↑
VLCSEnvCLSVavgacc inc
Val97.46%64.83%69.50%670.97%75.69%1
Ours97.81%66.98%69.50%70.97%76.32%0.63%↑
", + "bbox": [ + 200, + 643, + 797, + 775 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "Result We summarize our experimental results in Table 1. For each environment setup, we select the best model according to Algorithm 1 and validation accuracy. The results show that on all datasets, our selection criterion significantly outperforms the validation accuracy in average OOD accuracy. For a more detailed comparison, our method improves the OOD accuracy in most of the 12 setups. Our experiments demonstrate that our criterion can help select models with higher OOD accuracy. ", + "bbox": [ + 173, + 792, + 826, + 863 + ], + "page_idx": 7 + }, + { + "type": "image", + "img_path": "images/c514b7d4698237506faccba2638407d1bd68cc2a61ac79a424f03d7abc31b464.jpg", + "image_caption": [ + "Figure 1: The expansion function of the OOD generalization problem on Office-Home. The $\\mathbf { X } ^ { } -$ -axis stands for $\\mathcal { V } ( \\phi , \\mathcal { E } _ { a v a i l } )$ and the y-axis for $\\mathcal { V } ( \\phi , \\mathcal { E } _ { a l l } )$ . There are approximately 2 million points in each image, with each point representing a feature, and its color represents its informativeness. The solid red line stands for the expansion function under the corresponding $\\delta$ . When $\\delta$ increases, the expansion function decreases. When $\\delta = 0$ , no expansion function can make it learnable. " + ], + "image_footnote": [], + "bbox": [ + 191, + 121, + 810, + 234 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "One may wonder if the expansion function really exists and what it will look like for a real-world OOD generalization task. In this section, we consider the OOD dataset Office-Home [59]. We explicitly plot millions of features’ $\\mathcal { V } _ { \\rho } \\big ( \\phi , \\mathcal { E } _ { a v a i l } \\big )$ and $\\mathcal { V } _ { \\rho } \\big ( \\phi , \\mathcal { E } _ { a l l } \\big )$ with Total Variation $\\rho$ to see what the expansion function is like in this task. We take the architecture as ResNet-50 [23], and we trained thousands of models with more than five algorithms, obtaining about 2 million features. The results are in Figure 1. ", + "bbox": [ + 174, + 327, + 825, + 411 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "Existence of $s ( \\cdot )$ . When $\\delta = 0$ , some non-informative features are nearly 0-invariant across $\\mathcal { E } _ { a v a i l }$ but are varying across ${ \\mathcal { E } } _ { a l l }$ , so no expansion function can make this task learnable, i.e., this task is NOT $( s ( \\cdot ) , 0 )$ for any expansion function. But as $\\delta$ increases, only informative features are left, and now we can find appropriate $s ( \\cdot )$ to make it learnable. We can clearly realize from the figure that $s ( \\cdot )$ do exist when $\\delta \\geq 0 . 1 5$ . ", + "bbox": [ + 174, + 424, + 825, + 494 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "Trade-off between $s ( \\cdot )$ and $\\delta$ . The second phenomenon is that the slope of $s ( \\cdot )$ decreases as $\\delta$ increases, showing a trade-off between $s ( \\cdot )$ and $\\delta$ . Although this trade-off comes naturally from the definition of learnability, it has a deep meaning. As is shown in Section 4, $\\operatorname { e r r } ( f )$ is bounded by $O ( s ( \\varepsilon ) )$ where $\\varepsilon$ is the variation of the model. To make the bound tighter, a natural idea is to choose a flatter $s ( \\cdot )$ . However, a flatter $s ( \\cdot )$ corresponds to a larger $\\delta$ . Typically, learning a model to meet this higher informativeness requirement is more difficult, and it is possible that the algorithm achieves this by capturing more domain-specific features, which will therefore increase the variation of the model, $\\varepsilon$ . As a result, we are not sure whether $s ( \\varepsilon )$ will increase or decrease. We believe this is also the essence of model selection: i.e., to trade-off between the variation and informativeness of a model, which is done in Formula 11. ", + "bbox": [ + 173, + 508, + 825, + 647 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "7 More Related Works ", + "text_level": 1, + "bbox": [ + 174, + 665, + 379, + 683 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "Domain generalization [12, 39], or OOD generalization, has drawn much attention recently [21, 30]. The goal is to learn a model from several training domains and expect good performance on unseen test domains. [60, 64] offer a comprehensive survey. A popular solution is to extract domain-invariant feature representation. [45] and [49] proved that when the model is linear, the invariance under training domains can help discover invariant features on test domains. [5] introduces the invariant prediction into neural networks and proposes a practical objective function. After that, a lot of works arise from the view of causal discovery, distributional robustness and conditional independence [1, 7, 16, 15, 26, 32, 33, 43, 51, 61]. On the other hand, some works point out the weakness of existing methods from the theoretical and experimental perspectives [2, 21, 29, 41, 50]. ", + "bbox": [ + 173, + 696, + 825, + 821 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "The OOD generalization requires restrictions on how the target domains may differ. A straightforward approach is to define a set of test domains around the training domain using some distribution distance measure [6, 13, 19, 25, 51, 53, 54, 61]. Another feasible route is the causal framework which is robust to the test distributions caused by interventions[44, 46] on variables, e.g., [5, 24, 36, 37, 40, 47, 49, 52]. The principle of these methods is that a causal model is invariant and can achieve the minimal worstcase risk [4, 22, 44, 49]. Since the test distribution is unknown, additional assumptions are required for generalization analysis. [12, 18, 39] assume that the domains are generated from a hyper-distribution and measures the average risk estimation error bound. [3] derives a risk bound for any linear combination of training domains. For more related results in domain adaptation, a closed field where the test domains can be seen but are unlabeled, please see [9, 10, 28]. ", + "bbox": [ + 174, + 828, + 825, + 911 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "", + "bbox": [ + 174, + 92, + 825, + 147 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "8 Conclusion ", + "text_level": 1, + "bbox": [ + 174, + 166, + 299, + 184 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "In this paper, we take the first step towards a rigorous theoretical framework of OOD generalization. We propose a mathematical formulation to characterize the learnability of OOD generalization problem. Based on our framework, we prove generalization bounds and give guarantees for OOD generalization error. Inspired by our bound, we design a model selection criterion to check the model’s variation and validation accuracy simultaneously. Experiments show that our metric has a significant advantage over the traditional selection method. ", + "bbox": [ + 174, + 198, + 825, + 280 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "Acknowledgments and Disclosure of Funding ", + "text_level": 1, + "bbox": [ + 174, + 299, + 553, + 318 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "Authors are thankful to the anonymous reviewers for their helpful and constructive feedback. ", + "bbox": [ + 171, + 332, + 779, + 347 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "References ", + "text_level": 1, + "bbox": [ + 174, + 364, + 266, + 381 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "[1] Kartik Ahuja, Karthikeyan Shanmugam, Kush Varshney, and Amit Dhurandhar. Invariant risk minimization games. arXiv preprint arXiv:2002.04692, 2020. \n[2] Kartik Ahuja, Jun Wang, Amit Dhurandhar, Karthikeyan Shanmugam, and Kush R Varshney. 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Out-of-distribution generalization analysis via influence function. arXiv preprint arXiv:2101.08521, 2021. \n[64] Kaiyang Zhou, Ziwei Liu, Yu Qiao, Tao Xiang, and Chen Change Loy. Domain generalization: A survey. arXiv preprint arXiv:2103.02503, 2021. ", + "bbox": [ + 174, + 380, + 826, + 912 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "", + "bbox": [ + 171, + 50, + 828, + 920 + ], + "page_idx": 10 + }, + { + "type": "text", + "text": "", + "bbox": [ + 171, + 61, + 828, + 922 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": "", + "bbox": [ + 171, + 92, + 828, + 308 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "Checklist ", + "text_level": 1, + "bbox": [ + 173, + 332, + 254, + 348 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "1. For all authors... ", + "bbox": [ + 169, + 359, + 295, + 373 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "(a) Do the main claims made in the abstract and introduction accurately reflect the paper’s contributions and scope? [Yes] \n(b) Did you describe the limitations of your work? [Yes] See Section 5. \n(c) Did you discuss any potential negative societal impacts of your work? [N/A] \n(d) Have you read the ethics review guidelines and ensured that your paper conforms to them? [Yes] ", + "bbox": [ + 174, + 377, + 825, + 455 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "2. If you are including theoretical results... ", + "bbox": [ + 168, + 459, + 449, + 473 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "(a) Did you state the full set of assumptions of all theoretical results? [Yes] (b) Did you include complete proofs of all theoretical results? [Yes] ", + "bbox": [ + 174, + 478, + 669, + 508 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "3. If you ran experiments... ", + "bbox": [ + 168, + 513, + 348, + 527 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "(a) Did you include the code, data, and instructions needed to reproduce the main experimental results (either in the supplemental material or as a URL)? [Yes] \n(b) Did you specify all the training details (e.g., data splits, hyperparameters, how they were chosen)? [Yes] In the appendix. \n(c) Did you report error bars (e.g., with respect to the random seed after running experiments multiple times)? [N/A] \n(d) Did you include the total amount of compute and the type of resources used (e.g., type of GPUs, internal cluster, or cloud provider)? [Yes] ", + "bbox": [ + 174, + 531, + 826, + 650 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "4. If you are using existing assets (e.g., code, data, models) or curating/releasing new assets... ", + "bbox": [ + 166, + 655, + 782, + 669 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "(a) If your work uses existing assets, did you cite the creators? [Yes] We use [21] and we cite it in Section 6. \n(b) Did you mention the license of the assets? [N/A] \n(c) Did you include any new assets either in the supplemental material or as a URL? [No] \n(d) Did you discuss whether and how consent was obtained from people whose data you’re using/curating? [Yes] \n(e) Did you discuss whether the data you are using/curating contains personally identifiable information or offensive content? [N/A] ", + "bbox": [ + 173, + 672, + 826, + 794 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "5. If you used crowdsourcing or conducted research with human subjects... ", + "bbox": [ + 169, + 797, + 661, + 813 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "(a) Did you include the full text of instructions given to participants and screenshots, if applicable? [N/A] \n(b) Did you describe any potential participant risks, with links to Institutional Review Board (IRB) approvals, if applicable? [N/A] \n(c) Did you include the estimated hourly wage paid to participants and the total amount spent on participant compensation? 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So it is necessary to", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 116, + 505, + 128 + ], + "spans": [ + { + "bbox": [ + 105, + 116, + 505, + 128 + ], + "score": 1.0, + "content": "first formulate what OOD data can be generalized to, or, what is the relation between the available", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 127, + 506, + 140 + ], + "spans": [ + { + "bbox": [ + 105, + 127, + 186, + 140 + ], + "score": 1.0, + "content": "training domain set", + "type": "text" + }, + { + "bbox": [ + 187, + 128, + 212, + 138 + ], + "score": 0.91, + "content": "\\mathcal { E } _ { a v a i l }", + "type": "inline_equation" + }, + { + "bbox": [ + 212, + 127, + 316, + 140 + ], + "score": 1.0, + "content": "and the entire domain set", + "type": "text" + }, + { + "bbox": [ + 316, + 128, + 333, + 138 + ], + "score": 0.89, + "content": "{ \\mathcal { E } } _ { a l l }", + "type": "inline_equation" + }, + { + "bbox": [ + 333, + 127, + 506, + 140 + ], + "score": 1.0, + "content": ". Meanwhile, to what extent the invariance", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 137, + 430, + 151 + ], + "spans": [ + { + "bbox": [ + 105, + 137, + 164, + 151 + ], + "score": 1.0, + "content": "of features on", + "type": "text" + }, + { + "bbox": [ + 164, + 138, + 189, + 149 + ], + "score": 0.91, + "content": "\\mathcal { E } _ { a v a i l }", + "type": "inline_equation" + }, + { + "bbox": [ + 189, + 137, + 270, + 151 + ], + "score": 1.0, + "content": "can be preserved in", + "type": "text" + }, + { + "bbox": [ + 270, + 138, + 286, + 149 + ], + "score": 0.9, + "content": "{ \\mathcal { E } } _ { a l l }", + "type": "inline_equation" + }, + { + "bbox": [ + 286, + 137, + 430, + 151 + ], + "score": 1.0, + "content": "should be rigorously characterized.", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 3 + }, + { + "type": "text", + "bbox": [ + 107, + 154, + 505, + 264 + ], + "lines": [ + { + "bbox": [ + 105, + 153, + 505, + 168 + ], + "spans": [ + { + "bbox": [ + 105, + 153, + 505, + 168 + ], + "score": 1.0, + "content": "In this paper, we take the first step towards a general OOD framework by quantitatively formalizing", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 164, + 505, + 178 + ], + "spans": [ + { + "bbox": [ + 105, + 164, + 208, + 178 + ], + "score": 1.0, + "content": "the relationship between", + "type": "text" + }, + { + "bbox": [ + 208, + 166, + 234, + 177 + ], + "score": 0.91, + "content": "\\mathcal { E } _ { a v a i l }", + "type": "inline_equation" + }, + { + "bbox": [ + 234, + 164, + 252, + 178 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 252, + 166, + 268, + 177 + ], + "score": 0.92, + "content": "{ \\mathcal { E } } _ { a l l }", + "type": "inline_equation" + }, + { + "bbox": [ + 269, + 164, + 505, + 178 + ], + "score": 1.0, + "content": "in terms of the distributions of features and provide OOD", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 176, + 507, + 189 + ], + "spans": [ + { + "bbox": [ + 105, + 176, + 507, + 189 + ], + "score": 1.0, + "content": "generalization guarantees based on our quantification of the difficulty of OOD generalization problem.", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 106, + 188, + 505, + 199 + ], + "spans": [ + { + "bbox": [ + 106, + 188, + 505, + 199 + ], + "score": 1.0, + "content": "Specifically, we first rigorously formulate the intuition of invariant features used in previous works", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 198, + 506, + 210 + ], + "spans": [ + { + "bbox": [ + 105, + 198, + 506, + 210 + ], + "score": 1.0, + "content": "by introducing the “variation” and “informativeness” (Definition 3.1 and 3.2) of each feature. Our", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 208, + 506, + 222 + ], + "spans": [ + { + "bbox": [ + 105, + 208, + 506, + 222 + ], + "score": 1.0, + "content": "theoretical insight can then be informally stated as: for learnable OOD problems, if a feature is", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 219, + 507, + 234 + ], + "spans": [ + { + "bbox": [ + 105, + 219, + 361, + 234 + ], + "score": 1.0, + "content": "informative for the classification task as well as invariant over", + "type": "text" + }, + { + "bbox": [ + 362, + 221, + 387, + 231 + ], + "score": 0.9, + "content": "\\mathcal { E } _ { a v a i l }", + "type": "inline_equation" + }, + { + "bbox": [ + 387, + 219, + 507, + 234 + ], + "score": 1.0, + "content": ", then it is still invariant over", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 230, + 507, + 245 + ], + "spans": [ + { + "bbox": [ + 106, + 231, + 123, + 242 + ], + "score": 0.89, + "content": "{ \\mathcal { E } } _ { a l l }", + "type": "inline_equation" + }, + { + "bbox": [ + 123, + 230, + 352, + 245 + ], + "score": 1.0, + "content": ". In other words, invariance of informative features in", + "type": "text" + }, + { + "bbox": [ + 353, + 231, + 378, + 242 + ], + "score": 0.91, + "content": "\\mathcal { E } _ { a v a i l }", + "type": "inline_equation" + }, + { + "bbox": [ + 378, + 230, + 465, + 245 + ], + "score": 1.0, + "content": "can be preserved in", + "type": "text" + }, + { + "bbox": [ + 465, + 231, + 482, + 242 + ], + "score": 0.89, + "content": "{ \\mathcal { E } } _ { a l l }", + "type": "inline_equation" + }, + { + "bbox": [ + 482, + 230, + 507, + 245 + ], + "score": 1.0, + "content": ". We", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 241, + 505, + 254 + ], + "spans": [ + { + "bbox": [ + 105, + 241, + 505, + 254 + ], + "score": 1.0, + "content": "further introduce a class of functions, dubbed expansion function (Definition 3.3), to quantitatively", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 251, + 433, + 267 + ], + "spans": [ + { + "bbox": [ + 105, + 251, + 324, + 267 + ], + "score": 1.0, + "content": "characterize to what extent the variance of features on", + "type": "text" + }, + { + "bbox": [ + 324, + 253, + 349, + 264 + ], + "score": 0.91, + "content": "\\mathcal { E } _ { a v a i l }", + "type": "inline_equation" + }, + { + "bbox": [ + 349, + 251, + 412, + 267 + ], + "score": 1.0, + "content": "is amplified on", + "type": "text" + }, + { + "bbox": [ + 412, + 253, + 428, + 264 + ], + "score": 0.89, + "content": "{ \\mathcal { E } } _ { a l l }", + "type": "inline_equation" + }, + { + "bbox": [ + 429, + 251, + 433, + 267 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 11.5 + }, + { + "type": "text", + "bbox": [ + 107, + 269, + 505, + 356 + ], + "lines": [ + { + "bbox": [ + 105, + 268, + 505, + 282 + ], + "spans": [ + { + "bbox": [ + 105, + 268, + 505, + 282 + ], + "score": 1.0, + "content": "Based on our formulation, we derive theoretical guarantees on the OOD generalization error, i.e., the", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 280, + 507, + 294 + ], + "spans": [ + { + "bbox": [ + 105, + 280, + 273, + 294 + ], + "score": 1.0, + "content": "gap of largest error between the domain in", + "type": "text" + }, + { + "bbox": [ + 273, + 280, + 298, + 291 + ], + "score": 0.91, + "content": "\\mathcal { E } _ { a v a i l }", + "type": "inline_equation" + }, + { + "bbox": [ + 299, + 280, + 357, + 294 + ], + "score": 1.0, + "content": "and domain in", + "type": "text" + }, + { + "bbox": [ + 357, + 280, + 374, + 291 + ], + "score": 0.88, + "content": "{ \\mathcal { E } } _ { a l l }", + "type": "inline_equation" + }, + { + "bbox": [ + 374, + 280, + 507, + 294 + ], + "score": 1.0, + "content": ". Specifically, we prove the upper", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 291, + 505, + 303 + ], + "spans": [ + { + "bbox": [ + 105, + 291, + 505, + 303 + ], + "score": 1.0, + "content": "and lower bound of OOD generalization error in terms of the expansion function and the variation", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 301, + 506, + 316 + ], + "spans": [ + { + "bbox": [ + 105, + 301, + 203, + 316 + ], + "score": 1.0, + "content": "of learned features over", + "type": "text" + }, + { + "bbox": [ + 204, + 302, + 228, + 313 + ], + "score": 0.9, + "content": "\\mathcal { E } _ { a v a i l }", + "type": "inline_equation" + }, + { + "bbox": [ + 229, + 301, + 506, + 316 + ], + "score": 1.0, + "content": ". Our results theoretically confirm that 1) the expansion function can", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 311, + 505, + 326 + ], + "spans": [ + { + "bbox": [ + 105, + 311, + 505, + 326 + ], + "score": 1.0, + "content": "reflect the difficulty of OOD generalization problem, i.e., problems with more rapidly increasing", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 324, + 506, + 337 + ], + "spans": [ + { + "bbox": [ + 105, + 324, + 506, + 337 + ], + "score": 1.0, + "content": "expansion functions are harder and have worse generalization guarantees; 2) the generalization error", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 334, + 506, + 347 + ], + "spans": [ + { + "bbox": [ + 105, + 334, + 506, + 347 + ], + "score": 1.0, + "content": "gap can tend to zero when the variation of learned features tend to zero, so minimizing the variation", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 345, + 286, + 359 + ], + "spans": [ + { + "bbox": [ + 105, + 345, + 117, + 359 + ], + "score": 1.0, + "content": "in", + "type": "text" + }, + { + "bbox": [ + 117, + 345, + 142, + 357 + ], + "score": 0.91, + "content": "\\mathcal { E } _ { a v a i l }", + "type": "inline_equation" + }, + { + "bbox": [ + 142, + 345, + 286, + 359 + ], + "score": 1.0, + "content": "can reduce the generalization error.", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 20.5 + }, + { + "type": "text", + "bbox": [ + 107, + 362, + 505, + 471 + ], + "lines": [ + { + "bbox": [ + 105, + 361, + 505, + 374 + ], + "spans": [ + { + "bbox": [ + 105, + 361, + 505, + 374 + ], + "score": 1.0, + "content": "As pointed out by Gulrajani and Lopez-Paz [21], any OOD algorithm without a specified model", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 373, + 505, + 385 + ], + "spans": [ + { + "bbox": [ + 105, + 373, + 283, + 385 + ], + "score": 1.0, + "content": "selection criterion is not complete. Since", + "type": "text" + }, + { + "bbox": [ + 284, + 373, + 300, + 384 + ], + "score": 0.9, + "content": "{ \\mathcal { E } } _ { a l l }", + "type": "inline_equation" + }, + { + "bbox": [ + 300, + 373, + 505, + 385 + ], + "score": 1.0, + "content": "is unseen, hyper-parameters can only be chosen", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 382, + 502, + 398 + ], + "spans": [ + { + "bbox": [ + 105, + 382, + 160, + 398 + ], + "score": 1.0, + "content": "according to", + "type": "text" + }, + { + "bbox": [ + 160, + 384, + 185, + 395 + ], + "score": 0.9, + "content": "\\mathcal { E } _ { a v a i l }", + "type": "inline_equation" + }, + { + "bbox": [ + 186, + 382, + 477, + 398 + ], + "score": 1.0, + "content": ". Previous selection methods mainly focus on validation accuracy over", + "type": "text" + }, + { + "bbox": [ + 477, + 384, + 502, + 395 + ], + "score": 0.91, + "content": "\\mathcal { E } _ { a v a i l }", + "type": "inline_equation" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 394, + 506, + 408 + ], + "spans": [ + { + "bbox": [ + 105, + 394, + 506, + 408 + ], + "score": 1.0, + "content": "which is only a biased metric of OOD performance. On the contrary, a promising model selection", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 405, + 506, + 418 + ], + "spans": [ + { + "bbox": [ + 105, + 405, + 506, + 418 + ], + "score": 1.0, + "content": "method should instead be predictive of OOD performance. Inspired by our bounds, we propose a", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 417, + 505, + 428 + ], + "spans": [ + { + "bbox": [ + 106, + 417, + 505, + 428 + ], + "score": 1.0, + "content": "model selection method to select models with high validation accuracy and low variation, which", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 427, + 506, + 440 + ], + "spans": [ + { + "bbox": [ + 105, + 427, + 506, + 440 + ], + "score": 1.0, + "content": "corresponds to the upper bound of OOD error. The introduction of a model’s variation relieves", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 437, + 507, + 452 + ], + "spans": [ + { + "bbox": [ + 105, + 437, + 398, + 452 + ], + "score": 1.0, + "content": "the problem of classic selection methods, in which models that overfit", + "type": "text" + }, + { + "bbox": [ + 398, + 438, + 423, + 449 + ], + "score": 0.91, + "content": "\\mathcal { E } _ { a v a i l }", + "type": "inline_equation" + }, + { + "bbox": [ + 424, + 437, + 507, + 452 + ], + "score": 1.0, + "content": "tend to be selected.", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 449, + 506, + 461 + ], + "spans": [ + { + "bbox": [ + 105, + 449, + 506, + 461 + ], + "score": 1.0, + "content": "Experimental results show that our method can outperform baselines and select models with higher", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 460, + 170, + 473 + ], + "spans": [ + { + "bbox": [ + 106, + 460, + 170, + 473 + ], + "score": 1.0, + "content": "OOD accuracy.", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 29.5 + }, + { + "type": "text", + "bbox": [ + 107, + 493, + 357, + 504 + ], + "lines": [ + { + "bbox": [ + 106, + 491, + 360, + 507 + ], + "spans": [ + { + "bbox": [ + 106, + 491, + 360, + 507 + ], + "score": 1.0, + "content": "Contributions. We summarize our major contributions here:", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 35 + }, + { + "type": "text", + "bbox": [ + 106, + 518, + 506, + 562 + ], + "lines": [ + { + "bbox": [ + 105, + 518, + 506, + 530 + ], + "spans": [ + { + "bbox": [ + 105, + 518, + 506, + 530 + ], + "score": 1.0, + "content": "• We introduce a quantitative and rigorous formulation of OOD generalization problem that charac-", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 113, + 528, + 506, + 542 + ], + "spans": [ + { + "bbox": [ + 113, + 528, + 360, + 542 + ], + "score": 1.0, + "content": "terizes the relation of invariance over the training domain set", + "type": "text" + }, + { + "bbox": [ + 360, + 529, + 385, + 540 + ], + "score": 0.91, + "content": "\\mathcal { E } _ { a v a i l }", + "type": "inline_equation" + }, + { + "bbox": [ + 386, + 528, + 466, + 542 + ], + "score": 1.0, + "content": "and test domain set", + "type": "text" + }, + { + "bbox": [ + 466, + 529, + 483, + 540 + ], + "score": 0.89, + "content": "{ \\mathcal { E } } _ { a l l }", + "type": "inline_equation" + }, + { + "bbox": [ + 483, + 528, + 506, + 542 + ], + "score": 1.0, + "content": ". The", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 113, + 540, + 506, + 552 + ], + "spans": [ + { + "bbox": [ + 113, + 540, + 506, + 552 + ], + "score": 1.0, + "content": "core quantity in our characterization, the expansion function, determines the difficulty of an OOD", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 113, + 551, + 212, + 563 + ], + "spans": [ + { + "bbox": [ + 113, + 551, + 212, + 563 + ], + "score": 1.0, + "content": "generalization problem.", + "type": "text" + } + ], + "index": 39 + } + ], + "index": 37.5 + }, + { + "type": "text", + "bbox": [ + 107, + 574, + 504, + 607 + ], + "lines": [ + { + "bbox": [ + 105, + 573, + 506, + 587 + ], + "spans": [ + { + "bbox": [ + 105, + 573, + 506, + 587 + ], + "score": 1.0, + "content": "• We prove novel OOD generalization error bounds based on our formulation. The upper and lower", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 114, + 585, + 505, + 598 + ], + "spans": [ + { + "bbox": [ + 114, + 585, + 505, + 598 + ], + "score": 1.0, + "content": "bounds together indicate that the expansion function well characterizes the OOD generalization", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 114, + 596, + 321, + 608 + ], + "spans": [ + { + "bbox": [ + 114, + 596, + 321, + 608 + ], + "score": 1.0, + "content": "ability of features with different levels of variation.", + "type": "text" + } + ], + "index": 42 + } + ], + "index": 41 + }, + { + "type": "text", + "bbox": [ + 106, + 620, + 505, + 664 + ], + "lines": [ + { + "bbox": [ + 106, + 620, + 505, + 632 + ], + "spans": [ + { + "bbox": [ + 106, + 620, + 505, + 632 + ], + "score": 1.0, + "content": "• We design a model selection criterion that is inspired by our generalization bounds. Our criterion", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 114, + 631, + 505, + 643 + ], + "spans": [ + { + "bbox": [ + 114, + 631, + 505, + 643 + ], + "score": 1.0, + "content": "takes both the performance on training domains and the variation of models into consideration and", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 114, + 642, + 506, + 655 + ], + "spans": [ + { + "bbox": [ + 114, + 642, + 506, + 655 + ], + "score": 1.0, + "content": "is predictive of OOD performance according to our bounds. Experimental results demonstrate our", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 113, + 652, + 379, + 667 + ], + "spans": [ + { + "bbox": [ + 113, + 652, + 379, + 667 + ], + "score": 1.0, + "content": "selection criterion can choose models with higher OOD accuracy.", + "type": "text" + } + ], + "index": 46 + } + ], + "index": 44.5 + }, + { + "type": "text", + "bbox": [ + 108, + 678, + 505, + 722 + ], + "lines": [ + { + "bbox": [ + 106, + 677, + 505, + 691 + ], + "spans": [ + { + "bbox": [ + 106, + 677, + 505, + 691 + ], + "score": 1.0, + "content": "The rest of the paper is organized as follows: Section 2 is our preliminary. In Section 3, we give our", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 105, + 688, + 505, + 702 + ], + "spans": [ + { + "bbox": [ + 105, + 688, + 505, + 702 + ], + "score": 1.0, + "content": "theoretical formulation. Section 4 gives our generalization bound. We propose our model selection", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 105, + 700, + 506, + 712 + ], + "spans": [ + { + "bbox": [ + 105, + 700, + 506, + 712 + ], + "score": 1.0, + "content": "method in Section 5. In Section 6 we conduct experiments on expansion function and model selection.", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 106, + 711, + 426, + 722 + ], + "spans": [ + { + "bbox": [ + 106, + 711, + 426, + 722 + ], + "score": 1.0, + "content": "We review more related works in Section 7 and conclude our work in Section 8.", + "type": "text" + } + ], + "index": 50 + } + ], + "index": 48.5 + } + ], + "page_idx": 1, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 302, + 742, + 309, + 750 + ], + "lines": [ + { + "bbox": [ + 301, + 741, + 310, + 753 + ], + "spans": [ + { + "bbox": [ + 301, + 741, + 310, + 753 + ], + "score": 1.0, + "content": "2", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 73, + 505, + 149 + ], + "lines": [ + { + "bbox": [ + 106, + 73, + 505, + 85 + ], + "spans": [ + { + "bbox": [ + 106, + 73, + 505, + 85 + ], + "score": 1.0, + "content": "Though the idea of learning invariant features is intuitively reasonable, there is only limited theoretical", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 84, + 505, + 96 + ], + "spans": [ + { + "bbox": [ + 106, + 84, + 505, + 96 + ], + "score": 1.0, + "content": "understanding of what kind of invariance can guarantee OOD generalization. Clearly, generalization", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 95, + 505, + 107 + ], + "spans": [ + { + "bbox": [ + 105, + 95, + 505, + 107 + ], + "score": 1.0, + "content": "to an arbitrary out-of-distribution domain is impossible and in practice, the features can hardly be", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 105, + 506, + 119 + ], + "spans": [ + { + "bbox": [ + 105, + 105, + 212, + 119 + ], + "score": 1.0, + "content": "absolutely invariant from", + "type": "text" + }, + { + "bbox": [ + 213, + 106, + 237, + 117 + ], + "score": 0.91, + "content": "\\mathcal { E } _ { a v a i l }", + "type": "inline_equation" + }, + { + "bbox": [ + 238, + 105, + 250, + 119 + ], + "score": 1.0, + "content": "to", + "type": "text" + }, + { + "bbox": [ + 250, + 106, + 267, + 117 + ], + "score": 0.89, + "content": "{ \\mathcal { E } } _ { a l l }", + "type": "inline_equation" + }, + { + "bbox": [ + 267, + 105, + 506, + 119 + ], + "score": 1.0, + "content": "unless all the domains are identical. So it is necessary to", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 116, + 505, + 128 + ], + "spans": [ + { + "bbox": [ + 105, + 116, + 505, + 128 + ], + "score": 1.0, + "content": "first formulate what OOD data can be generalized to, or, what is the relation between the available", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 127, + 506, + 140 + ], + "spans": [ + { + "bbox": [ + 105, + 127, + 186, + 140 + ], + "score": 1.0, + "content": "training domain set", + "type": "text" + }, + { + "bbox": [ + 187, + 128, + 212, + 138 + ], + "score": 0.91, + "content": "\\mathcal { E } _ { a v a i l }", + "type": "inline_equation" + }, + { + "bbox": [ + 212, + 127, + 316, + 140 + ], + "score": 1.0, + "content": "and the entire domain set", + "type": "text" + }, + { + "bbox": [ + 316, + 128, + 333, + 138 + ], + "score": 0.89, + "content": "{ \\mathcal { E } } _ { a l l }", + "type": "inline_equation" + }, + { + "bbox": [ + 333, + 127, + 506, + 140 + ], + "score": 1.0, + "content": ". Meanwhile, to what extent the invariance", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 137, + 430, + 151 + ], + "spans": [ + { + "bbox": [ + 105, + 137, + 164, + 151 + ], + "score": 1.0, + "content": "of features on", + "type": "text" + }, + { + "bbox": [ + 164, + 138, + 189, + 149 + ], + "score": 0.91, + "content": "\\mathcal { E } _ { a v a i l }", + "type": "inline_equation" + }, + { + "bbox": [ + 189, + 137, + 270, + 151 + ], + "score": 1.0, + "content": "can be preserved in", + "type": "text" + }, + { + "bbox": [ + 270, + 138, + 286, + 149 + ], + "score": 0.9, + "content": "{ \\mathcal { E } } _ { a l l }", + "type": "inline_equation" + }, + { + "bbox": [ + 286, + 137, + 430, + 151 + ], + "score": 1.0, + "content": "should be rigorously characterized.", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 3, + "bbox_fs": [ + 105, + 73, + 506, + 151 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 154, + 505, + 264 + ], + "lines": [ + { + "bbox": [ + 105, + 153, + 505, + 168 + ], + "spans": [ + { + "bbox": [ + 105, + 153, + 505, + 168 + ], + "score": 1.0, + "content": "In this paper, we take the first step towards a general OOD framework by quantitatively formalizing", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 164, + 505, + 178 + ], + "spans": [ + { + "bbox": [ + 105, + 164, + 208, + 178 + ], + "score": 1.0, + "content": "the relationship between", + "type": "text" + }, + { + "bbox": [ + 208, + 166, + 234, + 177 + ], + "score": 0.91, + "content": "\\mathcal { E } _ { a v a i l }", + "type": "inline_equation" + }, + { + "bbox": [ + 234, + 164, + 252, + 178 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 252, + 166, + 268, + 177 + ], + "score": 0.92, + "content": "{ \\mathcal { E } } _ { a l l }", + "type": "inline_equation" + }, + { + "bbox": [ + 269, + 164, + 505, + 178 + ], + "score": 1.0, + "content": "in terms of the distributions of features and provide OOD", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 176, + 507, + 189 + ], + "spans": [ + { + "bbox": [ + 105, + 176, + 507, + 189 + ], + "score": 1.0, + "content": "generalization guarantees based on our quantification of the difficulty of OOD generalization problem.", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 106, + 188, + 505, + 199 + ], + "spans": [ + { + "bbox": [ + 106, + 188, + 505, + 199 + ], + "score": 1.0, + "content": "Specifically, we first rigorously formulate the intuition of invariant features used in previous works", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 198, + 506, + 210 + ], + "spans": [ + { + "bbox": [ + 105, + 198, + 506, + 210 + ], + "score": 1.0, + "content": "by introducing the “variation” and “informativeness” (Definition 3.1 and 3.2) of each feature. Our", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 208, + 506, + 222 + ], + "spans": [ + { + "bbox": [ + 105, + 208, + 506, + 222 + ], + "score": 1.0, + "content": "theoretical insight can then be informally stated as: for learnable OOD problems, if a feature is", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 219, + 507, + 234 + ], + "spans": [ + { + "bbox": [ + 105, + 219, + 361, + 234 + ], + "score": 1.0, + "content": "informative for the classification task as well as invariant over", + "type": "text" + }, + { + "bbox": [ + 362, + 221, + 387, + 231 + ], + "score": 0.9, + "content": "\\mathcal { E } _ { a v a i l }", + "type": "inline_equation" + }, + { + "bbox": [ + 387, + 219, + 507, + 234 + ], + "score": 1.0, + "content": ", then it is still invariant over", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 230, + 507, + 245 + ], + "spans": [ + { + "bbox": [ + 106, + 231, + 123, + 242 + ], + "score": 0.89, + "content": "{ \\mathcal { E } } _ { a l l }", + "type": "inline_equation" + }, + { + "bbox": [ + 123, + 230, + 352, + 245 + ], + "score": 1.0, + "content": ". In other words, invariance of informative features in", + "type": "text" + }, + { + "bbox": [ + 353, + 231, + 378, + 242 + ], + "score": 0.91, + "content": "\\mathcal { E } _ { a v a i l }", + "type": "inline_equation" + }, + { + "bbox": [ + 378, + 230, + 465, + 245 + ], + "score": 1.0, + "content": "can be preserved in", + "type": "text" + }, + { + "bbox": [ + 465, + 231, + 482, + 242 + ], + "score": 0.89, + "content": "{ \\mathcal { E } } _ { a l l }", + "type": "inline_equation" + }, + { + "bbox": [ + 482, + 230, + 507, + 245 + ], + "score": 1.0, + "content": ". We", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 241, + 505, + 254 + ], + "spans": [ + { + "bbox": [ + 105, + 241, + 505, + 254 + ], + "score": 1.0, + "content": "further introduce a class of functions, dubbed expansion function (Definition 3.3), to quantitatively", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 251, + 433, + 267 + ], + "spans": [ + { + "bbox": [ + 105, + 251, + 324, + 267 + ], + "score": 1.0, + "content": "characterize to what extent the variance of features on", + "type": "text" + }, + { + "bbox": [ + 324, + 253, + 349, + 264 + ], + "score": 0.91, + "content": "\\mathcal { E } _ { a v a i l }", + "type": "inline_equation" + }, + { + "bbox": [ + 349, + 251, + 412, + 267 + ], + "score": 1.0, + "content": "is amplified on", + "type": "text" + }, + { + "bbox": [ + 412, + 253, + 428, + 264 + ], + "score": 0.89, + "content": "{ \\mathcal { E } } _ { a l l }", + "type": "inline_equation" + }, + { + "bbox": [ + 429, + 251, + 433, + 267 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 11.5, + "bbox_fs": [ + 105, + 153, + 507, + 267 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 269, + 505, + 356 + ], + "lines": [ + { + "bbox": [ + 105, + 268, + 505, + 282 + ], + "spans": [ + { + "bbox": [ + 105, + 268, + 505, + 282 + ], + "score": 1.0, + "content": "Based on our formulation, we derive theoretical guarantees on the OOD generalization error, i.e., the", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 280, + 507, + 294 + ], + "spans": [ + { + "bbox": [ + 105, + 280, + 273, + 294 + ], + "score": 1.0, + "content": "gap of largest error between the domain in", + "type": "text" + }, + { + "bbox": [ + 273, + 280, + 298, + 291 + ], + "score": 0.91, + "content": "\\mathcal { E } _ { a v a i l }", + "type": "inline_equation" + }, + { + "bbox": [ + 299, + 280, + 357, + 294 + ], + "score": 1.0, + "content": "and domain in", + "type": "text" + }, + { + "bbox": [ + 357, + 280, + 374, + 291 + ], + "score": 0.88, + "content": "{ \\mathcal { E } } _ { a l l }", + "type": "inline_equation" + }, + { + "bbox": [ + 374, + 280, + 507, + 294 + ], + "score": 1.0, + "content": ". Specifically, we prove the upper", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 291, + 505, + 303 + ], + "spans": [ + { + "bbox": [ + 105, + 291, + 505, + 303 + ], + "score": 1.0, + "content": "and lower bound of OOD generalization error in terms of the expansion function and the variation", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 301, + 506, + 316 + ], + "spans": [ + { + "bbox": [ + 105, + 301, + 203, + 316 + ], + "score": 1.0, + "content": "of learned features over", + "type": "text" + }, + { + "bbox": [ + 204, + 302, + 228, + 313 + ], + "score": 0.9, + "content": "\\mathcal { E } _ { a v a i l }", + "type": "inline_equation" + }, + { + "bbox": [ + 229, + 301, + 506, + 316 + ], + "score": 1.0, + "content": ". Our results theoretically confirm that 1) the expansion function can", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 311, + 505, + 326 + ], + "spans": [ + { + "bbox": [ + 105, + 311, + 505, + 326 + ], + "score": 1.0, + "content": "reflect the difficulty of OOD generalization problem, i.e., problems with more rapidly increasing", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 324, + 506, + 337 + ], + "spans": [ + { + "bbox": [ + 105, + 324, + 506, + 337 + ], + "score": 1.0, + "content": "expansion functions are harder and have worse generalization guarantees; 2) the generalization error", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 334, + 506, + 347 + ], + "spans": [ + { + "bbox": [ + 105, + 334, + 506, + 347 + ], + "score": 1.0, + "content": "gap can tend to zero when the variation of learned features tend to zero, so minimizing the variation", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 345, + 286, + 359 + ], + "spans": [ + { + "bbox": [ + 105, + 345, + 117, + 359 + ], + "score": 1.0, + "content": "in", + "type": "text" + }, + { + "bbox": [ + 117, + 345, + 142, + 357 + ], + "score": 0.91, + "content": "\\mathcal { E } _ { a v a i l }", + "type": "inline_equation" + }, + { + "bbox": [ + 142, + 345, + 286, + 359 + ], + "score": 1.0, + "content": "can reduce the generalization error.", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 20.5, + "bbox_fs": [ + 105, + 268, + 507, + 359 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 362, + 505, + 471 + ], + "lines": [ + { + "bbox": [ + 105, + 361, + 505, + 374 + ], + "spans": [ + { + "bbox": [ + 105, + 361, + 505, + 374 + ], + "score": 1.0, + "content": "As pointed out by Gulrajani and Lopez-Paz [21], any OOD algorithm without a specified model", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 373, + 505, + 385 + ], + "spans": [ + { + "bbox": [ + 105, + 373, + 283, + 385 + ], + "score": 1.0, + "content": "selection criterion is not complete. Since", + "type": "text" + }, + { + "bbox": [ + 284, + 373, + 300, + 384 + ], + "score": 0.9, + "content": "{ \\mathcal { E } } _ { a l l }", + "type": "inline_equation" + }, + { + "bbox": [ + 300, + 373, + 505, + 385 + ], + "score": 1.0, + "content": "is unseen, hyper-parameters can only be chosen", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 382, + 502, + 398 + ], + "spans": [ + { + "bbox": [ + 105, + 382, + 160, + 398 + ], + "score": 1.0, + "content": "according to", + "type": "text" + }, + { + "bbox": [ + 160, + 384, + 185, + 395 + ], + "score": 0.9, + "content": "\\mathcal { E } _ { a v a i l }", + "type": "inline_equation" + }, + { + "bbox": [ + 186, + 382, + 477, + 398 + ], + "score": 1.0, + "content": ". Previous selection methods mainly focus on validation accuracy over", + "type": "text" + }, + { + "bbox": [ + 477, + 384, + 502, + 395 + ], + "score": 0.91, + "content": "\\mathcal { E } _ { a v a i l }", + "type": "inline_equation" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 394, + 506, + 408 + ], + "spans": [ + { + "bbox": [ + 105, + 394, + 506, + 408 + ], + "score": 1.0, + "content": "which is only a biased metric of OOD performance. On the contrary, a promising model selection", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 405, + 506, + 418 + ], + "spans": [ + { + "bbox": [ + 105, + 405, + 506, + 418 + ], + "score": 1.0, + "content": "method should instead be predictive of OOD performance. Inspired by our bounds, we propose a", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 417, + 505, + 428 + ], + "spans": [ + { + "bbox": [ + 106, + 417, + 505, + 428 + ], + "score": 1.0, + "content": "model selection method to select models with high validation accuracy and low variation, which", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 427, + 506, + 440 + ], + "spans": [ + { + "bbox": [ + 105, + 427, + 506, + 440 + ], + "score": 1.0, + "content": "corresponds to the upper bound of OOD error. The introduction of a model’s variation relieves", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 437, + 507, + 452 + ], + "spans": [ + { + "bbox": [ + 105, + 437, + 398, + 452 + ], + "score": 1.0, + "content": "the problem of classic selection methods, in which models that overfit", + "type": "text" + }, + { + "bbox": [ + 398, + 438, + 423, + 449 + ], + "score": 0.91, + "content": "\\mathcal { E } _ { a v a i l }", + "type": "inline_equation" + }, + { + "bbox": [ + 424, + 437, + 507, + 452 + ], + "score": 1.0, + "content": "tend to be selected.", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 449, + 506, + 461 + ], + "spans": [ + { + "bbox": [ + 105, + 449, + 506, + 461 + ], + "score": 1.0, + "content": "Experimental results show that our method can outperform baselines and select models with higher", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 460, + 170, + 473 + ], + "spans": [ + { + "bbox": [ + 106, + 460, + 170, + 473 + ], + "score": 1.0, + "content": "OOD accuracy.", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 29.5, + "bbox_fs": [ + 105, + 361, + 507, + 473 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 493, + 357, + 504 + ], + "lines": [ + { + "bbox": [ + 106, + 491, + 360, + 507 + ], + "spans": [ + { + "bbox": [ + 106, + 491, + 360, + 507 + ], + "score": 1.0, + "content": "Contributions. We summarize our major contributions here:", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 35, + "bbox_fs": [ + 106, + 491, + 360, + 507 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 518, + 506, + 562 + ], + "lines": [ + { + "bbox": [ + 105, + 518, + 506, + 530 + ], + "spans": [ + { + "bbox": [ + 105, + 518, + 506, + 530 + ], + "score": 1.0, + "content": "• We introduce a quantitative and rigorous formulation of OOD generalization problem that charac-", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 113, + 528, + 506, + 542 + ], + "spans": [ + { + "bbox": [ + 113, + 528, + 360, + 542 + ], + "score": 1.0, + "content": "terizes the relation of invariance over the training domain set", + "type": "text" + }, + { + "bbox": [ + 360, + 529, + 385, + 540 + ], + "score": 0.91, + "content": "\\mathcal { E } _ { a v a i l }", + "type": "inline_equation" + }, + { + "bbox": [ + 386, + 528, + 466, + 542 + ], + "score": 1.0, + "content": "and test domain set", + "type": "text" + }, + { + "bbox": [ + 466, + 529, + 483, + 540 + ], + "score": 0.89, + "content": "{ \\mathcal { E } } _ { a l l }", + "type": "inline_equation" + }, + { + "bbox": [ + 483, + 528, + 506, + 542 + ], + "score": 1.0, + "content": ". The", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 113, + 540, + 506, + 552 + ], + "spans": [ + { + "bbox": [ + 113, + 540, + 506, + 552 + ], + "score": 1.0, + "content": "core quantity in our characterization, the expansion function, determines the difficulty of an OOD", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 113, + 551, + 212, + 563 + ], + "spans": [ + { + "bbox": [ + 113, + 551, + 212, + 563 + ], + "score": 1.0, + "content": "generalization problem.", + "type": "text" + } + ], + "index": 39 + } + ], + "index": 37.5, + "bbox_fs": [ + 105, + 518, + 506, + 563 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 574, + 504, + 607 + ], + "lines": [ + { + "bbox": [ + 105, + 573, + 506, + 587 + ], + "spans": [ + { + "bbox": [ + 105, + 573, + 506, + 587 + ], + "score": 1.0, + "content": "• We prove novel OOD generalization error bounds based on our formulation. The upper and lower", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 114, + 585, + 505, + 598 + ], + "spans": [ + { + "bbox": [ + 114, + 585, + 505, + 598 + ], + "score": 1.0, + "content": "bounds together indicate that the expansion function well characterizes the OOD generalization", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 114, + 596, + 321, + 608 + ], + "spans": [ + { + "bbox": [ + 114, + 596, + 321, + 608 + ], + "score": 1.0, + "content": "ability of features with different levels of variation.", + "type": "text" + } + ], + "index": 42 + } + ], + "index": 41, + "bbox_fs": [ + 105, + 573, + 506, + 608 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 620, + 505, + 664 + ], + "lines": [ + { + "bbox": [ + 106, + 620, + 505, + 632 + ], + "spans": [ + { + "bbox": [ + 106, + 620, + 505, + 632 + ], + "score": 1.0, + "content": "• We design a model selection criterion that is inspired by our generalization bounds. Our criterion", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 114, + 631, + 505, + 643 + ], + "spans": [ + { + "bbox": [ + 114, + 631, + 505, + 643 + ], + "score": 1.0, + "content": "takes both the performance on training domains and the variation of models into consideration and", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 114, + 642, + 506, + 655 + ], + "spans": [ + { + "bbox": [ + 114, + 642, + 506, + 655 + ], + "score": 1.0, + "content": "is predictive of OOD performance according to our bounds. Experimental results demonstrate our", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 113, + 652, + 379, + 667 + ], + "spans": [ + { + "bbox": [ + 113, + 652, + 379, + 667 + ], + "score": 1.0, + "content": "selection criterion can choose models with higher OOD accuracy.", + "type": "text" + } + ], + "index": 46 + } + ], + "index": 44.5, + "bbox_fs": [ + 106, + 620, + 506, + 667 + ] + }, + { + "type": "text", + "bbox": [ + 108, + 678, + 505, + 722 + ], + "lines": [ + { + "bbox": [ + 106, + 677, + 505, + 691 + ], + "spans": [ + { + "bbox": [ + 106, + 677, + 505, + 691 + ], + "score": 1.0, + "content": "The rest of the paper is organized as follows: Section 2 is our preliminary. In Section 3, we give our", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 105, + 688, + 505, + 702 + ], + "spans": [ + { + "bbox": [ + 105, + 688, + 505, + 702 + ], + "score": 1.0, + "content": "theoretical formulation. Section 4 gives our generalization bound. We propose our model selection", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 105, + 700, + 506, + 712 + ], + "spans": [ + { + "bbox": [ + 105, + 700, + 506, + 712 + ], + "score": 1.0, + "content": "method in Section 5. 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Our framework can be easily extended to the case where the", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 300, + 506, + 313 + ], + "spans": [ + { + "bbox": [ + 105, + 300, + 506, + 313 + ], + "score": 1.0, + "content": "balanced assumption is removed, with an additional term corresponding to the imbalance adding to", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 311, + 214, + 324 + ], + "spans": [ + { + "bbox": [ + 106, + 311, + 214, + 324 + ], + "score": 1.0, + "content": "the generalization bounds.", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 14 + }, + { + "type": "title", + "bbox": [ + 106, + 338, + 355, + 352 + ], + "lines": [ + { + "bbox": [ + 104, + 337, + 356, + 354 + ], + "spans": [ + { + "bbox": [ + 104, + 337, + 356, + 354 + ], + "score": 1.0, + "content": "3 Framework of OOD Generalization Problem", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 18 + }, + { + "type": "text", + "bbox": [ + 106, + 362, + 506, + 473 + ], + "lines": [ + { + "bbox": [ + 105, + 362, + 505, + 376 + ], + "spans": [ + { + "bbox": [ + 105, + 362, + 505, + 376 + ], + "score": 1.0, + "content": "The main challenge of formalizing the OOD generalization problem is to mathematically describe", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 374, + 506, + 387 + ], + "spans": [ + { + "bbox": [ + 105, + 374, + 201, + 387 + ], + "score": 1.0, + "content": "the connection between", + "type": "text" + }, + { + "bbox": [ + 201, + 375, + 226, + 385 + ], + "score": 0.91, + "content": "\\mathcal { E } _ { a v a i l }", + "type": "inline_equation" + }, + { + "bbox": [ + 227, + 374, + 244, + 387 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 244, + 375, + 260, + 385 + ], + "score": 0.9, + "content": "{ \\mathcal { E } } _ { a l l }", + "type": "inline_equation" + }, + { + "bbox": [ + 261, + 374, + 506, + 387 + ], + "score": 1.0, + "content": "and how generalization depends on this relation. 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Our framework can be easily extended to the case where the", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 300, + 506, + 313 + ], + "spans": [ + { + "bbox": [ + 105, + 300, + 506, + 313 + ], + "score": 1.0, + "content": "balanced assumption is removed, with an additional term corresponding to the imbalance adding to", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 311, + 214, + 324 + ], + "spans": [ + { + "bbox": [ + 106, + 311, + 214, + 324 + ], + "score": 1.0, + "content": "the generalization bounds.", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 14, + "bbox_fs": [ + 105, + 245, + 507, + 324 + ] + }, + { + "type": "title", + "bbox": [ + 106, + 338, + 355, + 352 + ], + "lines": [ + { + "bbox": [ + 104, + 337, + 356, + 354 + ], + "spans": [ + { + "bbox": [ + 104, + 337, + 356, + 354 + ], + "score": 1.0, + "content": "3 Framework of OOD Generalization Problem", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 18 + }, + { + "type": "text", + "bbox": [ + 106, + 362, + 506, + 473 + ], + "lines": [ + { + "bbox": [ + 105, + 362, + 505, + 376 + ], + "spans": [ + { + "bbox": [ + 105, + 362, + 505, + 376 + ], + "score": 1.0, + "content": "The main challenge of formalizing the OOD generalization problem is to mathematically describe", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 374, + 506, + 387 + ], + "spans": [ + { + "bbox": [ + 105, + 374, + 201, + 387 + ], + "score": 1.0, + "content": "the connection between", + "type": "text" + }, + { + "bbox": [ + 201, + 375, + 226, + 385 + ], + "score": 0.91, + "content": "\\mathcal { E } _ { a v a i l }", + "type": "inline_equation" + }, + { + "bbox": [ + 227, + 374, + 244, + 387 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 244, + 375, + 260, + 385 + ], + "score": 0.9, + "content": "{ \\mathcal { E } } _ { a l l }", + "type": "inline_equation" + }, + { + "bbox": [ + 261, + 374, + 506, + 387 + ], + "score": 1.0, + "content": "and how generalization depends on this relation. 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Our framework is motivated by the understanding that, in an OOD generalization task,", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 418, + 506, + 431 + ], + "spans": [ + { + "bbox": [ + 105, + 418, + 265, + 431 + ], + "score": 1.0, + "content": "certain “property” of “good” features in", + "type": "text" + }, + { + "bbox": [ + 266, + 418, + 291, + 429 + ], + "score": 0.91, + "content": "\\mathcal { E } _ { a v a i l }", + "type": "inline_equation" + }, + { + "bbox": [ + 291, + 418, + 392, + 431 + ], + "score": 1.0, + "content": "should be “preserved” in", + "type": "text" + }, + { + "bbox": [ + 392, + 419, + 409, + 429 + ], + "score": 0.9, + "content": "{ \\mathcal { E } } _ { a l l }", + "type": "inline_equation" + }, + { + "bbox": [ + 409, + 418, + 506, + 431 + ], + "score": 1.0, + "content": "(the reason is described", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 428, + 506, + 442 + ], + "spans": [ + { + "bbox": [ + 105, + 428, + 506, + 442 + ], + "score": 1.0, + "content": "in Section 1). In Section 3.1, we will go into details on what we mean by “property” (variation,", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 439, + 505, + 452 + ], + "spans": [ + { + "bbox": [ + 105, + 439, + 505, + 452 + ], + "score": 1.0, + "content": "Definition 3.1), “good” (informativeness, Definition 3.2), and “preserved” (measured by expansion", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 450, + 506, + 464 + ], + "spans": [ + { + "bbox": [ + 105, + 450, + 506, + 464 + ], + "score": 1.0, + "content": "function). 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The first one is what", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 517, + 506, + 529 + ], + "spans": [ + { + "bbox": [ + 105, + 517, + 222, + 529 + ], + "score": 1.0, + "content": "we expect to be preserved in", + "type": "text" + }, + { + "bbox": [ + 222, + 518, + 238, + 528 + ], + "score": 0.9, + "content": "{ \\mathcal { E } } _ { a l l }", + "type": "inline_equation" + }, + { + "bbox": [ + 239, + 517, + 506, + 529 + ], + "score": 1.0, + "content": "and the second one characterizes what features will be considered.", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 527, + 505, + 542 + ], + "spans": [ + { + "bbox": [ + 105, + 527, + 171, + 542 + ], + "score": 1.0, + "content": "Specifically, let", + "type": "text" + }, + { + "bbox": [ + 171, + 529, + 203, + 541 + ], + "score": 0.92, + "content": "\\rho ( \\mathbb { P } , \\mathbb { Q } )", + "type": "inline_equation" + }, + { + "bbox": [ + 203, + 527, + 435, + 542 + ], + "score": 1.0, + "content": "be a symmetric “distance” of two distributions. 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The variation of feature", + "type": "text" + }, + { + "bbox": [ + 314, + 565, + 331, + 578 + ], + "score": 0.91, + "content": "\\phi ( \\cdot )", + "type": "inline_equation" + }, + { + "bbox": [ + 331, + 564, + 414, + 578 + ], + "score": 1.0, + "content": "across a domain set", + "type": "text" + }, + { + "bbox": [ + 415, + 567, + 421, + 575 + ], + "score": 0.84, + "content": "\\mathcal { E }", + "type": "inline_equation" + }, + { + "bbox": [ + 421, + 564, + 432, + 578 + ], + "score": 1.0, + "content": "is", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 35, + "bbox_fs": [ + 106, + 564, + 432, + 578 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 213, + 583, + 397, + 606 + ], + "lines": [ + { + "bbox": [ + 213, + 583, + 397, + 606 + ], + "spans": [ + { + "bbox": [ + 213, + 583, + 397, + 606 + ], + "score": 0.93, + "content": "\\mathcal { V } _ { \\rho } ( \\phi , \\mathcal { E } ) = \\operatorname* { m a x } _ { y \\in \\mathcal { V } } \\operatorname* { s u p } _ { e , e ^ { \\prime } \\in \\mathcal { E } } \\rho \\big ( \\mathbb { P } ( \\phi ^ { e } | y ) , \\mathbb { P } ( \\phi ^ { e ^ { \\prime } } | y ) \\big ) .", + "type": "interline_equation", + "image_path": "b5e6d90d4f6e707f43d0a5fd09cd50b46b59f7364145ae7e58a3a5a8d9599fbf.jpg" + } + ] + } + ], + "index": 36, + "virtual_lines": [ + { + "bbox": [ + 213, + 583, + 397, + 606 + ], + "spans": [], + "index": 36 + } + ] + }, + { + "type": "text", + "bbox": [ + 105, + 612, + 507, + 640 + ], + "lines": [ + { + "bbox": [ + 105, + 612, + 506, + 626 + ], + "spans": [ + { + "bbox": [ + 105, + 612, + 144, + 626 + ], + "score": 1.0, + "content": "A feature", + "type": "text" + }, + { + "bbox": [ + 144, + 613, + 161, + 624 + ], + "score": 0.92, + "content": "\\phi ( \\cdot )", + "type": "inline_equation" + }, + { + "bbox": [ + 161, + 612, + 171, + 626 + ], + "score": 1.0, + "content": "is", + "type": "text" + }, + { + "bbox": [ + 171, + 617, + 176, + 622 + ], + "score": 0.76, + "content": "\\varepsilon", + "type": "inline_equation" + }, + { + "bbox": [ + 176, + 612, + 243, + 626 + ], + "score": 1.0, + "content": "-invariant across", + "type": "text" + }, + { + "bbox": [ + 244, + 613, + 250, + 622 + ], + "score": 0.69, + "content": "\\mathcal { E }", + "type": "inline_equation" + }, + { + "bbox": [ + 251, + 612, + 255, + 626 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 255, + 612, + 312, + 624 + ], + "score": 0.89, + "content": "i f \\varepsilon \\geq \\mathcal { V } ( \\phi , \\mathcal { E } )", + "type": "inline_equation" + }, + { + "bbox": [ + 312, + 612, + 401, + 626 + ], + "score": 1.0, + "content": "(We omit the subscript", + "type": "text" + }, + { + "bbox": [ + 402, + 615, + 408, + 624 + ], + "score": 0.57, + "content": "\\rho", + "type": "inline_equation" + }, + { + "bbox": [ + 408, + 612, + 506, + 626 + ], + "score": 1.0, + "content": "in case of no ambiguity).", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 106, + 627, + 486, + 641 + ], + "spans": [ + { + "bbox": [ + 106, + 627, + 367, + 641 + ], + "score": 1.0, + "content": "Definition 3.2 (Informativeness). The informativeness of feature", + "type": "text" + }, + { + "bbox": [ + 368, + 627, + 385, + 640 + ], + "score": 0.94, + "content": "\\phi ( \\cdot )", + "type": "inline_equation" + }, + { + "bbox": [ + 385, + 627, + 468, + 641 + ], + "score": 1.0, + "content": "across a domain set", + "type": "text" + }, + { + "bbox": [ + 469, + 630, + 475, + 637 + ], + "score": 0.83, + "content": "\\mathcal { E }", + "type": "inline_equation" + }, + { + "bbox": [ + 475, + 627, + 486, + 641 + ], + "score": 1.0, + "content": "is", + "type": "text" + } + ], + "index": 38 + } + ], + "index": 37.5, + "bbox_fs": [ + 105, + 612, + 506, + 641 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 179, + 644, + 411, + 685 + ], + "lines": [ + { + "bbox": [ + 179, + 644, + 411, + 685 + ], + "spans": [ + { + "bbox": [ + 179, + 644, + 411, + 685 + ], + "score": 0.94, + "content": "\\mathcal { T } _ { \\rho } ( \\phi , \\mathcal { E } ) = \\frac { 1 } { K ( K - 1 ) } \\sum _ { \\stackrel { y \\neq y ^ { \\prime } } { y , y ^ { \\prime } \\in y } } \\operatorname* { m i n } _ { e \\in \\mathcal { E } } \\rho \\big ( \\mathbb { P } ( \\phi ^ { e } | y ) , \\mathbb { P } ( \\phi ^ { e } | y ^ { \\prime } ) \\big ) .", + "type": "interline_equation", + "image_path": "b5d2d9f29351ffd114b5530974aa3d8ba7d34f3c8275c029e6a978717a1c0731.jpg" + } + ] + } + ], + "index": 39.5, + "virtual_lines": [ + { + "bbox": [ + 179, + 644, + 411, + 664.5 + ], + "spans": [], + "index": 39 + }, + { + "bbox": [ + 179, + 664.5, + 411, + 685.0 + ], + "spans": [], + "index": 40 + } + ] + }, + { + "type": "text", + "bbox": [ + 105, + 690, + 330, + 703 + ], + "lines": [ + { + "bbox": [ + 104, + 689, + 330, + 704 + ], + "spans": [ + { + "bbox": [ + 104, + 689, + 145, + 704 + ], + "score": 1.0, + "content": "A feature", + "type": "text" + }, + { + "bbox": [ + 145, + 690, + 162, + 703 + ], + "score": 0.91, + "content": "\\phi ( \\cdot )", + "type": "inline_equation" + }, + { + "bbox": [ + 163, + 689, + 173, + 704 + ], + "score": 1.0, + "content": "is", + "type": "text" + }, + { + "bbox": [ + 173, + 691, + 179, + 700 + ], + "score": 0.77, + "content": "\\delta", + "type": "inline_equation" + }, + { + "bbox": [ + 179, + 689, + 258, + 704 + ], + "score": 1.0, + "content": "-informative across", + "type": "text" + }, + { + "bbox": [ + 258, + 693, + 264, + 700 + ], + "score": 0.81, + "content": "\\varepsilon", + "type": "inline_equation" + }, + { + "bbox": [ + 264, + 689, + 276, + 704 + ], + "score": 1.0, + "content": ", if", + "type": "text" + }, + { + "bbox": [ + 276, + 690, + 326, + 703 + ], + "score": 0.86, + "content": "\\quad \\delta \\leq { \\mathcal { I } } ( \\phi , { \\mathcal { E } } )", + "type": "inline_equation" + }, + { + "bbox": [ + 326, + 689, + 330, + 704 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 41 + } + ], + "index": 41, + "bbox_fs": [ + 104, + 689, + 330, + 704 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 72, + 505, + 128 + ], + "lines": [ + { + "bbox": [ + 105, + 72, + 506, + 85 + ], + "spans": [ + { + "bbox": [ + 105, + 72, + 162, + 85 + ], + "score": 1.0, + "content": "The variation", + "type": "text" + }, + { + "bbox": [ + 163, + 73, + 195, + 85 + ], + "score": 0.92, + "content": "\\mathcal { V } ( \\phi , \\mathcal { E } )", + "type": "inline_equation" + }, + { + "bbox": [ + 195, + 72, + 297, + 85 + ], + "score": 1.0, + "content": "measures the stability of", + "type": "text" + }, + { + "bbox": [ + 297, + 73, + 315, + 85 + ], + "score": 0.91, + "content": "\\phi ( \\cdot )", + "type": "inline_equation" + }, + { + "bbox": [ + 315, + 72, + 398, + 85 + ], + "score": 1.0, + "content": "over the domains in", + "type": "text" + }, + { + "bbox": [ + 398, + 73, + 405, + 83 + ], + "score": 0.79, + "content": "\\mathcal { E }", + "type": "inline_equation" + }, + { + "bbox": [ + 406, + 72, + 506, + 85 + ], + "score": 1.0, + "content": "and the informativeness", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 82, + 505, + 96 + ], + "spans": [ + { + "bbox": [ + 106, + 83, + 138, + 96 + ], + "score": 0.92, + "content": "{ \\mathcal { T } } ( \\phi , { \\mathcal { E } } )", + "type": "inline_equation" + }, + { + "bbox": [ + 138, + 82, + 231, + 96 + ], + "score": 1.0, + "content": "captures the ability of", + "type": "text" + }, + { + "bbox": [ + 231, + 84, + 249, + 96 + ], + "score": 0.9, + "content": "\\phi ( \\cdot )", + "type": "inline_equation" + }, + { + "bbox": [ + 249, + 82, + 505, + 96 + ], + "score": 1.0, + "content": "to distinguish different labels. We would like to highlight that", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 93, + 505, + 108 + ], + "spans": [ + { + "bbox": [ + 105, + 93, + 417, + 108 + ], + "score": 1.0, + "content": "the variation and informativeness are defined on each one-dimensional feature", + "type": "text" + }, + { + "bbox": [ + 418, + 95, + 435, + 106 + ], + "score": 0.9, + "content": "\\phi ( \\cdot )", + "type": "inline_equation" + }, + { + "bbox": [ + 435, + 93, + 505, + 108 + ], + "score": 1.0, + "content": ". Unlike previous", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 106, + 106, + 505, + 118 + ], + "spans": [ + { + "bbox": [ + 106, + 106, + 273, + 118 + ], + "score": 1.0, + "content": "distance between distributions defined in", + "type": "text" + }, + { + "bbox": [ + 273, + 106, + 280, + 116 + ], + "score": 0.76, + "content": "d", + "type": "inline_equation" + }, + { + "bbox": [ + 280, + 106, + 505, + 118 + ], + "score": 1.0, + "content": "-dimensional space, our definitions are more reasonable", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 106, + 117, + 347, + 128 + ], + "spans": [ + { + "bbox": [ + 106, + 117, + 347, + 128 + ], + "score": 1.0, + "content": "and practical, since it can be easily calculated and analyzed.", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 2 + }, + { + "type": "text", + "bbox": [ + 107, + 132, + 505, + 199 + ], + "lines": [ + { + "bbox": [ + 105, + 132, + 506, + 146 + ], + "spans": [ + { + "bbox": [ + 105, + 132, + 363, + 146 + ], + "score": 1.0, + "content": "We are now ready to introduce the core quantity for connecting", + "type": "text" + }, + { + "bbox": [ + 363, + 133, + 389, + 144 + ], + "score": 0.91, + "content": "\\mathcal { E } _ { a v a i l }", + "type": "inline_equation" + }, + { + "bbox": [ + 389, + 132, + 406, + 146 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 407, + 133, + 423, + 144 + ], + "score": 0.89, + "content": "{ \\mathcal { E } } _ { a l l }", + "type": "inline_equation" + }, + { + "bbox": [ + 424, + 132, + 506, + 146 + ], + "score": 1.0, + "content": ". Our motivation, as", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 144, + 505, + 155 + ], + "spans": [ + { + "bbox": [ + 106, + 144, + 505, + 155 + ], + "score": 1.0, + "content": "elaborated in the introduction section, is that, if a feature is informative for the classification task", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 154, + 506, + 168 + ], + "spans": [ + { + "bbox": [ + 105, + 154, + 182, + 168 + ], + "score": 1.0, + "content": "and invariant over", + "type": "text" + }, + { + "bbox": [ + 182, + 155, + 208, + 165 + ], + "score": 0.91, + "content": "\\mathcal { E } _ { a v a i l }", + "type": "inline_equation" + }, + { + "bbox": [ + 208, + 154, + 379, + 168 + ], + "score": 1.0, + "content": ", then to enable OOD generalization from", + "type": "text" + }, + { + "bbox": [ + 379, + 155, + 404, + 165 + ], + "score": 0.9, + "content": "\\mathcal { E } _ { a v a i l }", + "type": "inline_equation" + }, + { + "bbox": [ + 405, + 154, + 416, + 168 + ], + "score": 1.0, + "content": "to", + "type": "text" + }, + { + "bbox": [ + 416, + 155, + 433, + 165 + ], + "score": 0.89, + "content": "{ \\mathcal { E } } _ { a l l }", + "type": "inline_equation" + }, + { + "bbox": [ + 433, + 154, + 506, + 168 + ], + "score": 1.0, + "content": ", it should be still", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 164, + 505, + 179 + ], + "spans": [ + { + "bbox": [ + 105, + 164, + 165, + 179 + ], + "score": 1.0, + "content": "invariant over", + "type": "text" + }, + { + "bbox": [ + 165, + 166, + 182, + 176 + ], + "score": 0.9, + "content": "{ \\mathcal { E } } _ { a l l }", + "type": "inline_equation" + }, + { + "bbox": [ + 182, + 164, + 288, + 179 + ], + "score": 1.0, + "content": ". So the relation between", + "type": "text" + }, + { + "bbox": [ + 288, + 165, + 338, + 177 + ], + "score": 0.92, + "content": "\\mathcal { V } ( \\phi , \\mathcal { E } _ { a v a i l } )", + "type": "inline_equation" + }, + { + "bbox": [ + 338, + 164, + 357, + 179 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 357, + 165, + 398, + 177 + ], + "score": 0.92, + "content": "\\mathinner { \\gamma \\mathopen { \\left( \\phi , \\mathcal { E } _ { a l l } \\right) } }", + "type": "inline_equation" + }, + { + "bbox": [ + 399, + 164, + 505, + 179 + ], + "score": 1.0, + "content": "of an informative feature", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 176, + 506, + 189 + ], + "spans": [ + { + "bbox": [ + 105, + 176, + 506, + 189 + ], + "score": 1.0, + "content": "captures the feasibility and difficulty of OOD generalization. To quantitatively measure this relation,", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 106, + 188, + 263, + 199 + ], + "spans": [ + { + "bbox": [ + 106, + 188, + 263, + 199 + ], + "score": 1.0, + "content": "we define the following function class:", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 7.5 + }, + { + "type": "text", + "bbox": [ + 107, + 202, + 505, + 236 + ], + "lines": [ + { + "bbox": [ + 105, + 201, + 505, + 215 + ], + "spans": [ + { + "bbox": [ + 105, + 201, + 346, + 215 + ], + "score": 1.0, + "content": "Definition 3.3 (Expansion Function). We say a function", + "type": "text" + }, + { + "bbox": [ + 347, + 202, + 493, + 214 + ], + "score": 0.91, + "content": "s : \\mathbb { R } ^ { + } \\cup \\{ 0 \\} \\mathbb { R } ^ { + } \\cup \\{ 0 , + \\infty \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 494, + 201, + 505, + 215 + ], + "score": 1.0, + "content": "is", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 214, + 505, + 226 + ], + "spans": [ + { + "bbox": [ + 105, + 214, + 352, + 226 + ], + "score": 1.0, + "content": "an expansion function, iff the following properties hold: 1)", + "type": "text" + }, + { + "bbox": [ + 352, + 214, + 369, + 226 + ], + "score": 0.88, + "content": "s ( \\cdot )", + "type": "inline_equation" + }, + { + "bbox": [ + 369, + 214, + 505, + 226 + ], + "score": 1.0, + "content": "is monotonically increasing and", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 107, + 223, + 304, + 237 + ], + "spans": [ + { + "bbox": [ + 107, + 225, + 191, + 236 + ], + "score": 0.77, + "content": "s ( x ) \\geq x , \\forall x \\geq 0 ; 2 ,", + "type": "inline_equation" + }, + { + "bbox": [ + 191, + 223, + 194, + 237 + ], + "score": 1.0, + "content": ")", + "type": "text" + }, + { + "bbox": [ + 194, + 225, + 300, + 236 + ], + "score": 0.85, + "content": "\\begin{array} { r } { \\operatorname* { l i m } _ { x 0 ^ { + } } s ( x ) = s ( 0 ) = 0 } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 301, + 223, + 304, + 237 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 12 + }, + { + "type": "text", + "bbox": [ + 107, + 245, + 505, + 278 + ], + "lines": [ + { + "bbox": [ + 105, + 245, + 506, + 258 + ], + "spans": [ + { + "bbox": [ + 105, + 245, + 431, + 258 + ], + "score": 1.0, + "content": "This function class gives a full characterization of how the variation between", + "type": "text" + }, + { + "bbox": [ + 432, + 245, + 457, + 256 + ], + "score": 0.9, + "content": "\\mathcal { E } _ { a v a i l }", + "type": "inline_equation" + }, + { + "bbox": [ + 457, + 245, + 477, + 258 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 477, + 245, + 493, + 256 + ], + "score": 0.89, + "content": "{ \\mathcal { E } } _ { a l l }", + "type": "inline_equation" + }, + { + "bbox": [ + 494, + 245, + 506, + 258 + ], + "score": 1.0, + "content": "is", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 256, + 505, + 268 + ], + "spans": [ + { + "bbox": [ + 105, + 256, + 505, + 268 + ], + "score": 1.0, + "content": "related. Based on this function class, we can introduce our formulation of the learnability of OOD", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 267, + 211, + 279 + ], + "spans": [ + { + "bbox": [ + 105, + 267, + 211, + 279 + ], + "score": 1.0, + "content": "generalization as follows:", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 15 + }, + { + "type": "text", + "bbox": [ + 107, + 282, + 505, + 337 + ], + "lines": [ + { + "bbox": [ + 105, + 280, + 505, + 296 + ], + "spans": [ + { + "bbox": [ + 105, + 280, + 245, + 296 + ], + "score": 1.0, + "content": "Definition 3.4 (Learnability). Let", + "type": "text" + }, + { + "bbox": [ + 246, + 283, + 254, + 292 + ], + "score": 0.7, + "content": "\\Phi", + "type": "inline_equation" + }, + { + "bbox": [ + 255, + 280, + 357, + 296 + ], + "score": 1.0, + "content": "be the feature space and", + "type": "text" + }, + { + "bbox": [ + 357, + 284, + 364, + 294 + ], + "score": 0.64, + "content": "\\rho", + "type": "inline_equation" + }, + { + "bbox": [ + 364, + 280, + 505, + 296 + ], + "score": 1.0, + "content": "be a distribution distance. We say", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 293, + 505, + 306 + ], + "spans": [ + { + "bbox": [ + 105, + 293, + 258, + 306 + ], + "score": 1.0, + "content": "an OOD generalization problem from", + "type": "text" + }, + { + "bbox": [ + 258, + 294, + 283, + 304 + ], + "score": 0.89, + "content": "\\mathcal { E } _ { a v a i l }", + "type": "inline_equation" + }, + { + "bbox": [ + 283, + 293, + 294, + 306 + ], + "score": 1.0, + "content": "to", + "type": "text" + }, + { + "bbox": [ + 295, + 294, + 311, + 304 + ], + "score": 0.88, + "content": "{ \\mathcal { E } } _ { a l l }", + "type": "inline_equation" + }, + { + "bbox": [ + 312, + 293, + 505, + 306 + ], + "score": 1.0, + "content": "is learnable if there exists an expansion function", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 107, + 304, + 504, + 317 + ], + "spans": [ + { + "bbox": [ + 107, + 304, + 123, + 316 + ], + "score": 0.89, + "content": "s ( \\cdot )", + "type": "inline_equation" + }, + { + "bbox": [ + 123, + 304, + 142, + 317 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 142, + 304, + 168, + 315 + ], + "score": 0.9, + "content": "\\delta \\geq 0", + "type": "inline_equation" + }, + { + "bbox": [ + 168, + 304, + 243, + 317 + ], + "score": 1.0, + "content": ", such that: for all", + "type": "text" + }, + { + "bbox": [ + 244, + 304, + 271, + 315 + ], + "score": 0.91, + "content": "\\phi \\in \\Phi", + "type": "inline_equation" + }, + { + "bbox": [ + 271, + 304, + 313, + 317 + ], + "score": 1.0, + "content": "satisfying", + "type": "text" + }, + { + "bbox": [ + 314, + 304, + 386, + 316 + ], + "score": 0.91, + "content": "{ \\mathcal { T } } _ { \\rho } ( \\phi , { \\mathcal { E } } _ { a v a i l } ) \\geq \\delta", + "type": "inline_equation" + }, + { + "bbox": [ + 387, + 304, + 426, + 317 + ], + "score": 1.0, + "content": ", we have", + "type": "text" + }, + { + "bbox": [ + 427, + 304, + 504, + 316 + ], + "score": 0.91, + "content": "s ( \\mathcal { V } _ { \\rho } ( \\phi , \\mathcal { E } _ { a v a i l } ) ) \\geq", + "type": "inline_equation" + } + ], + "index": 19 + }, + { + "bbox": [ + 107, + 314, + 504, + 328 + ], + "spans": [ + { + "bbox": [ + 107, + 315, + 151, + 327 + ], + "score": 0.92, + "content": "\\mathcal { V } _ { \\rho } \\big ( \\phi , \\mathcal { E } _ { a l l } \\big )", + "type": "inline_equation" + }, + { + "bbox": [ + 151, + 314, + 189, + 328 + ], + "score": 1.0, + "content": ". If such", + "type": "text" + }, + { + "bbox": [ + 189, + 315, + 205, + 327 + ], + "score": 0.88, + "content": "s ( \\cdot )", + "type": "inline_equation" + }, + { + "bbox": [ + 206, + 314, + 225, + 328 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 226, + 316, + 232, + 325 + ], + "score": 0.71, + "content": "\\delta", + "type": "inline_equation" + }, + { + "bbox": [ + 232, + 314, + 377, + 328 + ], + "score": 1.0, + "content": "exist, we further call this problem", + "type": "text" + }, + { + "bbox": [ + 377, + 315, + 409, + 327 + ], + "score": 0.93, + "content": "( s ( \\cdot ) , \\delta )", + "type": "inline_equation" + }, + { + "bbox": [ + 410, + 314, + 481, + 328 + ], + "score": 1.0, + "content": "-learnable. If an", + "type": "text" + }, + { + "bbox": [ + 481, + 316, + 504, + 326 + ], + "score": 0.38, + "content": "o o D", + "type": "inline_equation" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 326, + 359, + 338 + ], + "spans": [ + { + "bbox": [ + 105, + 326, + 300, + 338 + ], + "score": 1.0, + "content": "generalization problem is not learnable, we call", + "type": "text" + }, + { + "bbox": [ + 300, + 327, + 307, + 336 + ], + "score": 0.5, + "content": "i t", + "type": "inline_equation" + }, + { + "bbox": [ + 307, + 326, + 359, + 338 + ], + "score": 1.0, + "content": "unlearnable.", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 19 + }, + { + "type": "text", + "bbox": [ + 105, + 346, + 482, + 358 + ], + "lines": [ + { + "bbox": [ + 106, + 345, + 484, + 359 + ], + "spans": [ + { + "bbox": [ + 106, + 345, + 484, + 359 + ], + "score": 1.0, + "content": "To understand the intuition and rationality of our formulation, several discussions are in order.", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 22 + }, + { + "type": "text", + "bbox": [ + 106, + 370, + 506, + 469 + ], + "lines": [ + { + "bbox": [ + 105, + 369, + 507, + 384 + ], + "spans": [ + { + "bbox": [ + 105, + 369, + 507, + 384 + ], + "score": 1.0, + "content": "Properties of the expansion function. In Definition 3.3, we highlight two properties of the ex-", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 381, + 507, + 394 + ], + "spans": [ + { + "bbox": [ + 105, + 381, + 507, + 394 + ], + "score": 1.0, + "content": "pansion function. The first property comes naturally from the monotonicity properties of variation:", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 392, + 507, + 406 + ], + "spans": [ + { + "bbox": [ + 105, + 392, + 123, + 406 + ], + "score": 1.0, + "content": "any", + "type": "text" + }, + { + "bbox": [ + 123, + 394, + 134, + 403 + ], + "score": 0.85, + "content": "\\varepsilon _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 134, + 392, + 232, + 406 + ], + "score": 1.0, + "content": "-invariant feature is also", + "type": "text" + }, + { + "bbox": [ + 233, + 394, + 243, + 403 + ], + "score": 0.86, + "content": "\\varepsilon _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 244, + 392, + 298, + 406 + ], + "score": 1.0, + "content": "-invariant for", + "type": "text" + }, + { + "bbox": [ + 298, + 393, + 331, + 403 + ], + "score": 0.9, + "content": "\\varepsilon _ { 2 } \\geq \\varepsilon _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 331, + 392, + 352, + 406 + ], + "score": 1.0, + "content": "; and", + "type": "text" + }, + { + "bbox": [ + 352, + 392, + 436, + 404 + ], + "score": 0.91, + "content": "\\mathcal { V } ( \\phi , \\mathcal { E } _ { 1 } ) \\leq \\mathcal { V } ( \\phi , \\mathcal { E } _ { 2 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 436, + 392, + 469, + 406 + ], + "score": 1.0, + "content": "for any", + "type": "text" + }, + { + "bbox": [ + 469, + 392, + 503, + 404 + ], + "score": 0.91, + "content": "{ \\mathcal { E } } _ { 1 } \\subseteq { \\mathcal { E } } _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 503, + 392, + 507, + 406 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 402, + 506, + 417 + ], + "spans": [ + { + "bbox": [ + 105, + 402, + 280, + 417 + ], + "score": 1.0, + "content": "The monotonicity also implies that larger", + "type": "text" + }, + { + "bbox": [ + 280, + 404, + 297, + 414 + ], + "score": 0.89, + "content": "{ \\mathcal { E } } _ { a l l }", + "type": "inline_equation" + }, + { + "bbox": [ + 297, + 402, + 374, + 417 + ], + "score": 1.0, + "content": "will induce larger", + "type": "text" + }, + { + "bbox": [ + 375, + 404, + 395, + 415 + ], + "score": 0.92, + "content": "s ( \\cdot ) ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 395, + 402, + 506, + 417 + ], + "score": 1.0, + "content": "and it is also harder to be", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 414, + 505, + 427 + ], + "spans": [ + { + "bbox": [ + 105, + 414, + 351, + 427 + ], + "score": 1.0, + "content": "generalized to. From this view, we can see that the scale of", + "type": "text" + }, + { + "bbox": [ + 352, + 414, + 368, + 426 + ], + "score": 0.91, + "content": "s ( \\cdot )", + "type": "inline_equation" + }, + { + "bbox": [ + 368, + 414, + 505, + 427 + ], + "score": 1.0, + "content": "can reflect the difficulty of OOD", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 425, + 506, + 438 + ], + "spans": [ + { + "bbox": [ + 105, + 425, + 506, + 438 + ], + "score": 1.0, + "content": "generalization. The second property is more crucial since it formulates the intuition that if an informa-", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 436, + 506, + 448 + ], + "spans": [ + { + "bbox": [ + 106, + 437, + 239, + 448 + ], + "score": 1.0, + "content": "tive feature is almost invariant in", + "type": "text" + }, + { + "bbox": [ + 240, + 436, + 264, + 447 + ], + "score": 0.92, + "content": "\\mathcal { E } _ { a v a i l }", + "type": "inline_equation" + }, + { + "bbox": [ + 265, + 437, + 383, + 448 + ], + "score": 1.0, + "content": ", it should remain invariant in", + "type": "text" + }, + { + "bbox": [ + 383, + 436, + 399, + 447 + ], + "score": 0.89, + "content": "{ \\mathcal { E } } _ { a l l }", + "type": "inline_equation" + }, + { + "bbox": [ + 400, + 437, + 506, + 448 + ], + "score": 1.0, + "content": ". Without this assumption,", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 446, + 506, + 460 + ], + "spans": [ + { + "bbox": [ + 105, + 446, + 506, + 460 + ], + "score": 1.0, + "content": "OOD generalization can never be guaranteed because we cannot predict whether an invariant and", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 104, + 456, + 356, + 472 + ], + "spans": [ + { + "bbox": [ + 104, + 456, + 195, + 472 + ], + "score": 1.0, + "content": "informative feature in", + "type": "text" + }, + { + "bbox": [ + 195, + 458, + 221, + 469 + ], + "score": 0.91, + "content": "\\mathcal { E } _ { a v a i l }", + "type": "inline_equation" + }, + { + "bbox": [ + 221, + 456, + 335, + 472 + ], + "score": 1.0, + "content": "will vary severely in unseen", + "type": "text" + }, + { + "bbox": [ + 336, + 458, + 352, + 469 + ], + "score": 0.9, + "content": "{ \\mathcal { E } } _ { a l l }", + "type": "inline_equation" + }, + { + "bbox": [ + 352, + 456, + 356, + 472 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 27 + }, + { + "type": "text", + "bbox": [ + 107, + 481, + 505, + 591 + ], + "lines": [ + { + "bbox": [ + 105, + 482, + 505, + 494 + ], + "spans": [ + { + "bbox": [ + 105, + 482, + 505, + 494 + ], + "score": 1.0, + "content": "Necessity of informativeness. We include a seemingly redundant quantity informativeness in the", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 492, + 506, + 505 + ], + "spans": [ + { + "bbox": [ + 105, + 492, + 506, + 505 + ], + "score": 1.0, + "content": "definition of learnability. However, this term is necessary because only informative features are", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 502, + 504, + 518 + ], + "spans": [ + { + "bbox": [ + 105, + 502, + 478, + 518 + ], + "score": 1.0, + "content": "responsible for the performance of classification. Non-informative but invariant features over", + "type": "text" + }, + { + "bbox": [ + 479, + 504, + 504, + 515 + ], + "score": 0.87, + "content": "\\mathcal { E } _ { a v a i l }", + "type": "inline_equation" + } + ], + "index": 34 + }, + { + "bbox": [ + 106, + 515, + 505, + 527 + ], + "spans": [ + { + "bbox": [ + 106, + 515, + 505, + 527 + ], + "score": 1.0, + "content": "may only capture some noise that is irrelevant to the classification problem, and we shall not expect", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 525, + 506, + 538 + ], + "spans": [ + { + "bbox": [ + 105, + 525, + 221, + 538 + ], + "score": 1.0, + "content": "the noise to be invariant over", + "type": "text" + }, + { + "bbox": [ + 222, + 525, + 238, + 536 + ], + "score": 0.89, + "content": "{ \\mathcal { E } } _ { a l l }", + "type": "inline_equation" + }, + { + "bbox": [ + 238, + 525, + 506, + 538 + ], + "score": 1.0, + "content": ". Moreover, we show in Figure 1 that in practice, many invariant but", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 537, + 506, + 549 + ], + "spans": [ + { + "bbox": [ + 105, + 537, + 182, + 549 + ], + "score": 1.0, + "content": "useless features in", + "type": "text" + }, + { + "bbox": [ + 182, + 537, + 208, + 548 + ], + "score": 0.91, + "content": "\\mathcal { E } _ { a v a i l }", + "type": "inline_equation" + }, + { + "bbox": [ + 208, + 537, + 259, + 549 + ], + "score": 1.0, + "content": "vary a lot in", + "type": "text" + }, + { + "bbox": [ + 260, + 537, + 276, + 547 + ], + "score": 0.9, + "content": "{ \\mathcal { E } } _ { a l l }", + "type": "inline_equation" + }, + { + "bbox": [ + 276, + 537, + 506, + 549 + ], + "score": 1.0, + "content": ", and adding the constraint of informativeness makes the", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 547, + 505, + 559 + ], + "spans": [ + { + "bbox": [ + 105, + 547, + 400, + 559 + ], + "score": 1.0, + "content": "expansion function reasonable. In addition, there are multiple choices of", + "type": "text" + }, + { + "bbox": [ + 400, + 547, + 433, + 559 + ], + "score": 0.92, + "content": "( s ( \\cdot ) , \\delta )", + "type": "inline_equation" + }, + { + "bbox": [ + 433, + 547, + 505, + 559 + ], + "score": 1.0, + "content": "to make an OOD", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 558, + 506, + 571 + ], + "spans": [ + { + "bbox": [ + 105, + 558, + 272, + 571 + ], + "score": 1.0, + "content": "generalization problem learnable: larger", + "type": "text" + }, + { + "bbox": [ + 273, + 559, + 279, + 568 + ], + "score": 0.69, + "content": "\\delta", + "type": "inline_equation" + }, + { + "bbox": [ + 280, + 558, + 426, + 571 + ], + "score": 1.0, + "content": "will filter out more features, and so", + "type": "text" + }, + { + "bbox": [ + 426, + 558, + 442, + 570 + ], + "score": 0.87, + "content": "\\dot { s } ( \\cdot )", + "type": "inline_equation" + }, + { + "bbox": [ + 443, + 558, + 506, + 571 + ], + "score": 1.0, + "content": "can be smaller", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 568, + 505, + 581 + ], + "spans": [ + { + "bbox": [ + 105, + 568, + 349, + 581 + ], + "score": 1.0, + "content": "(flatter). This multiplicity will result in a trade-off between", + "type": "text" + }, + { + "bbox": [ + 350, + 569, + 366, + 581 + ], + "score": 0.9, + "content": "s ( \\cdot )", + "type": "inline_equation" + }, + { + "bbox": [ + 366, + 568, + 385, + 581 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 385, + 569, + 390, + 579 + ], + "score": 0.76, + "content": "\\delta", + "type": "inline_equation" + }, + { + "bbox": [ + 391, + 568, + 505, + 581 + ], + "score": 1.0, + "content": ", which will be discussed in", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 579, + 157, + 592 + ], + "spans": [ + { + "bbox": [ + 105, + 579, + 157, + 592 + ], + "score": 1.0, + "content": "Section 6.2.", + "type": "text" + } + ], + "index": 41 + } + ], + "index": 36.5 + }, + { + "type": "text", + "bbox": [ + 106, + 603, + 505, + 692 + ], + "lines": [ + { + "bbox": [ + 105, + 602, + 505, + 617 + ], + "spans": [ + { + "bbox": [ + 105, + 602, + 505, + 617 + ], + "score": 1.0, + "content": "Two extreme cases: i.i.d. & unlearnable. To better understand the concept of learnability, we", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 614, + 506, + 628 + ], + "spans": [ + { + "bbox": [ + 105, + 614, + 451, + 628 + ], + "score": 1.0, + "content": "consider two extreme cases. (1) The first example is when all data from different", + "type": "text" + }, + { + "bbox": [ + 451, + 615, + 488, + 626 + ], + "score": 0.9, + "content": "\\textit { e } \\in \\mathcal { E } _ { a l l }", + "type": "inline_equation" + }, + { + "bbox": [ + 488, + 614, + 506, + 628 + ], + "score": 1.0, + "content": "are", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 625, + 506, + 638 + ], + "spans": [ + { + "bbox": [ + 105, + 625, + 431, + 638 + ], + "score": 1.0, + "content": "identically distributed, i.e., the classic supervised learning setting. This problem is", + "type": "text" + }, + { + "bbox": [ + 432, + 626, + 464, + 638 + ], + "score": 0.92, + "content": "( s ( \\cdot ) , 0 )", + "type": "inline_equation" + }, + { + "bbox": [ + 465, + 625, + 506, + 638 + ], + "score": 1.0, + "content": "-learnable", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 106, + 637, + 506, + 649 + ], + "spans": [ + { + "bbox": [ + 106, + 637, + 126, + 649 + ], + "score": 1.0, + "content": "with", + "type": "text" + }, + { + "bbox": [ + 126, + 637, + 164, + 649 + ], + "score": 0.93, + "content": "s ( x ) = x", + "type": "inline_equation" + }, + { + "bbox": [ + 164, + 637, + 506, + 649 + ], + "score": 1.0, + "content": ", implying no extra difficulty in OOD generalization. (2) As an example of unlearnable,", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 647, + 506, + 660 + ], + "spans": [ + { + "bbox": [ + 105, + 647, + 380, + 660 + ], + "score": 1.0, + "content": "consider the following case (modified from Colored MNIST [5]): For", + "type": "text" + }, + { + "bbox": [ + 381, + 648, + 423, + 659 + ], + "score": 0.92, + "content": "e \\in \\mathcal { E } _ { a v a i l }", + "type": "inline_equation" + }, + { + "bbox": [ + 423, + 647, + 506, + 660 + ], + "score": 1.0, + "content": ", images with label 0", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 657, + 506, + 673 + ], + "spans": [ + { + "bbox": [ + 105, + 657, + 436, + 673 + ], + "score": 1.0, + "content": "always has a red background while images with label 1 has a blue background. For", + "type": "text" + }, + { + "bbox": [ + 436, + 658, + 502, + 671 + ], + "score": 0.91, + "content": "e \\in \\mathcal { E } _ { a l l } \\ \\backslash \\ \\mathcal { E } _ { a v a i l }", + "type": "inline_equation" + }, + { + "bbox": [ + 503, + 657, + 506, + 673 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 105, + 669, + 506, + 682 + ], + "spans": [ + { + "bbox": [ + 105, + 669, + 345, + 682 + ], + "score": 1.0, + "content": "this relationship is entirely inverse. Since data from different", + "type": "text" + }, + { + "bbox": [ + 345, + 669, + 387, + 680 + ], + "score": 0.92, + "content": "e \\in \\mathcal { E } _ { a v a i l }", + "type": "inline_equation" + }, + { + "bbox": [ + 388, + 669, + 506, + 682 + ], + "score": 1.0, + "content": "are identically distributed but", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 105, + 680, + 507, + 693 + ], + "spans": [ + { + "bbox": [ + 105, + 680, + 186, + 693 + ], + "score": 1.0, + "content": "different from other", + "type": "text" + }, + { + "bbox": [ + 186, + 680, + 219, + 691 + ], + "score": 0.91, + "content": "e \\in \\mathcal { E } _ { a l l }", + "type": "inline_equation" + }, + { + "bbox": [ + 219, + 680, + 507, + 693 + ], + "score": 1.0, + "content": ", no expansion function can make it learnable, i.e., it is OOD-unlearnable.", + "type": "text" + } + ], + "index": 49 + } + ], + "index": 45.5 + } + ], + "page_idx": 3, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 106, + 701, + 505, + 722 + ], + "lines": [ + { + "bbox": [ + 118, + 699, + 506, + 714 + ], + "spans": [ + { + "bbox": [ + 118, + 699, + 241, + 714 + ], + "score": 1.0, + "content": "2When we talk about the scale of", + "type": "text" + }, + { + "bbox": [ + 241, + 703, + 255, + 712 + ], + "score": 0.89, + "content": "s ( \\cdot )", + "type": "inline_equation" + }, + { + "bbox": [ + 255, + 699, + 313, + 714 + ], + "score": 1.0, + "content": ", e.g. it is larger", + "type": "text" + }, + { + "bbox": [ + 313, + 703, + 317, + 711 + ], + "score": 0.4, + "content": "/", + "type": "inline_equation" + }, + { + "bbox": [ + 318, + 699, + 506, + 714 + ], + "score": 1.0, + "content": "smaller, we mean the comparison of two expansion", + "type": "text" + } + ] + }, + { + "bbox": [ + 106, + 712, + 302, + 723 + ], + "spans": [ + { + "bbox": [ + 106, + 712, + 302, + 723 + ], + "score": 1.0, + "content": "function, rather than the comparison along a function.", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 302, + 742, + 308, + 750 + ], + "lines": [ + { + "bbox": [ + 301, + 741, + 310, + 752 + ], + "spans": [ + { + "bbox": [ + 301, + 741, + 310, + 752 + ], + "score": 1.0, + "content": "", + "type": "text", + "height": 11, + "width": 9 + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 72, + 505, + 128 + ], + "lines": [ + { + "bbox": [ + 105, + 72, + 506, + 85 + ], + "spans": [ + { + "bbox": [ + 105, + 72, + 162, + 85 + ], + "score": 1.0, + "content": "The variation", + "type": "text" + }, + { + "bbox": [ + 163, + 73, + 195, + 85 + ], + "score": 0.92, + "content": "\\mathcal { V } ( \\phi , \\mathcal { E } )", + "type": "inline_equation" + }, + { + "bbox": [ + 195, + 72, + 297, + 85 + ], + "score": 1.0, + "content": "measures the stability of", + "type": "text" + }, + { + "bbox": [ + 297, + 73, + 315, + 85 + ], + "score": 0.91, + "content": "\\phi ( \\cdot )", + "type": "inline_equation" + }, + { + "bbox": [ + 315, + 72, + 398, + 85 + ], + "score": 1.0, + "content": "over the domains in", + "type": "text" + }, + { + "bbox": [ + 398, + 73, + 405, + 83 + ], + "score": 0.79, + "content": "\\mathcal { E }", + "type": "inline_equation" + }, + { + "bbox": [ + 406, + 72, + 506, + 85 + ], + "score": 1.0, + "content": "and the informativeness", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 82, + 505, + 96 + ], + "spans": [ + { + "bbox": [ + 106, + 83, + 138, + 96 + ], + "score": 0.92, + "content": "{ \\mathcal { T } } ( \\phi , { \\mathcal { E } } )", + "type": "inline_equation" + }, + { + "bbox": [ + 138, + 82, + 231, + 96 + ], + "score": 1.0, + "content": "captures the ability of", + "type": "text" + }, + { + "bbox": [ + 231, + 84, + 249, + 96 + ], + "score": 0.9, + "content": "\\phi ( \\cdot )", + "type": "inline_equation" + }, + { + "bbox": [ + 249, + 82, + 505, + 96 + ], + "score": 1.0, + "content": "to distinguish different labels. We would like to highlight that", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 93, + 505, + 108 + ], + "spans": [ + { + "bbox": [ + 105, + 93, + 417, + 108 + ], + "score": 1.0, + "content": "the variation and informativeness are defined on each one-dimensional feature", + "type": "text" + }, + { + "bbox": [ + 418, + 95, + 435, + 106 + ], + "score": 0.9, + "content": "\\phi ( \\cdot )", + "type": "inline_equation" + }, + { + "bbox": [ + 435, + 93, + 505, + 108 + ], + "score": 1.0, + "content": ". Unlike previous", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 106, + 106, + 505, + 118 + ], + "spans": [ + { + "bbox": [ + 106, + 106, + 273, + 118 + ], + "score": 1.0, + "content": "distance between distributions defined in", + "type": "text" + }, + { + "bbox": [ + 273, + 106, + 280, + 116 + ], + "score": 0.76, + "content": "d", + "type": "inline_equation" + }, + { + "bbox": [ + 280, + 106, + 505, + 118 + ], + "score": 1.0, + "content": "-dimensional space, our definitions are more reasonable", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 106, + 117, + 347, + 128 + ], + "spans": [ + { + "bbox": [ + 106, + 117, + 347, + 128 + ], + "score": 1.0, + "content": "and practical, since it can be easily calculated and analyzed.", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 2, + "bbox_fs": [ + 105, + 72, + 506, + 128 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 132, + 505, + 199 + ], + "lines": [ + { + "bbox": [ + 105, + 132, + 506, + 146 + ], + "spans": [ + { + "bbox": [ + 105, + 132, + 363, + 146 + ], + "score": 1.0, + "content": "We are now ready to introduce the core quantity for connecting", + "type": "text" + }, + { + "bbox": [ + 363, + 133, + 389, + 144 + ], + "score": 0.91, + "content": "\\mathcal { E } _ { a v a i l }", + "type": "inline_equation" + }, + { + "bbox": [ + 389, + 132, + 406, + 146 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 407, + 133, + 423, + 144 + ], + "score": 0.89, + "content": "{ \\mathcal { E } } _ { a l l }", + "type": "inline_equation" + }, + { + "bbox": [ + 424, + 132, + 506, + 146 + ], + "score": 1.0, + "content": ". Our motivation, as", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 144, + 505, + 155 + ], + "spans": [ + { + "bbox": [ + 106, + 144, + 505, + 155 + ], + "score": 1.0, + "content": "elaborated in the introduction section, is that, if a feature is informative for the classification task", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 154, + 506, + 168 + ], + "spans": [ + { + "bbox": [ + 105, + 154, + 182, + 168 + ], + "score": 1.0, + "content": "and invariant over", + "type": "text" + }, + { + "bbox": [ + 182, + 155, + 208, + 165 + ], + "score": 0.91, + "content": "\\mathcal { E } _ { a v a i l }", + "type": "inline_equation" + }, + { + "bbox": [ + 208, + 154, + 379, + 168 + ], + "score": 1.0, + "content": ", then to enable OOD generalization from", + "type": "text" + }, + { + "bbox": [ + 379, + 155, + 404, + 165 + ], + "score": 0.9, + "content": "\\mathcal { E } _ { a v a i l }", + "type": "inline_equation" + }, + { + "bbox": [ + 405, + 154, + 416, + 168 + ], + "score": 1.0, + "content": "to", + "type": "text" + }, + { + "bbox": [ + 416, + 155, + 433, + 165 + ], + "score": 0.89, + "content": "{ \\mathcal { E } } _ { a l l }", + "type": "inline_equation" + }, + { + "bbox": [ + 433, + 154, + 506, + 168 + ], + "score": 1.0, + "content": ", it should be still", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 164, + 505, + 179 + ], + "spans": [ + { + "bbox": [ + 105, + 164, + 165, + 179 + ], + "score": 1.0, + "content": "invariant over", + "type": "text" + }, + { + "bbox": [ + 165, + 166, + 182, + 176 + ], + "score": 0.9, + "content": "{ \\mathcal { E } } _ { a l l }", + "type": "inline_equation" + }, + { + "bbox": [ + 182, + 164, + 288, + 179 + ], + "score": 1.0, + "content": ". So the relation between", + "type": "text" + }, + { + "bbox": [ + 288, + 165, + 338, + 177 + ], + "score": 0.92, + "content": "\\mathcal { V } ( \\phi , \\mathcal { E } _ { a v a i l } )", + "type": "inline_equation" + }, + { + "bbox": [ + 338, + 164, + 357, + 179 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 357, + 165, + 398, + 177 + ], + "score": 0.92, + "content": "\\mathinner { \\gamma \\mathopen { \\left( \\phi , \\mathcal { E } _ { a l l } \\right) } }", + "type": "inline_equation" + }, + { + "bbox": [ + 399, + 164, + 505, + 179 + ], + "score": 1.0, + "content": "of an informative feature", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 176, + 506, + 189 + ], + "spans": [ + { + "bbox": [ + 105, + 176, + 506, + 189 + ], + "score": 1.0, + "content": "captures the feasibility and difficulty of OOD generalization. To quantitatively measure this relation,", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 106, + 188, + 263, + 199 + ], + "spans": [ + { + "bbox": [ + 106, + 188, + 263, + 199 + ], + "score": 1.0, + "content": "we define the following function class:", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 7.5, + "bbox_fs": [ + 105, + 132, + 506, + 199 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 202, + 505, + 236 + ], + "lines": [ + { + "bbox": [ + 105, + 201, + 505, + 215 + ], + "spans": [ + { + "bbox": [ + 105, + 201, + 346, + 215 + ], + "score": 1.0, + "content": "Definition 3.3 (Expansion Function). We say a function", + "type": "text" + }, + { + "bbox": [ + 347, + 202, + 493, + 214 + ], + "score": 0.91, + "content": "s : \\mathbb { R } ^ { + } \\cup \\{ 0 \\} \\mathbb { R } ^ { + } \\cup \\{ 0 , + \\infty \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 494, + 201, + 505, + 215 + ], + "score": 1.0, + "content": "is", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 214, + 505, + 226 + ], + "spans": [ + { + "bbox": [ + 105, + 214, + 352, + 226 + ], + "score": 1.0, + "content": "an expansion function, iff the following properties hold: 1)", + "type": "text" + }, + { + "bbox": [ + 352, + 214, + 369, + 226 + ], + "score": 0.88, + "content": "s ( \\cdot )", + "type": "inline_equation" + }, + { + "bbox": [ + 369, + 214, + 505, + 226 + ], + "score": 1.0, + "content": "is monotonically increasing and", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 107, + 223, + 304, + 237 + ], + "spans": [ + { + "bbox": [ + 107, + 225, + 191, + 236 + ], + "score": 0.77, + "content": "s ( x ) \\geq x , \\forall x \\geq 0 ; 2 ,", + "type": "inline_equation" + }, + { + "bbox": [ + 191, + 223, + 194, + 237 + ], + "score": 1.0, + "content": ")", + "type": "text" + }, + { + "bbox": [ + 194, + 225, + 300, + 236 + ], + "score": 0.85, + "content": "\\begin{array} { r } { \\operatorname* { l i m } _ { x 0 ^ { + } } s ( x ) = s ( 0 ) = 0 } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 301, + 223, + 304, + 237 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 12, + "bbox_fs": [ + 105, + 201, + 505, + 237 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 245, + 505, + 278 + ], + "lines": [ + { + "bbox": [ + 105, + 245, + 506, + 258 + ], + "spans": [ + { + "bbox": [ + 105, + 245, + 431, + 258 + ], + "score": 1.0, + "content": "This function class gives a full characterization of how the variation between", + "type": "text" + }, + { + "bbox": [ + 432, + 245, + 457, + 256 + ], + "score": 0.9, + "content": "\\mathcal { E } _ { a v a i l }", + "type": "inline_equation" + }, + { + "bbox": [ + 457, + 245, + 477, + 258 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 477, + 245, + 493, + 256 + ], + "score": 0.89, + "content": "{ \\mathcal { E } } _ { a l l }", + "type": "inline_equation" + }, + { + "bbox": [ + 494, + 245, + 506, + 258 + ], + "score": 1.0, + "content": "is", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 256, + 505, + 268 + ], + "spans": [ + { + "bbox": [ + 105, + 256, + 505, + 268 + ], + "score": 1.0, + "content": "related. Based on this function class, we can introduce our formulation of the learnability of OOD", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 267, + 211, + 279 + ], + "spans": [ + { + "bbox": [ + 105, + 267, + 211, + 279 + ], + "score": 1.0, + "content": "generalization as follows:", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 15, + "bbox_fs": [ + 105, + 245, + 506, + 279 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 282, + 505, + 337 + ], + "lines": [ + { + "bbox": [ + 105, + 280, + 505, + 296 + ], + "spans": [ + { + "bbox": [ + 105, + 280, + 245, + 296 + ], + "score": 1.0, + "content": "Definition 3.4 (Learnability). Let", + "type": "text" + }, + { + "bbox": [ + 246, + 283, + 254, + 292 + ], + "score": 0.7, + "content": "\\Phi", + "type": "inline_equation" + }, + { + "bbox": [ + 255, + 280, + 357, + 296 + ], + "score": 1.0, + "content": "be the feature space and", + "type": "text" + }, + { + "bbox": [ + 357, + 284, + 364, + 294 + ], + "score": 0.64, + "content": "\\rho", + "type": "inline_equation" + }, + { + "bbox": [ + 364, + 280, + 505, + 296 + ], + "score": 1.0, + "content": "be a distribution distance. We say", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 293, + 505, + 306 + ], + "spans": [ + { + "bbox": [ + 105, + 293, + 258, + 306 + ], + "score": 1.0, + "content": "an OOD generalization problem from", + "type": "text" + }, + { + "bbox": [ + 258, + 294, + 283, + 304 + ], + "score": 0.89, + "content": "\\mathcal { E } _ { a v a i l }", + "type": "inline_equation" + }, + { + "bbox": [ + 283, + 293, + 294, + 306 + ], + "score": 1.0, + "content": "to", + "type": "text" + }, + { + "bbox": [ + 295, + 294, + 311, + 304 + ], + "score": 0.88, + "content": "{ \\mathcal { E } } _ { a l l }", + "type": "inline_equation" + }, + { + "bbox": [ + 312, + 293, + 505, + 306 + ], + "score": 1.0, + "content": "is learnable if there exists an expansion function", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 107, + 304, + 504, + 317 + ], + "spans": [ + { + "bbox": [ + 107, + 304, + 123, + 316 + ], + "score": 0.89, + "content": "s ( \\cdot )", + "type": "inline_equation" + }, + { + "bbox": [ + 123, + 304, + 142, + 317 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 142, + 304, + 168, + 315 + ], + "score": 0.9, + "content": "\\delta \\geq 0", + "type": "inline_equation" + }, + { + "bbox": [ + 168, + 304, + 243, + 317 + ], + "score": 1.0, + "content": ", such that: for all", + "type": "text" + }, + { + "bbox": [ + 244, + 304, + 271, + 315 + ], + "score": 0.91, + "content": "\\phi \\in \\Phi", + "type": "inline_equation" + }, + { + "bbox": [ + 271, + 304, + 313, + 317 + ], + "score": 1.0, + "content": "satisfying", + "type": "text" + }, + { + "bbox": [ + 314, + 304, + 386, + 316 + ], + "score": 0.91, + "content": "{ \\mathcal { T } } _ { \\rho } ( \\phi , { \\mathcal { E } } _ { a v a i l } ) \\geq \\delta", + "type": "inline_equation" + }, + { + "bbox": [ + 387, + 304, + 426, + 317 + ], + "score": 1.0, + "content": ", we have", + "type": "text" + }, + { + "bbox": [ + 427, + 304, + 504, + 316 + ], + "score": 0.91, + "content": "s ( \\mathcal { V } _ { \\rho } ( \\phi , \\mathcal { E } _ { a v a i l } ) ) \\geq", + "type": "inline_equation" + } + ], + "index": 19 + }, + { + "bbox": [ + 107, + 314, + 504, + 328 + ], + "spans": [ + { + "bbox": [ + 107, + 315, + 151, + 327 + ], + "score": 0.92, + "content": "\\mathcal { V } _ { \\rho } \\big ( \\phi , \\mathcal { E } _ { a l l } \\big )", + "type": "inline_equation" + }, + { + "bbox": [ + 151, + 314, + 189, + 328 + ], + "score": 1.0, + "content": ". If such", + "type": "text" + }, + { + "bbox": [ + 189, + 315, + 205, + 327 + ], + "score": 0.88, + "content": "s ( \\cdot )", + "type": "inline_equation" + }, + { + "bbox": [ + 206, + 314, + 225, + 328 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 226, + 316, + 232, + 325 + ], + "score": 0.71, + "content": "\\delta", + "type": "inline_equation" + }, + { + "bbox": [ + 232, + 314, + 377, + 328 + ], + "score": 1.0, + "content": "exist, we further call this problem", + "type": "text" + }, + { + "bbox": [ + 377, + 315, + 409, + 327 + ], + "score": 0.93, + "content": "( s ( \\cdot ) , \\delta )", + "type": "inline_equation" + }, + { + "bbox": [ + 410, + 314, + 481, + 328 + ], + "score": 1.0, + "content": "-learnable. If an", + "type": "text" + }, + { + "bbox": [ + 481, + 316, + 504, + 326 + ], + "score": 0.38, + "content": "o o D", + "type": "inline_equation" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 326, + 359, + 338 + ], + "spans": [ + { + "bbox": [ + 105, + 326, + 300, + 338 + ], + "score": 1.0, + "content": "generalization problem is not learnable, we call", + "type": "text" + }, + { + "bbox": [ + 300, + 327, + 307, + 336 + ], + "score": 0.5, + "content": "i t", + "type": "inline_equation" + }, + { + "bbox": [ + 307, + 326, + 359, + 338 + ], + "score": 1.0, + "content": "unlearnable.", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 19, + "bbox_fs": [ + 105, + 280, + 505, + 338 + ] + }, + { + "type": "text", + "bbox": [ + 105, + 346, + 482, + 358 + ], + "lines": [ + { + "bbox": [ + 106, + 345, + 484, + 359 + ], + "spans": [ + { + "bbox": [ + 106, + 345, + 484, + 359 + ], + "score": 1.0, + "content": "To understand the intuition and rationality of our formulation, several discussions are in order.", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 22, + "bbox_fs": [ + 106, + 345, + 484, + 359 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 370, + 506, + 469 + ], + "lines": [ + { + "bbox": [ + 105, + 369, + 507, + 384 + ], + "spans": [ + { + "bbox": [ + 105, + 369, + 507, + 384 + ], + "score": 1.0, + "content": "Properties of the expansion function. In Definition 3.3, we highlight two properties of the ex-", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 381, + 507, + 394 + ], + "spans": [ + { + "bbox": [ + 105, + 381, + 507, + 394 + ], + "score": 1.0, + "content": "pansion function. The first property comes naturally from the monotonicity properties of variation:", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 392, + 507, + 406 + ], + "spans": [ + { + "bbox": [ + 105, + 392, + 123, + 406 + ], + "score": 1.0, + "content": "any", + "type": "text" + }, + { + "bbox": [ + 123, + 394, + 134, + 403 + ], + "score": 0.85, + "content": "\\varepsilon _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 134, + 392, + 232, + 406 + ], + "score": 1.0, + "content": "-invariant feature is also", + "type": "text" + }, + { + "bbox": [ + 233, + 394, + 243, + 403 + ], + "score": 0.86, + "content": "\\varepsilon _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 244, + 392, + 298, + 406 + ], + "score": 1.0, + "content": "-invariant for", + "type": "text" + }, + { + "bbox": [ + 298, + 393, + 331, + 403 + ], + "score": 0.9, + "content": "\\varepsilon _ { 2 } \\geq \\varepsilon _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 331, + 392, + 352, + 406 + ], + "score": 1.0, + "content": "; and", + "type": "text" + }, + { + "bbox": [ + 352, + 392, + 436, + 404 + ], + "score": 0.91, + "content": "\\mathcal { V } ( \\phi , \\mathcal { E } _ { 1 } ) \\leq \\mathcal { V } ( \\phi , \\mathcal { E } _ { 2 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 436, + 392, + 469, + 406 + ], + "score": 1.0, + "content": "for any", + "type": "text" + }, + { + "bbox": [ + 469, + 392, + 503, + 404 + ], + "score": 0.91, + "content": "{ \\mathcal { E } } _ { 1 } \\subseteq { \\mathcal { E } } _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 503, + 392, + 507, + 406 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 402, + 506, + 417 + ], + "spans": [ + { + "bbox": [ + 105, + 402, + 280, + 417 + ], + "score": 1.0, + "content": "The monotonicity also implies that larger", + "type": "text" + }, + { + "bbox": [ + 280, + 404, + 297, + 414 + ], + "score": 0.89, + "content": "{ \\mathcal { E } } _ { a l l }", + "type": "inline_equation" + }, + { + "bbox": [ + 297, + 402, + 374, + 417 + ], + "score": 1.0, + "content": "will induce larger", + "type": "text" + }, + { + "bbox": [ + 375, + 404, + 395, + 415 + ], + "score": 0.92, + "content": "s ( \\cdot ) ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 395, + 402, + 506, + 417 + ], + "score": 1.0, + "content": "and it is also harder to be", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 414, + 505, + 427 + ], + "spans": [ + { + "bbox": [ + 105, + 414, + 351, + 427 + ], + "score": 1.0, + "content": "generalized to. From this view, we can see that the scale of", + "type": "text" + }, + { + "bbox": [ + 352, + 414, + 368, + 426 + ], + "score": 0.91, + "content": "s ( \\cdot )", + "type": "inline_equation" + }, + { + "bbox": [ + 368, + 414, + 505, + 427 + ], + "score": 1.0, + "content": "can reflect the difficulty of OOD", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 425, + 506, + 438 + ], + "spans": [ + { + "bbox": [ + 105, + 425, + 506, + 438 + ], + "score": 1.0, + "content": "generalization. The second property is more crucial since it formulates the intuition that if an informa-", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 436, + 506, + 448 + ], + "spans": [ + { + "bbox": [ + 106, + 437, + 239, + 448 + ], + "score": 1.0, + "content": "tive feature is almost invariant in", + "type": "text" + }, + { + "bbox": [ + 240, + 436, + 264, + 447 + ], + "score": 0.92, + "content": "\\mathcal { E } _ { a v a i l }", + "type": "inline_equation" + }, + { + "bbox": [ + 265, + 437, + 383, + 448 + ], + "score": 1.0, + "content": ", it should remain invariant in", + "type": "text" + }, + { + "bbox": [ + 383, + 436, + 399, + 447 + ], + "score": 0.89, + "content": "{ \\mathcal { E } } _ { a l l }", + "type": "inline_equation" + }, + { + "bbox": [ + 400, + 437, + 506, + 448 + ], + "score": 1.0, + "content": ". Without this assumption,", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 446, + 506, + 460 + ], + "spans": [ + { + "bbox": [ + 105, + 446, + 506, + 460 + ], + "score": 1.0, + "content": "OOD generalization can never be guaranteed because we cannot predict whether an invariant and", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 104, + 456, + 356, + 472 + ], + "spans": [ + { + "bbox": [ + 104, + 456, + 195, + 472 + ], + "score": 1.0, + "content": "informative feature in", + "type": "text" + }, + { + "bbox": [ + 195, + 458, + 221, + 469 + ], + "score": 0.91, + "content": "\\mathcal { E } _ { a v a i l }", + "type": "inline_equation" + }, + { + "bbox": [ + 221, + 456, + 335, + 472 + ], + "score": 1.0, + "content": "will vary severely in unseen", + "type": "text" + }, + { + "bbox": [ + 336, + 458, + 352, + 469 + ], + "score": 0.9, + "content": "{ \\mathcal { E } } _ { a l l }", + "type": "inline_equation" + }, + { + "bbox": [ + 352, + 456, + 356, + 472 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 27, + "bbox_fs": [ + 104, + 369, + 507, + 472 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 481, + 505, + 591 + ], + "lines": [ + { + "bbox": [ + 105, + 482, + 505, + 494 + ], + "spans": [ + { + "bbox": [ + 105, + 482, + 505, + 494 + ], + "score": 1.0, + "content": "Necessity of informativeness. We include a seemingly redundant quantity informativeness in the", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 492, + 506, + 505 + ], + "spans": [ + { + "bbox": [ + 105, + 492, + 506, + 505 + ], + "score": 1.0, + "content": "definition of learnability. However, this term is necessary because only informative features are", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 502, + 504, + 518 + ], + "spans": [ + { + "bbox": [ + 105, + 502, + 478, + 518 + ], + "score": 1.0, + "content": "responsible for the performance of classification. Non-informative but invariant features over", + "type": "text" + }, + { + "bbox": [ + 479, + 504, + 504, + 515 + ], + "score": 0.87, + "content": "\\mathcal { E } _ { a v a i l }", + "type": "inline_equation" + } + ], + "index": 34 + }, + { + "bbox": [ + 106, + 515, + 505, + 527 + ], + "spans": [ + { + "bbox": [ + 106, + 515, + 505, + 527 + ], + "score": 1.0, + "content": "may only capture some noise that is irrelevant to the classification problem, and we shall not expect", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 525, + 506, + 538 + ], + "spans": [ + { + "bbox": [ + 105, + 525, + 221, + 538 + ], + "score": 1.0, + "content": "the noise to be invariant over", + "type": "text" + }, + { + "bbox": [ + 222, + 525, + 238, + 536 + ], + "score": 0.89, + "content": "{ \\mathcal { E } } _ { a l l }", + "type": "inline_equation" + }, + { + "bbox": [ + 238, + 525, + 506, + 538 + ], + "score": 1.0, + "content": ". Moreover, we show in Figure 1 that in practice, many invariant but", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 537, + 506, + 549 + ], + "spans": [ + { + "bbox": [ + 105, + 537, + 182, + 549 + ], + "score": 1.0, + "content": "useless features in", + "type": "text" + }, + { + "bbox": [ + 182, + 537, + 208, + 548 + ], + "score": 0.91, + "content": "\\mathcal { E } _ { a v a i l }", + "type": "inline_equation" + }, + { + "bbox": [ + 208, + 537, + 259, + 549 + ], + "score": 1.0, + "content": "vary a lot in", + "type": "text" + }, + { + "bbox": [ + 260, + 537, + 276, + 547 + ], + "score": 0.9, + "content": "{ \\mathcal { E } } _ { a l l }", + "type": "inline_equation" + }, + { + "bbox": [ + 276, + 537, + 506, + 549 + ], + "score": 1.0, + "content": ", and adding the constraint of informativeness makes the", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 547, + 505, + 559 + ], + "spans": [ + { + "bbox": [ + 105, + 547, + 400, + 559 + ], + "score": 1.0, + "content": "expansion function reasonable. In addition, there are multiple choices of", + "type": "text" + }, + { + "bbox": [ + 400, + 547, + 433, + 559 + ], + "score": 0.92, + "content": "( s ( \\cdot ) , \\delta )", + "type": "inline_equation" + }, + { + "bbox": [ + 433, + 547, + 505, + 559 + ], + "score": 1.0, + "content": "to make an OOD", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 558, + 506, + 571 + ], + "spans": [ + { + "bbox": [ + 105, + 558, + 272, + 571 + ], + "score": 1.0, + "content": "generalization problem learnable: larger", + "type": "text" + }, + { + "bbox": [ + 273, + 559, + 279, + 568 + ], + "score": 0.69, + "content": "\\delta", + "type": "inline_equation" + }, + { + "bbox": [ + 280, + 558, + 426, + 571 + ], + "score": 1.0, + "content": "will filter out more features, and so", + "type": "text" + }, + { + "bbox": [ + 426, + 558, + 442, + 570 + ], + "score": 0.87, + "content": "\\dot { s } ( \\cdot )", + "type": "inline_equation" + }, + { + "bbox": [ + 443, + 558, + 506, + 571 + ], + "score": 1.0, + "content": "can be smaller", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 568, + 505, + 581 + ], + "spans": [ + { + "bbox": [ + 105, + 568, + 349, + 581 + ], + "score": 1.0, + "content": "(flatter). This multiplicity will result in a trade-off between", + "type": "text" + }, + { + "bbox": [ + 350, + 569, + 366, + 581 + ], + "score": 0.9, + "content": "s ( \\cdot )", + "type": "inline_equation" + }, + { + "bbox": [ + 366, + 568, + 385, + 581 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 385, + 569, + 390, + 579 + ], + "score": 0.76, + "content": "\\delta", + "type": "inline_equation" + }, + { + "bbox": [ + 391, + 568, + 505, + 581 + ], + "score": 1.0, + "content": ", which will be discussed in", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 579, + 157, + 592 + ], + "spans": [ + { + "bbox": [ + 105, + 579, + 157, + 592 + ], + "score": 1.0, + "content": "Section 6.2.", + "type": "text" + } + ], + "index": 41 + } + ], + "index": 36.5, + "bbox_fs": [ + 105, + 482, + 506, + 592 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 603, + 505, + 692 + ], + "lines": [ + { + "bbox": [ + 105, + 602, + 505, + 617 + ], + "spans": [ + { + "bbox": [ + 105, + 602, + 505, + 617 + ], + "score": 1.0, + "content": "Two extreme cases: i.i.d. & unlearnable. To better understand the concept of learnability, we", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 614, + 506, + 628 + ], + "spans": [ + { + "bbox": [ + 105, + 614, + 451, + 628 + ], + "score": 1.0, + "content": "consider two extreme cases. (1) The first example is when all data from different", + "type": "text" + }, + { + "bbox": [ + 451, + 615, + 488, + 626 + ], + "score": 0.9, + "content": "\\textit { e } \\in \\mathcal { E } _ { a l l }", + "type": "inline_equation" + }, + { + "bbox": [ + 488, + 614, + 506, + 628 + ], + "score": 1.0, + "content": "are", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 625, + 506, + 638 + ], + "spans": [ + { + "bbox": [ + 105, + 625, + 431, + 638 + ], + "score": 1.0, + "content": "identically distributed, i.e., the classic supervised learning setting. This problem is", + "type": "text" + }, + { + "bbox": [ + 432, + 626, + 464, + 638 + ], + "score": 0.92, + "content": "( s ( \\cdot ) , 0 )", + "type": "inline_equation" + }, + { + "bbox": [ + 465, + 625, + 506, + 638 + ], + "score": 1.0, + "content": "-learnable", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 106, + 637, + 506, + 649 + ], + "spans": [ + { + "bbox": [ + 106, + 637, + 126, + 649 + ], + "score": 1.0, + "content": "with", + "type": "text" + }, + { + "bbox": [ + 126, + 637, + 164, + 649 + ], + "score": 0.93, + "content": "s ( x ) = x", + "type": "inline_equation" + }, + { + "bbox": [ + 164, + 637, + 506, + 649 + ], + "score": 1.0, + "content": ", implying no extra difficulty in OOD generalization. (2) As an example of unlearnable,", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 647, + 506, + 660 + ], + "spans": [ + { + "bbox": [ + 105, + 647, + 380, + 660 + ], + "score": 1.0, + "content": "consider the following case (modified from Colored MNIST [5]): For", + "type": "text" + }, + { + "bbox": [ + 381, + 648, + 423, + 659 + ], + "score": 0.92, + "content": "e \\in \\mathcal { E } _ { a v a i l }", + "type": "inline_equation" + }, + { + "bbox": [ + 423, + 647, + 506, + 660 + ], + "score": 1.0, + "content": ", images with label 0", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 657, + 506, + 673 + ], + "spans": [ + { + "bbox": [ + 105, + 657, + 436, + 673 + ], + "score": 1.0, + "content": "always has a red background while images with label 1 has a blue background. For", + "type": "text" + }, + { + "bbox": [ + 436, + 658, + 502, + 671 + ], + "score": 0.91, + "content": "e \\in \\mathcal { E } _ { a l l } \\ \\backslash \\ \\mathcal { E } _ { a v a i l }", + "type": "inline_equation" + }, + { + "bbox": [ + 503, + 657, + 506, + 673 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 105, + 669, + 506, + 682 + ], + "spans": [ + { + "bbox": [ + 105, + 669, + 345, + 682 + ], + "score": 1.0, + "content": "this relationship is entirely inverse. Since data from different", + "type": "text" + }, + { + "bbox": [ + 345, + 669, + 387, + 680 + ], + "score": 0.92, + "content": "e \\in \\mathcal { E } _ { a v a i l }", + "type": "inline_equation" + }, + { + "bbox": [ + 388, + 669, + 506, + 682 + ], + "score": 1.0, + "content": "are identically distributed but", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 105, + 680, + 507, + 693 + ], + "spans": [ + { + "bbox": [ + 105, + 680, + 186, + 693 + ], + "score": 1.0, + "content": "different from other", + "type": "text" + }, + { + "bbox": [ + 186, + 680, + 219, + 691 + ], + "score": 0.91, + "content": "e \\in \\mathcal { E } _ { a l l }", + "type": "inline_equation" + }, + { + "bbox": [ + 219, + 680, + 507, + 693 + ], + "score": 1.0, + "content": ", no expansion function can make it learnable, i.e., it is OOD-unlearnable.", + "type": "text" + } + ], + "index": 49 + } + ], + "index": 45.5, + "bbox_fs": [ + 105, + 602, + 507, + 693 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 106, + 72, + 505, + 106 + ], + "lines": [ + { + "bbox": [ + 105, + 72, + 506, + 86 + ], + "spans": [ + { + "bbox": [ + 105, + 72, + 506, + 86 + ], + "score": 1.0, + "content": "The unlearnability of this case also coincides with our intuition: Without prior knowledge, it is not", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 83, + 506, + 97 + ], + "spans": [ + { + "bbox": [ + 105, + 83, + 506, + 97 + ], + "score": 1.0, + "content": "clear from merely the training data, whether the task is to distinguish digit 0 from 1, or to distinguish", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 106, + 95, + 413, + 107 + ], + "spans": [ + { + "bbox": [ + 106, + 95, + 306, + 107 + ], + "score": 1.0, + "content": "color red from blue. As a result, generalization to", + "type": "text" + }, + { + "bbox": [ + 306, + 95, + 322, + 105 + ], + "score": 0.93, + "content": "{ \\mathcal { E } } _ { a l l }", + "type": "inline_equation" + }, + { + "bbox": [ + 323, + 95, + 413, + 107 + ], + "score": 1.0, + "content": "cannot be guaranteed.", + "type": "text" + } + ], + "index": 2 + } + ], + "index": 1 + }, + { + "type": "title", + "bbox": [ + 107, + 120, + 239, + 135 + ], + "lines": [ + { + "bbox": [ + 104, + 120, + 240, + 137 + ], + "spans": [ + { + "bbox": [ + 104, + 120, + 240, + 137 + ], + "score": 1.0, + "content": "4 Generalization Bound", + "type": "text" + } + ], + "index": 3 + } + ], + "index": 3 + }, + { + "type": "text", + "bbox": [ + 106, + 145, + 504, + 168 + ], + "lines": [ + { + "bbox": [ + 104, + 144, + 506, + 159 + ], + "spans": [ + { + "bbox": [ + 104, + 144, + 375, + 159 + ], + "score": 1.0, + "content": "In this section, we consider an OOD generalization problem from", + "type": "text" + }, + { + "bbox": [ + 375, + 146, + 400, + 156 + ], + "score": 0.91, + "content": "\\mathcal { E } _ { a v a i l }", + "type": "inline_equation" + }, + { + "bbox": [ + 401, + 144, + 412, + 159 + ], + "score": 1.0, + "content": "to", + "type": "text" + }, + { + "bbox": [ + 412, + 146, + 428, + 156 + ], + "score": 0.9, + "content": "{ \\mathcal { E } } _ { a l l }", + "type": "inline_equation" + }, + { + "bbox": [ + 428, + 144, + 506, + 159 + ], + "score": 1.0, + "content": ", and our goal is to", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 155, + 394, + 169 + ], + "spans": [ + { + "bbox": [ + 105, + 155, + 308, + 169 + ], + "score": 1.0, + "content": "analyze the OOD generalization error of classifier", + "type": "text" + }, + { + "bbox": [ + 308, + 156, + 348, + 168 + ], + "score": 0.93, + "content": "f = g \\circ h", + "type": "inline_equation" + }, + { + "bbox": [ + 349, + 155, + 394, + 169 + ], + "score": 1.0, + "content": "defined by", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 4.5 + }, + { + "type": "interline_equation", + "bbox": [ + 222, + 172, + 367, + 186 + ], + "lines": [ + { + "bbox": [ + 222, + 172, + 367, + 186 + ], + "spans": [ + { + "bbox": [ + 222, + 172, + 367, + 186 + ], + "score": 0.92, + "content": "\\mathrm { e r r } ( f ) = \\mathcal { L } ( \\mathcal { E } _ { a l l } , f ) - \\mathcal { L } ( \\mathcal { E } _ { a v a i l } , f ) ,", + "type": "interline_equation", + "image_path": "f61601e575911b7b38e2ff41bc91c80f1ca3bdfad17110177fbe71a235a390ce.jpg" + } + ] + } + ], + "index": 6, + "virtual_lines": [ + { + "bbox": [ + 222, + 172, + 367, + 186 + ], + "spans": [], + "index": 6 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 189, + 505, + 245 + ], + "lines": [ + { + "bbox": [ + 105, + 190, + 506, + 203 + ], + "spans": [ + { + "bbox": [ + 105, + 190, + 245, + 203 + ], + "score": 1.0, + "content": "where we assume the loss function", + "type": "text" + }, + { + "bbox": [ + 245, + 190, + 267, + 201 + ], + "score": 0.91, + "content": "l ( \\cdot , \\cdot )", + "type": "inline_equation" + }, + { + "bbox": [ + 268, + 190, + 326, + 203 + ], + "score": 1.0, + "content": "is bounded by", + "type": "text" + }, + { + "bbox": [ + 327, + 190, + 350, + 202 + ], + "score": 0.94, + "content": "[ 0 , C ]", + "type": "inline_equation" + }, + { + "bbox": [ + 351, + 190, + 506, + 203 + ], + "score": 1.0, + "content": ". We prove two upper bounds (4.1, 4.2)", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 200, + 506, + 213 + ], + "spans": [ + { + "bbox": [ + 105, + 200, + 244, + 213 + ], + "score": 1.0, + "content": "as well as a lower bound (4.3) for", + "type": "text" + }, + { + "bbox": [ + 244, + 201, + 270, + 213 + ], + "score": 0.88, + "content": "\\operatorname { e r r } ( f )", + "type": "inline_equation" + }, + { + "bbox": [ + 270, + 200, + 506, + 213 + ], + "score": 1.0, + "content": "based on our formulation. Our bounds together provide a", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 211, + 506, + 225 + ], + "spans": [ + { + "bbox": [ + 105, + 211, + 506, + 225 + ], + "score": 1.0, + "content": "complete characterization of the difficulty of OOD generalization. Since we expect that an invariant", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 222, + 505, + 235 + ], + "spans": [ + { + "bbox": [ + 105, + 222, + 352, + 235 + ], + "score": 1.0, + "content": "classifier can generalize to unseen domains, we hope to bound", + "type": "text" + }, + { + "bbox": [ + 353, + 223, + 378, + 234 + ], + "score": 0.85, + "content": "\\operatorname { e r r } ( f )", + "type": "inline_equation" + }, + { + "bbox": [ + 379, + 222, + 505, + 235 + ], + "score": 1.0, + "content": "in terms of the certain variation", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 232, + 485, + 246 + ], + "spans": [ + { + "bbox": [ + 105, + 232, + 117, + 246 + ], + "score": 1.0, + "content": "of", + "type": "text" + }, + { + "bbox": [ + 118, + 234, + 124, + 245 + ], + "score": 0.84, + "content": "f", + "type": "inline_equation" + }, + { + "bbox": [ + 125, + 232, + 365, + 246 + ], + "score": 1.0, + "content": ". To this end, we define the variation and informativeness of", + "type": "text" + }, + { + "bbox": [ + 365, + 235, + 372, + 245 + ], + "score": 0.84, + "content": "f", + "type": "inline_equation" + }, + { + "bbox": [ + 372, + 232, + 485, + 246 + ], + "score": 1.0, + "content": "in terms of its features, i.e.,", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 9 + }, + { + "type": "interline_equation", + "bbox": [ + 213, + 248, + 396, + 295 + ], + "lines": [ + { + "bbox": [ + 213, + 248, + 396, + 295 + ], + "spans": [ + { + "bbox": [ + 213, + 248, + 396, + 295 + ], + "score": 0.92, + "content": "\\begin{array} { r l r } { \\mathcal { V } ^ { \\operatorname* { s u p } } ( h , \\mathcal { E } _ { a v a i l } ) } & { \\triangleq } & { \\underset { \\beta \\in S ^ { d - 1 } } { \\operatorname* { s u p } } \\mathcal { V } ( \\beta ^ { \\top } h , \\mathcal { E } _ { a v a i l } ) , } \\\\ { \\mathcal { T } ^ { \\operatorname* { i n f } } ( h , \\mathcal { E } _ { a v a i l } ) } & { \\triangleq } & { \\underset { \\beta \\in S ^ { d - 1 } } { \\operatorname* { i n f } } \\mathcal { T } ( \\beta ^ { \\top } h , \\mathcal { E } _ { a v a i l } ) , } \\end{array}", + "type": "interline_equation", + "image_path": "37ed4f7a0dc7d2319bdc5086f2478da0e7cc992b3f71f92313ccf9703cd17289.jpg" + } + ] + } + ], + "index": 13, + "virtual_lines": [ + { + "bbox": [ + 213, + 248, + 396, + 263.6666666666667 + ], + "spans": [], + "index": 12 + }, + { + "bbox": [ + 213, + 263.6666666666667, + 396, + 279.33333333333337 + ], + "spans": [], + "index": 13 + }, + { + "bbox": [ + 213, + 279.33333333333337, + 396, + 295.00000000000006 + ], + "spans": [], + "index": 14 + } + ] + }, + { + "type": "text", + "bbox": [ + 105, + 298, + 505, + 313 + ], + "lines": [ + { + "bbox": [ + 105, + 296, + 507, + 315 + ], + "spans": [ + { + "bbox": [ + 105, + 296, + 133, + 315 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 133, + 299, + 218, + 312 + ], + "score": 0.93, + "content": "( \\beta ^ { \\top } h ) ( x ) = \\beta ^ { \\top } h ( x )", + "type": "inline_equation" + }, + { + "bbox": [ + 219, + 296, + 280, + 315 + ], + "score": 1.0, + "content": "is a feature and", + "type": "text" + }, + { + "bbox": [ + 281, + 299, + 404, + 312 + ], + "score": 0.93, + "content": "S ^ { d - 1 } = \\{ \\beta \\in \\mathbb { R } ^ { d } : \\| \\beta \\| _ { 2 } = 1 \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 405, + 296, + 446, + 315 + ], + "score": 1.0, + "content": "is the unit", + "type": "text" + }, + { + "bbox": [ + 446, + 300, + 474, + 312 + ], + "score": 0.91, + "content": "( d - 1 )", + "type": "inline_equation" + }, + { + "bbox": [ + 474, + 296, + 507, + 315 + ], + "score": 1.0, + "content": "-sphere.", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 15 + }, + { + "type": "text", + "bbox": [ + 106, + 322, + 505, + 411 + ], + "lines": [ + { + "bbox": [ + 105, + 322, + 506, + 336 + ], + "spans": [ + { + "bbox": [ + 105, + 322, + 506, + 336 + ], + "score": 1.0, + "content": "Necessity of using supremum over linear combination. One seemingly plausible definition of", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 334, + 506, + 347 + ], + "spans": [ + { + "bbox": [ + 105, + 334, + 218, + 347 + ], + "score": 1.0, + "content": "the variation of a classifier", + "type": "text" + }, + { + "bbox": [ + 218, + 335, + 226, + 346 + ], + "score": 0.84, + "content": "f", + "type": "inline_equation" + }, + { + "bbox": [ + 226, + 334, + 352, + 347 + ], + "score": 1.0, + "content": "can be the supremum over all", + "type": "text" + }, + { + "bbox": [ + 352, + 334, + 437, + 346 + ], + "score": 0.91, + "content": "\\mathcal { V } ( \\phi _ { i } , \\mathcal { E } _ { a v a i l } ) , i \\in [ d ]", + "type": "inline_equation" + }, + { + "bbox": [ + 437, + 334, + 506, + 347 + ], + "score": 1.0, + "content": ". However, as is", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 346, + 505, + 357 + ], + "spans": [ + { + "bbox": [ + 106, + 346, + 505, + 357 + ], + "score": 1.0, + "content": "shown in Appendix 1, it is possible that two high-dimensional joint distributions have close marginal", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 355, + 505, + 368 + ], + "spans": [ + { + "bbox": [ + 105, + 355, + 505, + 368 + ], + "score": 1.0, + "content": "distribution in each dimension, while they do not overlap. In other words, there exist cases where", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 107, + 365, + 506, + 380 + ], + "spans": [ + { + "bbox": [ + 107, + 367, + 206, + 379 + ], + "score": 0.91, + "content": "\\mathcal { V } ( \\phi _ { i } , \\mathcal { E } _ { a l l } ) = 0 , \\forall i \\in [ d ]", + "type": "inline_equation" + }, + { + "bbox": [ + 206, + 365, + 337, + 380 + ], + "score": 1.0, + "content": "but after applying the top model", + "type": "text" + }, + { + "bbox": [ + 338, + 369, + 344, + 379 + ], + "score": 0.81, + "content": "g", + "type": "inline_equation" + }, + { + "bbox": [ + 344, + 365, + 365, + 380 + ], + "score": 1.0, + "content": "over", + "type": "text" + }, + { + "bbox": [ + 365, + 367, + 375, + 378 + ], + "score": 0.87, + "content": "\\phi _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 376, + 365, + 506, + 380 + ], + "score": 1.0, + "content": "’s, the distribution varies a lot in", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 375, + 507, + 393 + ], + "spans": [ + { + "bbox": [ + 106, + 379, + 131, + 389 + ], + "score": 0.86, + "content": "\\mathcal { E } _ { a v a i l }", + "type": "inline_equation" + }, + { + "bbox": [ + 132, + 375, + 422, + 393 + ], + "score": 1.0, + "content": ". Our definition comes from the simple idea that the class of top model", + "type": "text" + }, + { + "bbox": [ + 423, + 378, + 430, + 388 + ], + "score": 0.82, + "content": "\\mathcal { G }", + "type": "inline_equation" + }, + { + "bbox": [ + 430, + 375, + 507, + 393 + ], + "score": 1.0, + "content": "is at least a linear", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 104, + 388, + 503, + 401 + ], + "spans": [ + { + "bbox": [ + 104, + 388, + 486, + 401 + ], + "score": 1.0, + "content": "space, so we should at least consider the variation of every (normalized) linear combination of", + "type": "text" + }, + { + "bbox": [ + 486, + 388, + 503, + 401 + ], + "score": 0.9, + "content": "h ( \\cdot )", + "type": "inline_equation" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 399, + 390, + 411 + ], + "spans": [ + { + "bbox": [ + 105, + 399, + 390, + 411 + ], + "score": 1.0, + "content": "With this, we can guarantee the joint distribution distance is still small.", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 19.5 + }, + { + "type": "text", + "bbox": [ + 106, + 413, + 505, + 461 + ], + "lines": [ + { + "bbox": [ + 105, + 412, + 506, + 426 + ], + "spans": [ + { + "bbox": [ + 105, + 412, + 395, + 426 + ], + "score": 1.0, + "content": "Theorem 4.1 (Main Theorem). Suppose we have learned a classifier", + "type": "text" + }, + { + "bbox": [ + 396, + 414, + 463, + 426 + ], + "score": 0.93, + "content": "f ( x ) = g ( h ( x ) )", + "type": "inline_equation" + }, + { + "bbox": [ + 463, + 412, + 506, + 426 + ], + "score": 1.0, + "content": "such that", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 424, + 506, + 439 + ], + "spans": [ + { + "bbox": [ + 106, + 426, + 145, + 437 + ], + "score": 0.87, + "content": "\\forall e \\in \\mathcal { E } _ { a l l }", + "type": "inline_equation" + }, + { + "bbox": [ + 145, + 424, + 163, + 439 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 163, + 426, + 195, + 438 + ], + "score": 0.72, + "content": "\\forall y \\in \\mathcal { V } ;", + "type": "inline_equation" + }, + { + "bbox": [ + 194, + 425, + 290, + 438 + ], + "score": 0.84, + "content": ", p _ { h ^ { e } | Y ^ { e } } ( h | y ) \\in L ^ { 2 } ( \\mathbb R ^ { d } )", + "type": "inline_equation" + }, + { + "bbox": [ + 291, + 424, + 506, + 439 + ], + "score": 1.0, + "content": ". Denote the characteristic function of random variable", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 107, + 437, + 506, + 452 + ], + "spans": [ + { + "bbox": [ + 107, + 438, + 132, + 450 + ], + "score": 0.89, + "content": "h ^ { e } | Y ^ { e }", + "type": "inline_equation" + }, + { + "bbox": [ + 133, + 437, + 146, + 452 + ], + "score": 1.0, + "content": "as", + "type": "text" + }, + { + "bbox": [ + 146, + 438, + 311, + 450 + ], + "score": 0.88, + "content": "\\hat { p } _ { h ^ { e } | Y ^ { e } } ( t | y ) = \\mathbb { E } [ \\exp \\{ i \\langle t , h ^ { e } \\rangle \\} | Y ^ { e } = y ]", + "type": "inline_equation" + }, + { + "bbox": [ + 312, + 437, + 444, + 452 + ], + "score": 1.0, + "content": ". Assume the hypothetical space", + "type": "text" + }, + { + "bbox": [ + 444, + 438, + 453, + 448 + ], + "score": 0.82, + "content": "\\mathcal { F }", + "type": "inline_equation" + }, + { + "bbox": [ + 454, + 437, + 506, + 452 + ], + "score": 1.0, + "content": "satisfies the", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 104, + 448, + 430, + 461 + ], + "spans": [ + { + "bbox": [ + 104, + 448, + 251, + 461 + ], + "score": 1.0, + "content": "following regularity conditions that", + "type": "text" + }, + { + "bbox": [ + 251, + 449, + 426, + 460 + ], + "score": 0.9, + "content": "\\exists \\alpha , M _ { 1 } , M _ { 2 } > 0 , \\forall f \\in \\mathcal { F } , \\forall e \\in \\mathcal { E } _ { a l l } , y \\in \\mathcal { V }", + "type": "inline_equation" + }, + { + "bbox": [ + 427, + 448, + 430, + 461 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 25.5 + }, + { + "type": "interline_equation", + "bbox": [ + 154, + 464, 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\\mathcal { E } _ { a l l } )", + "type": "inline_equation" + }, + { + "bbox": [ + 168, + 494, + 179, + 510 + ], + "score": 1.0, + "content": "is", + "type": "text" + }, + { + "bbox": [ + 179, + 495, + 263, + 509 + ], + "score": 0.88, + "content": "\\left( s ( \\cdot ) , \\mathcal { T } ^ { i n f } ( h , \\mathcal { E } _ { a v a i l } ) \\right)", + "type": "inline_equation" + }, + { + "bbox": [ + 263, + 494, + 332, + 510 + ], + "score": 1.0, + "content": "-learnable under", + "type": "text" + }, + { + "bbox": [ + 333, + 496, + 341, + 506 + ], + "score": 0.8, + "content": "\\Phi", + "type": "inline_equation" + }, + { + "bbox": [ + 341, + 494, + 424, + 510 + ], + "score": 1.0, + "content": "with Total Variation", + "type": "text" + }, + { + "bbox": [ + 424, + 495, + 434, + 507 + ], + "score": 0.88, + "content": "\\rho ^ { 3 }", + "type": "inline_equation" + }, + { + "bbox": [ + 434, + 494, + 493, + 510 + ], + "score": 1.0, + "content": ", then we have", + "type": "text" + } + ], + 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1.0, + "content": "The above theorem holds for a general classifier learned by any algorithms. Due to its generality, we", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 572, + 506, + 583 + ], + "spans": [ + { + "bbox": [ + 106, + 572, + 506, + 583 + ], + "score": 1.0, + "content": "need to introduce some technical regularity conditions on the density function. The assumption (4)", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 582, + 506, + 595 + ], + "spans": [ + { + "bbox": [ + 105, + 582, + 506, + 595 + ], + "score": 1.0, + "content": "assume the decay rate of density and its characteristic function, which is common in the literature,", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 104, + 592, + 506, + 605 + ], + "spans": [ + { + "bbox": [ + 104, + 592, + 506, + 605 + ], + "score": 1.0, + "content": "e.g. [14]. This theorem demonstrates that, the generalization error can be bounded by a function of", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 604, + 506, + 617 + ], + "spans": [ + { + "bbox": [ + 105, + 604, + 171, + 617 + ], + "score": 1.0, + "content": "the variation of", + "type": "text" + }, + { + "bbox": [ + 172, + 605, + 178, + 614 + ], + "score": 0.77, + "content": "h", + "type": "inline_equation" + }, + { + "bbox": [ + 179, + 604, + 506, + 617 + ], + "score": 1.0, + "content": ", and it converges to 0 as the variation approaches to 0. Under some special but", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 614, + 506, + 629 + ], + "spans": [ + { + "bbox": [ + 105, + 614, + 243, + 629 + ], + "score": 1.0, + "content": "typical case where the top model", + "type": "text" + }, + { + "bbox": [ + 244, + 617, + 250, + 627 + ], + "score": 0.82, + "content": "g", + "type": "inline_equation" + }, + { + "bbox": [ + 251, + 614, + 506, + 629 + ], + "score": 1.0, + "content": "is linear, we can further show that even without the regularity", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 626, + 412, + 639 + ], + "spans": [ + { + "bbox": [ + 105, + 626, + 412, + 639 + ], + "score": 1.0, + "content": "conditions in Theorem 4.1, we have a much better (linear) convergence rate.", + "type": "text" + } + ], + "index": 38 + } + ], + "index": 35 + }, + { + "type": "text", + "bbox": [ + 106, + 641, + 504, + 665 + ], + "lines": [ + { + "bbox": [ + 102, + 635, + 509, + 660 + ], + "spans": [ + { + "bbox": [ + 102, + 635, + 370, + 660 + ], + "score": 1.0, + "content": "Theorem 4.2 (Linear Top Model). Consider any loss satisfying", + "type": "text" + }, + { + "bbox": [ + 370, + 640, + 480, + 655 + ], + "score": 0.92, + "content": "\\begin{array} { r } { \\ell ( \\hat { y } , y ) = \\sum _ { k = 1 } ^ { K } \\ell _ { 0 } ( \\hat { y } _ { k } , y _ { k } ) } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 480, + 635, + 509, + 660 + ], + "score": 1.0, + "content": ".4 For", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 652, + 277, + 667 + ], + "spans": [ + { + "bbox": [ + 105, + 652, + 250, + 667 + ], + "score": 1.0, + "content": "any classifier with linear top model", + "type": "text" + }, + { + "bbox": [ + 250, + 655, + 256, + 665 + ], + "score": 0.68, + "content": "g", + "type": "inline_equation" + }, + { + "bbox": [ + 256, + 652, + 277, + 667 + ], + "score": 1.0, + "content": ", i.e.,", + "type": "text" + } + ], + "index": 40 + } + ], + "index": 39.5 + }, + { + "type": "interline_equation", + "bbox": [ + 195, + 668, + 396, + 683 + ], + "lines": [ + { + "bbox": [ + 195, + 668, + 396, + 683 + ], + "spans": [ + { + "bbox": [ + 195, + 668, + 396, + 683 + ], + "score": 0.9, + "content": "f ( x ) = A h ( x ) + b w i t h A \\in \\mathbb { R } ^ { K \\times d } , b \\in \\mathbb { R } ^ { K } ,", + "type": "interline_equation", + "image_path": "fd483c8cfb69d41c51abea8bfcbef01bb15ff1bdd62fa9a0e20893ef0103bc4c.jpg" + } + ] + } + ], + "index": 41, + "virtual_lines": [ + { + "bbox": [ + 195, + 668, + 396, + 683 + ], + "spans": [], + "index": 41 + } + ] + } + ], + "page_idx": 4, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 106, + 690, + 506, + 722 + ], + "lines": [ + { + "bbox": [ + 118, + 688, + 477, + 703 + ], + "spans": [ + { + "bbox": [ + 118, + 688, + 196, + 703 + ], + "score": 1.0, + "content": "3For two distribution", + "type": "text" + }, + { + "bbox": [ + 196, + 691, + 214, + 701 + ], + "score": 0.84, + "content": "\\mathbb { P } , \\mathbb { Q }", + "type": "inline_equation" + }, + { + "bbox": [ + 214, + 688, + 335, + 703 + ], + "score": 1.0, + "content": "with probability density function", + "type": "text" + }, + { + "bbox": [ + 335, + 691, + 349, + 701 + ], + "score": 0.25, + "content": "p , q", + "type": "inline_equation" + }, + { + "bbox": [ + 341, + 689, + 474, + 702 + ], + "score": 0.9, + "content": "\\begin{array} { r } { , q , \\rho ( \\mathbb { P } , \\mathbb { Q } ) = \\frac { 1 } { 2 } \\int _ { x } | p ( x ) - q ( x ) | \\mathrm { d } x } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 474, + 688, + 477, + 703 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ] + }, + { + "bbox": [ + 118, + 699, + 506, + 714 + ], + "spans": [ + { + "bbox": [ + 118, + 699, + 506, + 714 + ], + "score": 1.0, + "content": "4This decomposition is a technical assumption to make the proof more convenient. Truncated square loss or", + "type": "text" + } + ] + }, + { + "bbox": [ + 106, + 711, + 279, + 723 + ], + "spans": [ + { + "bbox": [ + 106, + 711, + 279, + 723 + ], + "score": 1.0, + "content": "Truncated absolute loss satisfy this assumption.", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 302, + 741, + 309, + 750 + ], + "lines": [ + { + "bbox": [ + 302, + 741, + 309, + 752 + ], + "spans": [ + { + "bbox": [ + 302, + 741, + 309, + 752 + ], + "score": 1.0, + "content": "5", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 106, + 72, + 505, + 106 + ], + "lines": [ + { + "bbox": [ + 105, + 72, + 506, + 86 + ], + "spans": [ + { + "bbox": [ + 105, + 72, + 506, + 86 + ], + "score": 1.0, + "content": "The unlearnability of this case also coincides with our intuition: Without prior knowledge, it is not", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 83, + 506, + 97 + ], + "spans": [ + { + "bbox": [ + 105, + 83, + 506, + 97 + ], + "score": 1.0, + "content": "clear from merely the training data, whether the task is to distinguish digit 0 from 1, or to distinguish", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 106, + 95, + 413, + 107 + ], + "spans": [ + { + "bbox": [ + 106, + 95, + 306, + 107 + ], + "score": 1.0, + "content": "color red from blue. As a result, generalization to", + "type": "text" + }, + { + "bbox": [ + 306, + 95, + 322, + 105 + ], + "score": 0.93, + "content": "{ \\mathcal { E } } _ { a l l }", + "type": "inline_equation" + }, + { + "bbox": [ + 323, + 95, + 413, + 107 + ], + "score": 1.0, + "content": "cannot be guaranteed.", + "type": "text" + } + ], + "index": 2 + } + ], + "index": 1, + "bbox_fs": [ + 105, + 72, + 506, + 107 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 120, + 239, + 135 + ], + "lines": [ + { + "bbox": [ + 104, + 120, + 240, + 137 + ], + "spans": [ + { + "bbox": [ + 104, + 120, + 240, + 137 + ], + "score": 1.0, + "content": "4 Generalization Bound", + "type": "text" + } + ], + "index": 3 + } + ], + "index": 3 + }, + { + "type": "text", + "bbox": [ + 106, + 145, + 504, + 168 + ], + "lines": [ + { + "bbox": [ + 104, + 144, + 506, + 159 + ], + "spans": [ + { + "bbox": [ + 104, + 144, + 375, + 159 + ], + "score": 1.0, + "content": "In this section, we consider an OOD generalization problem from", + "type": "text" + }, + { + "bbox": [ + 375, + 146, + 400, + 156 + ], + "score": 0.91, + "content": "\\mathcal { E } _ { a v a i l }", + "type": "inline_equation" + }, + { + "bbox": [ + 401, + 144, + 412, + 159 + ], + "score": 1.0, + "content": "to", + "type": "text" + }, + { + "bbox": [ + 412, + 146, + 428, + 156 + ], + "score": 0.9, + "content": "{ \\mathcal { E } } _ { a l l }", + "type": "inline_equation" + }, + { + "bbox": [ + 428, + 144, + 506, + 159 + ], + "score": 1.0, + "content": ", and our goal is to", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 155, + 394, + 169 + ], + "spans": [ + { + "bbox": [ + 105, + 155, + 308, + 169 + ], + "score": 1.0, + "content": "analyze the OOD generalization error of classifier", + "type": "text" + }, + { + "bbox": [ + 308, + 156, + 348, + 168 + ], + "score": 0.93, + "content": "f = g \\circ h", + "type": "inline_equation" + }, + { + "bbox": [ + 349, + 155, + 394, + 169 + ], + "score": 1.0, + "content": "defined by", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 4.5, + "bbox_fs": [ + 104, + 144, + 506, + 169 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 222, + 172, + 367, + 186 + ], + "lines": [ + { + "bbox": [ + 222, + 172, + 367, + 186 + ], + "spans": [ + { + "bbox": [ + 222, + 172, + 367, + 186 + ], + "score": 0.92, + "content": "\\mathrm { e r r } ( f ) = \\mathcal { L } ( \\mathcal { E } _ { a l l } , f ) - \\mathcal { L } ( \\mathcal { E } _ { a v a i l } , f ) ,", + "type": "interline_equation", + "image_path": "f61601e575911b7b38e2ff41bc91c80f1ca3bdfad17110177fbe71a235a390ce.jpg" + } + ] + } + ], + "index": 6, + "virtual_lines": [ + { + "bbox": [ + 222, + 172, + 367, + 186 + ], + "spans": [], + "index": 6 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 189, + 505, + 245 + ], + "lines": [ + { + "bbox": [ + 105, + 190, + 506, + 203 + ], + "spans": [ + { + "bbox": [ + 105, + 190, + 245, + 203 + ], + "score": 1.0, + "content": "where we assume the loss function", + "type": "text" + }, + { + "bbox": [ + 245, + 190, + 267, + 201 + ], + "score": 0.91, + "content": "l ( \\cdot , \\cdot )", + "type": "inline_equation" + }, + { + "bbox": [ + 268, + 190, + 326, + 203 + ], + "score": 1.0, + "content": "is bounded by", + "type": "text" + }, + { + "bbox": [ + 327, + 190, + 350, + 202 + ], + "score": 0.94, + "content": "[ 0 , C ]", + "type": "inline_equation" + }, + { + "bbox": [ + 351, + 190, + 506, + 203 + ], + "score": 1.0, + "content": ". We prove two upper bounds (4.1, 4.2)", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 200, + 506, + 213 + ], + "spans": [ + { + "bbox": [ + 105, + 200, + 244, + 213 + ], + "score": 1.0, + "content": "as well as a lower bound (4.3) for", + "type": "text" + }, + { + "bbox": [ + 244, + 201, + 270, + 213 + ], + "score": 0.88, + "content": "\\operatorname { e r r } ( f )", + "type": "inline_equation" + }, + { + "bbox": [ + 270, + 200, + 506, + 213 + ], + "score": 1.0, + "content": "based on our formulation. Our bounds together provide a", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 211, + 506, + 225 + ], + "spans": [ + { + "bbox": [ + 105, + 211, + 506, + 225 + ], + "score": 1.0, + "content": "complete characterization of the difficulty of OOD generalization. Since we expect that an invariant", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 222, + 505, + 235 + ], + "spans": [ + { + "bbox": [ + 105, + 222, + 352, + 235 + ], + "score": 1.0, + "content": "classifier can generalize to unseen domains, we hope to bound", + "type": "text" + }, + { + "bbox": [ + 353, + 223, + 378, + 234 + ], + "score": 0.85, + "content": "\\operatorname { e r r } ( f )", + "type": "inline_equation" + }, + { + "bbox": [ + 379, + 222, + 505, + 235 + ], + "score": 1.0, + "content": "in terms of the certain variation", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 232, + 485, + 246 + ], + "spans": [ + { + "bbox": [ + 105, + 232, + 117, + 246 + ], + "score": 1.0, + "content": "of", + "type": "text" + }, + { + "bbox": [ + 118, + 234, + 124, + 245 + ], + "score": 0.84, + "content": "f", + "type": "inline_equation" + }, + { + "bbox": [ + 125, + 232, + 365, + 246 + ], + "score": 1.0, + "content": ". To this end, we define the variation and informativeness of", + "type": "text" + }, + { + "bbox": [ + 365, + 235, + 372, + 245 + ], + "score": 0.84, + "content": "f", + "type": "inline_equation" + }, + { + "bbox": [ + 372, + 232, + 485, + 246 + ], + "score": 1.0, + "content": "in terms of its features, i.e.,", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 9, + "bbox_fs": [ + 105, + 190, + 506, + 246 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 213, + 248, + 396, + 295 + ], + "lines": [ + { + "bbox": [ + 213, + 248, + 396, + 295 + ], + "spans": [ + { + "bbox": [ + 213, + 248, + 396, + 295 + ], + "score": 0.92, + "content": "\\begin{array} { r l r } { \\mathcal { V } ^ { \\operatorname* { s u p } } ( h , \\mathcal { E } _ { a v a i l } ) } & { \\triangleq } & { \\underset { \\beta \\in S ^ { d - 1 } } { \\operatorname* { s u p } } \\mathcal { V } ( \\beta ^ { \\top } h , \\mathcal { E } _ { a v a i l } ) , } \\\\ { \\mathcal { T } ^ { \\operatorname* { i n f } } ( h , \\mathcal { E } _ { a v a i l } ) } & { \\triangleq } & { \\underset { \\beta \\in S ^ { d - 1 } } { \\operatorname* { i n f } } \\mathcal { T } ( \\beta ^ { \\top } h , \\mathcal { E } _ { a v a i l } ) , } \\end{array}", + "type": "interline_equation", + "image_path": "37ed4f7a0dc7d2319bdc5086f2478da0e7cc992b3f71f92313ccf9703cd17289.jpg" + } + ] + } + ], + "index": 13, + "virtual_lines": [ + { + "bbox": [ + 213, + 248, + 396, + 263.6666666666667 + ], + "spans": [], + "index": 12 + }, + { + "bbox": [ + 213, + 263.6666666666667, + 396, + 279.33333333333337 + ], + "spans": [], + "index": 13 + }, + { + "bbox": [ + 213, + 279.33333333333337, + 396, + 295.00000000000006 + ], + "spans": [], + "index": 14 + } + ] + }, + { + "type": "text", + "bbox": [ + 105, + 298, + 505, + 313 + ], + "lines": [ + { + "bbox": [ + 105, + 296, + 507, + 315 + ], + "spans": [ + { + "bbox": [ + 105, + 296, + 133, + 315 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 133, + 299, + 218, + 312 + ], + "score": 0.93, + "content": "( \\beta ^ { \\top } h ) ( x ) = \\beta ^ { \\top } h ( x )", + "type": "inline_equation" + }, + { + "bbox": [ + 219, + 296, + 280, + 315 + ], + "score": 1.0, + "content": "is a feature and", + "type": "text" + }, + { + "bbox": [ + 281, + 299, + 404, + 312 + ], + "score": 0.93, + "content": "S ^ { d - 1 } = \\{ \\beta \\in \\mathbb { R } ^ { d } : \\| \\beta \\| _ { 2 } = 1 \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 405, + 296, + 446, + 315 + ], + "score": 1.0, + "content": "is the unit", + "type": "text" + }, + { + "bbox": [ + 446, + 300, + 474, + 312 + ], + "score": 0.91, + "content": "( d - 1 )", + "type": "inline_equation" + }, + { + "bbox": [ + 474, + 296, + 507, + 315 + ], + "score": 1.0, + "content": "-sphere.", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 15, + "bbox_fs": [ + 105, + 296, + 507, + 315 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 322, + 505, + 411 + ], + "lines": [ + { + "bbox": [ + 105, + 322, + 506, + 336 + ], + "spans": [ + { + "bbox": [ + 105, + 322, + 506, + 336 + ], + "score": 1.0, + "content": "Necessity of using supremum over linear combination. One seemingly plausible definition of", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 334, + 506, + 347 + ], + "spans": [ + { + "bbox": [ + 105, + 334, + 218, + 347 + ], + "score": 1.0, + "content": "the variation of a classifier", + "type": "text" + }, + { + "bbox": [ + 218, + 335, + 226, + 346 + ], + "score": 0.84, + "content": "f", + "type": "inline_equation" + }, + { + "bbox": [ + 226, + 334, + 352, + 347 + ], + "score": 1.0, + "content": "can be the supremum over all", + "type": "text" + }, + { + "bbox": [ + 352, + 334, + 437, + 346 + ], + "score": 0.91, + "content": "\\mathcal { V } ( \\phi _ { i } , \\mathcal { E } _ { a v a i l } ) , i \\in [ d ]", + "type": "inline_equation" + }, + { + "bbox": [ + 437, + 334, + 506, + 347 + ], + "score": 1.0, + "content": ". However, as is", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 346, + 505, + 357 + ], + "spans": [ + { + "bbox": [ + 106, + 346, + 505, + 357 + ], + "score": 1.0, + "content": "shown in Appendix 1, it is possible that two high-dimensional joint distributions have close marginal", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 355, + 505, + 368 + ], + "spans": [ + { + "bbox": [ + 105, + 355, + 505, + 368 + ], + "score": 1.0, + "content": "distribution in each dimension, while they do not overlap. In other words, there exist cases where", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 107, + 365, + 506, + 380 + ], + "spans": [ + { + "bbox": [ + 107, + 367, + 206, + 379 + ], + "score": 0.91, + "content": "\\mathcal { V } ( \\phi _ { i } , \\mathcal { E } _ { a l l } ) = 0 , \\forall i \\in [ d ]", + "type": "inline_equation" + }, + { + "bbox": [ + 206, + 365, + 337, + 380 + ], + "score": 1.0, + "content": "but after applying the top model", + "type": "text" + }, + { + "bbox": [ + 338, + 369, + 344, + 379 + ], + "score": 0.81, + "content": "g", + "type": "inline_equation" + }, + { + "bbox": [ + 344, + 365, + 365, + 380 + ], + "score": 1.0, + "content": "over", + "type": "text" + }, + { + "bbox": [ + 365, + 367, + 375, + 378 + ], + "score": 0.87, + "content": "\\phi _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 376, + 365, + 506, + 380 + ], + "score": 1.0, + "content": "’s, the distribution varies a lot in", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 375, + 507, + 393 + ], + "spans": [ + { + "bbox": [ + 106, + 379, + 131, + 389 + ], + "score": 0.86, + "content": "\\mathcal { E } _ { a v a i l }", + "type": "inline_equation" + }, + { + "bbox": [ + 132, + 375, + 422, + 393 + ], + "score": 1.0, + "content": ". Our definition comes from the simple idea that the class of top model", + "type": "text" + }, + { + "bbox": [ + 423, + 378, + 430, + 388 + ], + "score": 0.82, + "content": "\\mathcal { G }", + "type": "inline_equation" + }, + { + "bbox": [ + 430, + 375, + 507, + 393 + ], + "score": 1.0, + "content": "is at least a linear", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 104, + 388, + 503, + 401 + ], + "spans": [ + { + "bbox": [ + 104, + 388, + 486, + 401 + ], + "score": 1.0, + "content": "space, so we should at least consider the variation of every (normalized) linear combination of", + "type": "text" + }, + { + "bbox": [ + 486, + 388, + 503, + 401 + ], + "score": 0.9, + "content": "h ( \\cdot )", + "type": "inline_equation" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 399, + 390, + 411 + ], + "spans": [ + { + "bbox": [ + 105, + 399, + 390, + 411 + ], + "score": 1.0, + "content": "With this, we can guarantee the joint distribution distance is still small.", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 19.5, + "bbox_fs": [ + 104, + 322, + 507, + 411 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 413, + 505, + 461 + ], + "lines": [ + { + "bbox": [ + 105, + 412, + 506, + 426 + ], + "spans": [ + { + "bbox": [ + 105, + 412, + 395, + 426 + ], + "score": 1.0, + "content": "Theorem 4.1 (Main Theorem). Suppose we have learned a classifier", + "type": "text" + }, + { + "bbox": [ + 396, + 414, + 463, + 426 + ], + "score": 0.93, + "content": "f ( x ) = g ( h ( x ) )", + "type": "inline_equation" + }, + { + "bbox": [ + 463, + 412, + 506, + 426 + ], + "score": 1.0, + "content": "such that", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 424, + 506, + 439 + ], + "spans": [ + { + "bbox": [ + 106, + 426, + 145, + 437 + ], + "score": 0.87, + "content": "\\forall e \\in \\mathcal { E } _ { a l l }", + "type": "inline_equation" + }, + { + "bbox": [ + 145, + 424, + 163, + 439 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 163, + 426, + 195, + 438 + ], + "score": 0.72, + "content": "\\forall y \\in \\mathcal { V } ;", + "type": "inline_equation" + }, + { + "bbox": [ + 194, + 425, + 290, + 438 + ], + "score": 0.84, + "content": ", p _ { h ^ { e } | Y ^ { e } } ( h | y ) \\in L ^ { 2 } ( \\mathbb R ^ { d } )", + "type": "inline_equation" + }, + { + "bbox": [ + 291, + 424, + 506, + 439 + ], + "score": 1.0, + "content": ". Denote the characteristic function of random variable", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 107, + 437, + 506, + 452 + ], + "spans": [ + { + "bbox": [ + 107, + 438, + 132, + 450 + ], + "score": 0.89, + "content": "h ^ { e } | Y ^ { e }", + "type": "inline_equation" + }, + { + "bbox": [ + 133, + 437, + 146, + 452 + ], + "score": 1.0, + "content": "as", + "type": "text" + }, + { + "bbox": [ + 146, + 438, + 311, + 450 + ], + "score": 0.88, + "content": "\\hat { p } _ { h ^ { e } | Y ^ { e } } ( t | y ) = \\mathbb { E } [ \\exp \\{ i \\langle t , h ^ { e } \\rangle \\} | Y ^ { e } = y ]", + "type": "inline_equation" + }, + { + "bbox": [ + 312, + 437, + 444, + 452 + ], + "score": 1.0, + "content": ". Assume the hypothetical space", + "type": "text" + }, + { + "bbox": [ + 444, + 438, + 453, + 448 + ], + "score": 0.82, + "content": "\\mathcal { F }", + "type": "inline_equation" + }, + { + "bbox": [ + 454, + 437, + 506, + 452 + ], + "score": 1.0, + "content": "satisfies the", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 104, + 448, + 430, + 461 + ], + "spans": [ + { + "bbox": [ + 104, + 448, + 251, + 461 + ], + "score": 1.0, + "content": "following regularity conditions that", + "type": "text" + }, + { + "bbox": [ + 251, + 449, + 426, + 460 + ], + "score": 0.9, + "content": "\\exists \\alpha , M _ { 1 } , M _ { 2 } > 0 , \\forall f \\in \\mathcal { F } , \\forall e \\in \\mathcal { E } _ { a l l } , y \\in \\mathcal { V }", + "type": "inline_equation" + }, + { + "bbox": [ + 427, + 448, + 430, + 461 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 25.5, + "bbox_fs": [ + 104, + 412, + 506, + 461 + ] + }, + { + "type": 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"content": "I f ( \\mathcal { E } _ { a v a i l } , \\mathcal { E } _ { a l l } )", + "type": "inline_equation" + }, + { + "bbox": [ + 168, + 494, + 179, + 510 + ], + "score": 1.0, + "content": "is", + "type": "text" + }, + { + "bbox": [ + 179, + 495, + 263, + 509 + ], + "score": 0.88, + "content": "\\left( s ( \\cdot ) , \\mathcal { T } ^ { i n f } ( h , \\mathcal { E } _ { a v a i l } ) \\right)", + "type": "inline_equation" + }, + { + "bbox": [ + 263, + 494, + 332, + 510 + ], + "score": 1.0, + "content": "-learnable under", + "type": "text" + }, + { + "bbox": [ + 333, + 496, + 341, + 506 + ], + "score": 0.8, + "content": "\\Phi", + "type": "inline_equation" + }, + { + "bbox": [ + 341, + 494, + 424, + 510 + ], + "score": 1.0, + "content": "with Total Variation", + "type": "text" + }, + { + "bbox": [ + 424, + 495, + 434, + 507 + ], + "score": 0.88, + "content": "\\rho ^ { 3 }", + "type": "inline_equation" + }, + { + "bbox": [ + 434, + 494, + 493, + 510 + ], + "score": 1.0, + "content": ", then we have", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 29, + "bbox_fs": [ + 106, + 494, + 493, + 510 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 214, + 513, + 377, + 536 + ], + "lines": [ + { + "bbox": [ + 214, + 513, + 377, + 536 + ], + "spans": [ + { + "bbox": [ + 214, + 513, + 377, + 536 + ], + "score": 0.93, + "content": "\\operatorname { e r r } ( f ) \\leq O \\left( s \\left( \\mathcal { V } _ { \\rho } ^ { s u p } ( h , \\mathcal { E } _ { a v a i l } ) \\right) ^ { \\frac { \\alpha ^ { 2 } } { ( \\alpha + d ) ^ { 2 } } } \\right) .", + "type": "interline_equation", + "image_path": "47d72154cd89cf0c6271de3f68baa90e658deea6245d64bec4f750c758b38256.jpg" + } + ] + } + ], + "index": 30, + "virtual_lines": [ + { + "bbox": [ + 214, + 513, + 377, + 536 + ], + "spans": [], + "index": 30 + } + ] + }, + { + "type": "text", + "bbox": [ + 108, + 540, + 397, + 552 + ], + "lines": [ + { + "bbox": [ + 105, + 539, + 398, + 554 + ], + "spans": [ + 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506, + 637 + ], + "lines": [ + { + "bbox": [ + 105, + 560, + 506, + 573 + ], + "spans": [ + { + "bbox": [ + 105, + 560, + 506, + 573 + ], + "score": 1.0, + "content": "The above theorem holds for a general classifier learned by any algorithms. Due to its generality, we", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 572, + 506, + 583 + ], + "spans": [ + { + "bbox": [ + 106, + 572, + 506, + 583 + ], + "score": 1.0, + "content": "need to introduce some technical regularity conditions on the density function. The assumption (4)", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 582, + 506, + 595 + ], + "spans": [ + { + "bbox": [ + 105, + 582, + 506, + 595 + ], + "score": 1.0, + "content": "assume the decay rate of density and its characteristic function, which is common in the literature,", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 104, + 592, + 506, + 605 + ], + "spans": [ + { + "bbox": [ + 104, + 592, + 506, + 605 + ], + "score": 1.0, + "content": "e.g. [14]. This theorem demonstrates that, the generalization error can be bounded by a function of", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 604, + 506, + 617 + ], + "spans": [ + { + "bbox": [ + 105, + 604, + 171, + 617 + ], + "score": 1.0, + "content": "the variation of", + "type": "text" + }, + { + "bbox": [ + 172, + 605, + 178, + 614 + ], + "score": 0.77, + "content": "h", + "type": "inline_equation" + }, + { + "bbox": [ + 179, + 604, + 506, + 617 + ], + "score": 1.0, + "content": ", and it converges to 0 as the variation approaches to 0. Under some special but", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 614, + 506, + 629 + ], + "spans": [ + { + "bbox": [ + 105, + 614, + 243, + 629 + ], + "score": 1.0, + "content": "typical case where the top model", + "type": "text" + }, + { + "bbox": [ + 244, + 617, + 250, + 627 + ], + "score": 0.82, + "content": "g", + "type": "inline_equation" + }, + { + "bbox": [ + 251, + 614, + 506, + 629 + ], + "score": 1.0, + "content": "is linear, we can further show that even without the regularity", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 626, + 412, + 639 + ], + "spans": [ + { + "bbox": [ + 105, + 626, + 412, + 639 + ], + "score": 1.0, + "content": "conditions in Theorem 4.1, we have a much better (linear) convergence rate.", + "type": "text" + } + ], + "index": 38 + } + ], + "index": 35, + "bbox_fs": [ + 104, + 560, + 506, + 639 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 641, + 504, + 665 + ], + "lines": [ + { + "bbox": [ + 102, + 635, + 509, + 660 + ], + "spans": [ + { + "bbox": [ + 102, + 635, + 370, + 660 + ], + "score": 1.0, + "content": "Theorem 4.2 (Linear Top Model). Consider any loss satisfying", + "type": "text" + }, + { + "bbox": [ + 370, + 640, + 480, + 655 + ], + "score": 0.92, + "content": "\\begin{array} { r } { \\ell ( \\hat { y } , y ) = \\sum _ { k = 1 } ^ { K } \\ell _ { 0 } ( \\hat { y } _ { k } , y _ { k } ) } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 480, + 635, + 509, + 660 + ], + "score": 1.0, + "content": ".4 For", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 652, + 277, + 667 + ], + "spans": [ + { + "bbox": [ + 105, + 652, + 250, + 667 + ], + "score": 1.0, + "content": "any classifier with linear top model", + "type": "text" + }, + { + "bbox": [ + 250, + 655, + 256, + 665 + ], + "score": 0.68, + "content": "g", + "type": "inline_equation" + }, + { + "bbox": [ + 256, + 652, + 277, + 667 + ], + "score": 1.0, + "content": ", i.e.,", + "type": "text" + } + ], + "index": 40 + } + ], + "index": 39.5, + "bbox_fs": [ + 102, + 635, + 509, + 667 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 195, + 668, + 396, + 683 + ], + "lines": [ + { + "bbox": [ + 195, + 668, + 396, + 683 + ], + "spans": [ + { + "bbox": [ + 195, + 668, + 396, + 683 + ], + "score": 0.9, + "content": "f ( x ) = A h ( x ) + b w i t h A \\in \\mathbb { R } ^ { K \\times d } , b \\in \\mathbb { R } ^ { K } ,", + "type": "interline_equation", + "image_path": "fd483c8cfb69d41c51abea8bfcbef01bb15ff1bdd62fa9a0e20893ef0103bc4c.jpg" + } + ] + } + ], + "index": 41, + "virtual_lines": [ + { + "bbox": [ + 195, + 668, + 396, + 683 + ], + "spans": [], + "index": 41 + } + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 105, + 72, + 488, + 86 + ], + "lines": [ + { + "bbox": [ + 106, + 70, + 489, + 87 + ], + "spans": [ + { + "bbox": [ + 106, + 72, + 167, + 85 + ], + "score": 0.9, + "content": "i f ( \\mathcal { E } _ { a v a i l } , \\mathcal { E } _ { a l l } )", + "type": "inline_equation" + }, + { + "bbox": [ + 167, + 70, + 178, + 87 + ], + "score": 1.0, + "content": "is", + "type": "text" + }, + { + "bbox": [ + 178, + 72, + 263, + 85 + ], + "score": 0.89, + "content": "\\left( s ( \\cdot ) , \\mathcal { T } ^ { i n f } ( h , \\mathcal { E } _ { a v a i l } ) \\right)", + "type": "inline_equation" + }, + { + "bbox": [ + 263, + 70, + 332, + 87 + ], + "score": 1.0, + "content": "-learnable under", + "type": "text" + }, + { + "bbox": [ + 332, + 73, + 340, + 83 + ], + "score": 0.81, + "content": "\\Phi", + "type": "inline_equation" + }, + { + "bbox": [ + 341, + 70, + 423, + 87 + ], + "score": 1.0, + "content": "with Total Variation", + "type": "text" + }, + { + "bbox": [ + 424, + 75, + 430, + 84 + ], + "score": 0.69, + "content": "\\rho", + "type": "inline_equation" + }, + { + "bbox": [ + 430, + 70, + 489, + 87 + ], + "score": 1.0, + "content": ", then we have", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "interline_equation", + "bbox": [ + 226, + 90, + 364, + 111 + ], + "lines": [ + { + "bbox": [ + 226, + 90, + 364, + 111 + ], + "spans": [ + { + "bbox": [ + 226, + 90, + 364, + 111 + ], + "score": 0.93, + "content": "\\mathrm { e r r } ( f ) \\leq O \\Bigl ( s \\bigl ( \\mathcal { V } ^ { s u p } ( h , \\mathcal { E } _ { a v a i l } ) \\bigr ) \\Bigr ) .", + "type": "interline_equation", + "image_path": "bff65b7e956cd09791c5ed2c8933cda9684cbbae3c4dc554a897bdb02f4ed4da.jpg" + } + ] + } + ], + "index": 1, + "virtual_lines": [ + { + "bbox": [ + 226, + 90, + 364, + 111 + ], + "spans": [], + "index": 1 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 115, + 253, + 127 + ], + "lines": [ + { + "bbox": [ + 105, + 114, + 254, + 129 + ], + "spans": [ + { + "bbox": [ + 105, + 114, + 128, + 129 + ], + "score": 1.0, + "content": "Here", + "type": "text" + }, + { + "bbox": [ + 128, + 115, + 147, + 128 + ], + "score": 0.89, + "content": "O ( \\cdot )", + "type": "inline_equation" + }, + { + "bbox": [ + 148, + 114, + 241, + 129 + ], + "score": 1.0, + "content": "depends only on d and", + "type": "text" + }, + { + "bbox": [ + 242, + 116, + 250, + 126 + ], + "score": 0.78, + "content": "C", + "type": "inline_equation" + }, + { + "bbox": [ + 250, + 114, + 254, + 129 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 2 + } + ], + "index": 2 + }, + { + "type": "text", + "bbox": [ + 106, + 138, + 506, + 227 + ], + "lines": [ + { + "bbox": [ + 105, + 138, + 505, + 151 + ], + "spans": [ + { + "bbox": [ + 105, + 138, + 505, + 151 + ], + "score": 1.0, + "content": "Discussion. Theorem 4.1 shows that, for any model, the generalization gap depends largely on the", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 104, + 148, + 507, + 163 + ], + "spans": [ + { + "bbox": [ + 104, + 148, + 225, + 163 + ], + "score": 1.0, + "content": "model’s variation captured by", + "type": "text" + }, + { + "bbox": [ + 226, + 150, + 285, + 161 + ], + "score": 0.92, + "content": "\\mathcal { V } ^ { \\mathrm { s u p } } ( h , \\mathcal { E } _ { a v a i l } )", + "type": "inline_equation" + }, + { + "bbox": [ + 286, + 148, + 507, + 163 + ], + "score": 1.0, + "content": ". The result is irrelevant to the algorithm and provides a", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 160, + 506, + 173 + ], + "spans": [ + { + "bbox": [ + 105, + 160, + 275, + 173 + ], + "score": 1.0, + "content": "guarantee for the generalization gap from", + "type": "text" + }, + { + "bbox": [ + 275, + 161, + 300, + 172 + ], + "score": 0.91, + "content": "\\mathcal { E } _ { a v a i l }", + "type": "inline_equation" + }, + { + "bbox": [ + 300, + 160, + 311, + 173 + ], + "score": 1.0, + "content": "to", + "type": "text" + }, + { + "bbox": [ + 312, + 160, + 328, + 171 + ], + "score": 0.89, + "content": "{ \\mathcal { E } } _ { a l l }", + "type": "inline_equation" + }, + { + "bbox": [ + 328, + 160, + 448, + 173 + ], + "score": 1.0, + "content": ", so long as the learned model", + "type": "text" + }, + { + "bbox": [ + 449, + 161, + 456, + 172 + ], + "score": 0.84, + "content": "f", + "type": "inline_equation" + }, + { + "bbox": [ + 456, + 160, + 506, + 173 + ], + "score": 1.0, + "content": "is invariant,", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 104, + 170, + 507, + 185 + ], + "spans": [ + { + "bbox": [ + 104, + 170, + 124, + 185 + ], + "score": 1.0, + "content": "i.e.", + "type": "text" + }, + { + "bbox": [ + 124, + 171, + 183, + 183 + ], + "score": 0.92, + "content": "\\mathcal { V } ^ { \\mathrm { s u p } } ( h , \\mathcal { E } _ { a v a i l } )", + "type": "inline_equation" + }, + { + "bbox": [ + 184, + 170, + 252, + 185 + ], + "score": 1.0, + "content": "is small. When", + "type": "text" + }, + { + "bbox": [ + 252, + 172, + 268, + 183 + ], + "score": 0.91, + "content": "s ( \\cdot )", + "type": "inline_equation" + }, + { + "bbox": [ + 268, + 170, + 396, + 185 + ], + "score": 1.0, + "content": "is fixed, a model with smaller", + "type": "text" + }, + { + "bbox": [ + 396, + 171, + 455, + 183 + ], + "score": 0.91, + "content": "\\mathcal { V } ^ { \\mathrm { s u p } } ( h , \\mathcal { E } _ { a v a i l } )", + "type": "inline_equation" + }, + { + "bbox": [ + 456, + 170, + 507, + 185 + ], + "score": 1.0, + "content": "results in a", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 104, + 181, + 506, + 196 + ], + "spans": [ + { + "bbox": [ + 104, + 181, + 388, + 196 + ], + "score": 1.0, + "content": "smaller gap, which matches our understanding that invariant features in", + "type": "text" + }, + { + "bbox": [ + 388, + 183, + 413, + 194 + ], + "score": 0.91, + "content": "\\mathcal { E } _ { a v a i l }", + "type": "inline_equation" + }, + { + "bbox": [ + 413, + 181, + 506, + 196 + ], + "score": 1.0, + "content": "are somehow invariant", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 104, + 191, + 506, + 207 + ], + "spans": [ + { + "bbox": [ + 104, + 191, + 117, + 207 + ], + "score": 1.0, + "content": "in", + "type": "text" + }, + { + "bbox": [ + 117, + 193, + 133, + 204 + ], + "score": 0.92, + "content": "{ \\mathcal { E } } _ { a l l }", + "type": "inline_equation" + }, + { + "bbox": [ + 133, + 191, + 164, + 207 + ], + "score": 1.0, + "content": ". When", + "type": "text" + }, + { + "bbox": [ + 164, + 193, + 224, + 205 + ], + "score": 0.92, + "content": "\\mathcal { V } ^ { \\mathrm { s u p } } ( h , \\mathcal { E } _ { a v a i l } )", + "type": "inline_equation" + }, + { + "bbox": [ + 224, + 191, + 506, + 207 + ], + "score": 1.0, + "content": "is fixed, more difficult generalization will generate a larger expansion", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 203, + 504, + 217 + ], + "spans": [ + { + "bbox": [ + 105, + 203, + 475, + 217 + ], + "score": 1.0, + "content": "function, which leads to a larger gap. For the Gaussian class with bounded mean and variance,", + "type": "text" + }, + { + "bbox": [ + 475, + 204, + 504, + 215 + ], + "score": 0.89, + "content": "\\alpha \\gg d", + "type": "inline_equation" + } + ], + "index": 9 + }, + { + "bbox": [ + 106, + 215, + 267, + 227 + ], + "spans": [ + { + "bbox": [ + 106, + 215, + 267, + 227 + ], + "score": 1.0, + "content": "and the convergent rate is almost linear.", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 6.5 + }, + { + "type": "text", + "bbox": [ + 106, + 231, + 505, + 287 + ], + "lines": [ + { + "bbox": [ + 106, + 231, + 506, + 244 + ], + "spans": [ + { + "bbox": [ + 106, + 231, + 248, + 244 + ], + "score": 1.0, + "content": "However, without any constraint to", + "type": "text" + }, + { + "bbox": [ + 248, + 234, + 254, + 243 + ], + "score": 0.79, + "content": "g", + "type": "inline_equation" + }, + { + "bbox": [ + 254, + 231, + 506, + 244 + ], + "score": 1.0, + "content": ", the convergent rate might be small. Theorem 4.2 then offers a", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 242, + 505, + 254 + ], + "spans": [ + { + "bbox": [ + 105, + 242, + 433, + 254 + ], + "score": 1.0, + "content": "generalization bound with a linear convergent rate under mild assumptions when", + "type": "text" + }, + { + "bbox": [ + 434, + 244, + 441, + 254 + ], + "score": 0.79, + "content": "g", + "type": "inline_equation" + }, + { + "bbox": [ + 441, + 242, + 505, + 254 + ], + "score": 1.0, + "content": "is linear, which", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 253, + 505, + 266 + ], + "spans": [ + { + "bbox": [ + 105, + 253, + 505, + 266 + ], + "score": 1.0, + "content": "is common in reality. It relaxes the concentration assumption (Formula 4) and asks only for the", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 264, + 505, + 276 + ], + "spans": [ + { + "bbox": [ + 106, + 264, + 505, + 276 + ], + "score": 1.0, + "content": "integrability of the density. The convergent rate is identical to the convergent rate of the expansion", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 275, + 369, + 287 + ], + "spans": [ + { + "bbox": [ + 106, + 275, + 198, + 287 + ], + "score": 1.0, + "content": "function, showing that", + "type": "text" + }, + { + "bbox": [ + 199, + 275, + 214, + 287 + ], + "score": 0.9, + "content": "s ( \\cdot )", + "type": "inline_equation" + }, + { + "bbox": [ + 215, + 275, + 369, + 287 + ], + "score": 1.0, + "content": "captures the generalization quite well.", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 13 + }, + { + "type": "text", + "bbox": [ + 106, + 297, + 505, + 365 + ], + "lines": [ + { + "bbox": [ + 106, + 298, + 505, + 309 + ], + "spans": [ + { + "bbox": [ + 106, + 298, + 505, + 309 + ], + "score": 1.0, + "content": "Proof Sketch of Theorem 4.1. The proof of the main result, Theorem 4.1, is decomposed into the", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 308, + 506, + 321 + ], + "spans": [ + { + "bbox": [ + 105, + 308, + 249, + 321 + ], + "score": 1.0, + "content": "following steps. First, we transform", + "type": "text" + }, + { + "bbox": [ + 249, + 309, + 275, + 321 + ], + "score": 0.88, + "content": "\\operatorname { e r r } ( f )", + "type": "inline_equation" + }, + { + "bbox": [ + 275, + 308, + 506, + 321 + ], + "score": 1.0, + "content": "into the Total Variation of joint distributions of features in", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 318, + 506, + 332 + ], + "spans": [ + { + "bbox": [ + 105, + 318, + 506, + 332 + ], + "score": 1.0, + "content": "different domains (Step 1). To bound the Total Variation, it is sufficient to bound the distance of the", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 331, + 505, + 342 + ], + "spans": [ + { + "bbox": [ + 105, + 331, + 505, + 342 + ], + "score": 1.0, + "content": "corresponding Fourier transform, and further, it is equivalent to bound the Radon transform of joint", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 339, + 506, + 356 + ], + "spans": [ + { + "bbox": [ + 105, + 339, + 300, + 356 + ], + "score": 1.0, + "content": "distributions (Step 2). Eventually, we show that", + "type": "text" + }, + { + "bbox": [ + 300, + 341, + 373, + 354 + ], + "score": 0.92, + "content": "\\mathcal { V } ^ { \\mathrm { s u p } } ( \\beta ^ { \\top } h , \\mathcal { E } _ { a v a i l } )", + "type": "inline_equation" + }, + { + "bbox": [ + 373, + 339, + 506, + 356 + ], + "score": 1.0, + "content": "can be used to bound the Radon", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 353, + 285, + 365 + ], + "spans": [ + { + "bbox": [ + 106, + 353, + 285, + 365 + ], + "score": 1.0, + "content": "transform, which finishes the proof (Step 3).", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 18.5 + }, + { + "type": "text", + "bbox": [ + 106, + 368, + 339, + 380 + ], + "lines": [ + { + "bbox": [ + 106, + 368, + 340, + 381 + ], + "spans": [ + { + "bbox": [ + 106, + 368, + 340, + 381 + ], + "score": 1.0, + "content": "Step 1. The OOD generalization error can be bounded as:", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 22 + }, + { + "type": "interline_equation", + "bbox": [ + 142, + 384, + 449, + 416 + ], + "lines": [ + { + "bbox": [ + 142, + 384, + 449, + 416 + ], + "spans": [ + { + "bbox": [ + 142, + 384, + 449, + 416 + ], + "score": 0.93, + "content": "\\mathrm { e r r } ( f ) \\le \\operatorname* { s u p } _ { ( e , e ^ { \\prime } ) \\in ( \\mathcal { E } _ { a v a i l } , \\mathcal { E } _ { a l l } ) } \\frac { C } { K } \\sum _ { y \\in \\mathcal { Y } } \\int _ { h \\in \\mathbb { R } ^ { d } } \\big | p _ { h ^ { e } | Y ^ { e } } ( h | y ) - p _ { h ^ { e ^ { \\prime } } | Y ^ { e ^ { \\prime } } } ( h | y ) ) \\big | \\mathrm { d } h .", + "type": "interline_equation", + "image_path": "9c22961cf79642649ac7052d3371c49427c7a1ab896566ae2ddd688515f2d0dd.jpg" + } + ] + } + ], + "index": 24, + "virtual_lines": [ + { + "bbox": [ + 142, + 384, + 449, + 394.6666666666667 + ], + "spans": [], + "index": 23 + }, + { + "bbox": [ + 142, + 394.6666666666667, + 449, + 405.33333333333337 + ], + "spans": [], + "index": 24 + }, + { + "bbox": [ + 142, + 405.33333333333337, + 449, + 416.00000000000006 + ], + "spans": [], + "index": 25 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 425, + 381, + 437 + ], + "lines": [ + { + "bbox": [ + 106, + 425, + 381, + 438 + ], + "spans": [ + { + "bbox": [ + 106, + 425, + 381, + 438 + ], + "score": 1.0, + "content": "Step 2. According to the assumption (4), the dominant term in (7) is", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 26 + }, + { + "type": "interline_equation", + "bbox": [ + 172, + 440, + 417, + 469 + ], + "lines": [ + { + "bbox": [ + 172, + 440, + 417, + 469 + ], + "spans": [ + { + "bbox": [ + 172, + 440, + 417, + 469 + ], + "score": 0.92, + "content": "\\int _ { | h | \\leq r _ { 1 } } \\Big | \\int _ { | t | \\leq r _ { 2 } } e ^ { - i \\langle h , t \\rangle } \\big ( \\hat { p } _ { h ^ { e } | Y ^ { e } } ( t | y ) - \\hat { p } _ { h ^ { e ^ { \\prime } } | Y ^ { e ^ { \\prime } } } ( t | y ) \\big ) \\big ) \\mathrm { d } t \\Big | \\mathrm { d } t ,", + "type": "interline_equation", + "image_path": "8d84abc42db5ac0bce20ba33a361ed49a07b3c2d298320a7da3642158de97201.jpg" + } + ] + } + ], + "index": 28, + "virtual_lines": [ + { + "bbox": [ + 172, + 440, + 417, + 449.6666666666667 + ], + "spans": [], + "index": 27 + }, + { + "bbox": [ + 172, + 449.6666666666667, + 417, + 459.33333333333337 + ], + "spans": [], + "index": 28 + }, + { + "bbox": [ + 172, + 459.33333333333337, + 417, + 469.00000000000006 + ], + "spans": [], + "index": 29 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 473, + 506, + 497 + ], + "lines": [ + { + "bbox": [ + 105, + 473, + 505, + 487 + ], + "spans": [ + { + "bbox": [ + 105, + 473, + 134, + 487 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 135, + 476, + 145, + 485 + ], + "score": 0.84, + "content": "r _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 145, + 473, + 165, + 487 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 166, + 476, + 176, + 485 + ], + "score": 0.83, + "content": "r _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 176, + 473, + 349, + 487 + ], + "score": 1.0, + "content": "are well-selected scalars that depend on", + "type": "text" + }, + { + "bbox": [ + 350, + 473, + 423, + 487 + ], + "score": 0.9, + "content": "s \\big ( \\mathcal { V } _ { \\rho } ^ { \\mathrm { s u p } } ( h , \\mathcal { E } _ { a v a i l } ) \\big )", + "type": "inline_equation" + }, + { + "bbox": [ + 423, + 473, + 505, + 487 + ], + "score": 1.0, + "content": ". By the Projection", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 484, + 419, + 498 + ], + "spans": [ + { + "bbox": [ + 105, + 484, + 419, + 498 + ], + "score": 1.0, + "content": "Theorem [31, 42] and the Fourier Inversion Formula, (8) is bounded above by", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 30.5 + }, + { + "type": "interline_equation", + "bbox": [ + 204, + 501, + 386, + 528 + ], + "lines": [ + { + "bbox": [ + 204, + 501, + 386, + 528 + ], + "spans": [ + { + "bbox": [ + 204, + 501, + 386, + 528 + ], + "score": 0.94, + "content": "O ( r _ { 1 } ^ { d } r _ { 2 } ^ { d } ) \\times \\int _ { u \\in \\mathbb { R } } \\big | \\mathscr { R } _ { e ^ { \\prime } } ( \\beta , u ) - \\mathscr { R } _ { e } ( \\beta , u ) \\big | \\mathrm { d } u ,", + "type": "interline_equation", + "image_path": "49950396e3d8096d0baf981fde2f0be38a8c185cda445644133fe0a23d73c6a7.jpg" + } + ] + } + ], + "index": 32.5, + "virtual_lines": [ + { + "bbox": [ + 204, + 501, + 386, + 514.5 + ], + "spans": [], + "index": 32 + }, + { + "bbox": [ + 204, + 514.5, + 386, + 528.0 + ], + "spans": [], + "index": 33 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 532, + 327, + 545 + ], + "lines": [ + { + "bbox": [ + 105, + 529, + 327, + 546 + ], + "spans": [ + { + "bbox": [ + 105, + 529, + 133, + 546 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 133, + 532, + 171, + 544 + ], + "score": 0.93, + "content": "\\mathcal { R } _ { e } ( \\beta , u )", + "type": "inline_equation" + }, + { + "bbox": [ + 171, + 529, + 276, + 546 + ], + "score": 1.0, + "content": "is the Radon transform of", + "type": "text" + }, + { + "bbox": [ + 277, + 532, + 324, + 545 + ], + "score": 0.93, + "content": "p _ { h ^ { e } | Y ^ { e } } ( t | y )", + "type": "inline_equation" + }, + { + "bbox": [ + 324, + 529, + 327, + 546 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 34 + }, + { + "type": "text", + "bbox": [ + 106, + 550, + 505, + 584 + ], + "lines": [ + { + "bbox": [ + 105, + 549, + 506, + 564 + ], + "spans": [ + { + "bbox": [ + 105, + 549, + 351, + 564 + ], + "score": 1.0, + "content": "Step 3. The right-hand side of Formula 8 can be bounded by", + "type": "text" + }, + { + "bbox": [ + 351, + 549, + 460, + 564 + ], + "score": 0.9, + "content": "O \\big ( r _ { 1 } ^ { d } r _ { 2 } ^ { d } s \\big ( \\mathcal { V } _ { \\rho } ^ { \\mathrm { s u p } } ( h , \\mathcal { E } _ { a v a i l } ) \\big ) \\big )", + "type": "inline_equation" + }, + { + "bbox": [ + 461, + 549, + 506, + 564 + ], + "score": 1.0, + "content": ". We finish", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 561, + 506, + 574 + ], + "spans": [ + { + "bbox": [ + 105, + 561, + 241, + 574 + ], + "score": 1.0, + "content": "the proof by selecting appropriate", + "type": "text" + }, + { + "bbox": [ + 241, + 563, + 252, + 573 + ], + "score": 0.85, + "content": "r _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 252, + 561, + 269, + 574 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 269, + 563, + 280, + 573 + ], + "score": 0.85, + "content": "r _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 280, + 561, + 506, + 574 + ], + "score": 1.0, + "content": "to balance the rate of the dominant term and other minor", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 572, + 395, + 585 + ], + "spans": [ + { + "bbox": [ + 105, + 572, + 395, + 585 + ], + "score": 1.0, + "content": "terms. For more details, please see Appendix 2 for the complete proofs.", + "type": "text" + } + ], + "index": 37 + } + ], + "index": 36 + }, + { + "type": "text", + "bbox": [ + 107, + 588, + 277, + 600 + ], + "lines": [ + { + "bbox": [ + 105, + 586, + 274, + 603 + ], + "spans": [ + { + "bbox": [ + 105, + 586, + 249, + 603 + ], + "score": 1.0, + "content": "Now we turn to the lower bound of", + "type": "text" + }, + { + "bbox": [ + 249, + 588, + 274, + 601 + ], + "score": 0.89, + "content": "\\operatorname { e r r } ( f )", + "type": "inline_equation" + } + ], + "index": 38 + } + ], + "index": 38 + }, + { + "type": "text", + "bbox": [ + 106, + 603, + 505, + 672 + ], + "lines": [ + { + "bbox": [ + 105, + 601, + 505, + 617 + ], + "spans": [ + { + "bbox": [ + 105, + 601, + 316, + 617 + ], + "score": 1.0, + "content": "Theorem 4.3 (Lower Bound). Consider 0-1 loss:", + "type": "text" + }, + { + "bbox": [ + 316, + 603, + 398, + 615 + ], + "score": 0.9, + "content": "\\ell ( \\hat { y } , y ) = \\mathbb { I } ( \\hat { y } \\neq y )", + "type": "inline_equation" + }, + { + "bbox": [ + 398, + 601, + 440, + 617 + ], + "score": 1.0, + "content": ". For any", + "type": "text" + }, + { + "bbox": [ + 441, + 604, + 468, + 614 + ], + "score": 0.9, + "content": "\\delta > 0", + "type": "inline_equation" + }, + { + "bbox": [ + 468, + 601, + 505, + 617 + ], + "score": 1.0, + "content": "and any", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 613, + 507, + 641 + ], + "spans": [ + { + "bbox": [ + 105, + 615, + 122, + 641 + ], + "score": 1.0, + "content": "exps.t.", + "type": "text" + }, + { + "bbox": [ + 123, + 628, + 240, + 639 + ], + "score": 0.88, + "content": "k x \\leq s ( x ) < + \\infty , x \\in [ 0 , t ]", + "type": "inline_equation" + }, + { + "bbox": [ + 240, + 615, + 405, + 630 + ], + "score": 0.79, + "content": "\\begin{array} { r } { s _ { + } ^ { \\prime } ( 0 ) \\triangleq \\operatorname* { l i m } _ { x \\to 0 ^ { + } } \\frac { s ( x ) - s ( 0 ) } { x } \\in ( 1 , + \\infty ) } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 316, + 613, + 507, + 630 + ], + "score": 1.0, + "content": "s(x)−s(0) ∈ (1, +∞); 2) exists k > 1, t > 0,", + "type": "text" + }, + { + "bbox": [ + 338, + 629, + 351, + 639 + ], + "score": 0.83, + "content": "C _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 384, + 629, + 407, + 638 + ], + "score": 0.45, + "content": "O O D", + "type": "inline_equation" + }, + { + "bbox": [ + 407, + 615, + 447, + 641 + ], + "score": 1.0, + "content": "generaliz", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 107, + 638, + 506, + 652 + ], + "spans": [ + { + "bbox": [ + 107, + 639, + 158, + 650 + ], + "score": 0.87, + "content": "( \\mathcal { E } _ { a v a i l } , \\mathcal { E } _ { a l l } )", + "type": "inline_equation" + }, + { + "bbox": [ + 158, + 638, + 187, + 652 + ], + "score": 1.0, + "content": "that is", + "type": "text" + }, + { + "bbox": [ + 187, + 639, + 219, + 651 + ], + "score": 0.9, + "content": "( s ( \\cdot ) , \\delta )", + "type": "inline_equation" + }, + { + "bbox": [ + 220, + 638, + 371, + 652 + ], + "score": 1.0, + "content": "-learnable under linear feature space", + "type": "text" + }, + { + "bbox": [ + 371, + 639, + 380, + 649 + ], + "score": 0.66, + "content": "\\Phi", + "type": "inline_equation" + }, + { + "bbox": [ + 380, + 638, + 443, + 652 + ], + "score": 1.0, + "content": "w.r.t symmetric", + "type": "text" + }, + { + "bbox": [ + 443, + 639, + 457, + 649 + ], + "score": 0.34, + "content": "K L", + "type": "inline_equation" + }, + { + "bbox": [ + 457, + 638, + 506, + 652 + ], + "score": 1.0, + "content": "-divergence", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 106, + 649, + 506, + 662 + ], + "spans": [ + { + "bbox": [ + 106, + 652, + 113, + 661 + ], + "score": 0.52, + "content": "\\rho", + "type": "inline_equation" + }, + { + "bbox": [ + 113, + 649, + 136, + 662 + ], + "score": 1.0, + "content": ", s.t.", + "type": "text" + }, + { + "bbox": [ + 136, + 649, + 187, + 662 + ], + "score": 0.89, + "content": "\\forall \\varepsilon \\ \\in \\ [ 0 , \\frac { t } { 2 } ] .", + "type": "inline_equation" + }, + { + "bbox": [ + 187, + 649, + 284, + 662 + ], + "score": 1.0, + "content": ", the optimal classifier", + "type": "text" + }, + { + "bbox": [ + 284, + 650, + 292, + 661 + ], + "score": 0.83, + "content": "f", + "type": "inline_equation" + }, + { + "bbox": [ + 292, + 649, + 336, + 662 + ], + "score": 1.0, + "content": "satisfying", + "type": "text" + }, + { + "bbox": [ + 337, + 650, + 420, + 662 + ], + "score": 0.92, + "content": "\\mathcal { V } ^ { s u p } ( h , \\mathcal { E } _ { a v a i l } ) = \\varepsilon", + "type": "inline_equation" + }, + { + "bbox": [ + 420, + 649, + 506, + 662 + ], + "score": 1.0, + "content": "will have the OOD", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 660, + 263, + 673 + ], + "spans": [ + { + "bbox": [ + 105, + 660, + 263, + 673 + ], + "score": 1.0, + "content": "generalization error lower bounded by", + "type": "text" + } + ], + "index": 43 + } + ], + "index": 41 + }, + { + "type": "interline_equation", + "bbox": [ + 228, + 676, + 363, + 690 + ], + "lines": [ + { + "bbox": [ + 228, + 676, + 363, + 690 + ], + "spans": [ + { + "bbox": [ + 228, + 676, + 363, + 690 + ], + "score": 0.9, + "content": "\\mathrm { e r r } ( f ) \\geq C _ { 0 } \\cdot s ( \\mathcal { V } ^ { s u p } ( h , \\mathcal { E } _ { a v a i l } ) ) .", + "type": "interline_equation", + "image_path": "b71427ee15ff225ca02625e3250ad817c12b27937efe9a03a49e7ec1eee529f3.jpg" + } + ] + } + ], + "index": 44, + "virtual_lines": [ + { + "bbox": [ + 228, + 676, + 363, + 690 + ], + "spans": [], + "index": 44 + } + ] + }, + { + "type": "text", + "bbox": [ + 105, + 699, + 504, + 723 + ], + "lines": [ + { + "bbox": [ + 105, + 699, + 505, + 712 + ], + "spans": [ + { + "bbox": [ + 105, + 699, + 210, + 712 + ], + "score": 1.0, + "content": "Theorem 4.3 shows that", + "type": "text" + }, + { + "bbox": [ + 210, + 700, + 236, + 712 + ], + "score": 0.9, + "content": "\\operatorname { e r r } ( f )", + "type": "inline_equation" + }, + { + "bbox": [ + 237, + 699, + 325, + 712 + ], + "score": 1.0, + "content": "of optimal classifier", + "type": "text" + }, + { + "bbox": [ + 326, + 700, + 333, + 711 + ], + "score": 0.82, + "content": "f", + "type": "inline_equation" + }, + { + "bbox": [ + 333, + 699, + 505, + 712 + ], + "score": 1.0, + "content": "is lower bounded by its variation. Here", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 710, + 505, + 723 + ], + "spans": [ + { + "bbox": [ + 105, + 710, + 293, + 723 + ], + "score": 1.0, + "content": "“optimal” means the classifier that minimize", + "type": "text" + }, + { + "bbox": [ + 294, + 711, + 342, + 723 + ], + "score": 0.92, + "content": "\\mathcal { L } ( f , \\mathcal { E } _ { a v a i l } )", + "type": "inline_equation" + }, + { + "bbox": [ + 343, + 710, + 505, + 723 + ], + "score": 1.0, + "content": ". Altogether, the above three theorems", + "type": "text" + } + ], + "index": 46 + } + ], + "index": 45.5 + } + ], + "page_idx": 5, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 302, + 741, + 309, + 750 + ], + "lines": [ + { + "bbox": [ + 302, + 741, + 310, + 752 + ], + "spans": [ + { + "bbox": [ + 302, + 741, + 310, + 752 + ], + "score": 1.0, + "content": "6", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 105, + 72, + 488, + 86 + ], + "lines": [ + { + "bbox": [ + 106, + 70, + 489, + 87 + ], + "spans": [ + { + "bbox": [ + 106, + 72, + 167, + 85 + ], + "score": 0.9, + "content": "i f ( \\mathcal { E } _ { a v a i l } , \\mathcal { E } _ { a l l } )", + "type": "inline_equation" + }, + { + "bbox": [ + 167, + 70, + 178, + 87 + ], + "score": 1.0, + "content": "is", + "type": "text" + }, + { + "bbox": [ + 178, + 72, + 263, + 85 + ], + "score": 0.89, + "content": "\\left( s ( \\cdot ) , \\mathcal { T } ^ { i n f } ( h , \\mathcal { E } _ { a v a i l } ) \\right)", + "type": "inline_equation" + }, + { + "bbox": [ + 263, + 70, + 332, + 87 + ], + "score": 1.0, + "content": "-learnable under", + "type": "text" + }, + { + "bbox": [ + 332, + 73, + 340, + 83 + ], + "score": 0.81, + "content": "\\Phi", + "type": "inline_equation" + }, + { + "bbox": [ + 341, + 70, + 423, + 87 + ], + "score": 1.0, + "content": "with Total Variation", + "type": "text" + }, + { + "bbox": [ + 424, + 75, + 430, + 84 + ], + "score": 0.69, + "content": "\\rho", + "type": "inline_equation" + }, + { + "bbox": [ + 430, + 70, + 489, + 87 + ], + "score": 1.0, + "content": ", then we have", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0, + "bbox_fs": [ + 106, + 70, + 489, + 87 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 226, + 90, + 364, + 111 + ], + "lines": [ + { + "bbox": [ + 226, + 90, + 364, + 111 + ], + "spans": [ + { + "bbox": [ + 226, + 90, + 364, + 111 + ], + "score": 0.93, + "content": "\\mathrm { e r r } ( f ) \\leq O \\Bigl ( s \\bigl ( \\mathcal { V } ^ { s u p } ( h , \\mathcal { E } _ { a v a i l } ) \\bigr ) \\Bigr ) .", + "type": "interline_equation", + "image_path": "bff65b7e956cd09791c5ed2c8933cda9684cbbae3c4dc554a897bdb02f4ed4da.jpg" + } + ] + } + ], + "index": 1, + "virtual_lines": [ + { + "bbox": [ + 226, + 90, + 364, + 111 + ], + "spans": [], + "index": 1 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 115, + 253, + 127 + ], + "lines": [ + { + "bbox": [ + 105, + 114, + 254, + 129 + ], + "spans": [ + { + "bbox": [ + 105, + 114, + 128, + 129 + ], + "score": 1.0, + "content": "Here", + "type": "text" + }, + { + "bbox": [ + 128, + 115, + 147, + 128 + ], + "score": 0.89, + "content": "O ( \\cdot )", + "type": "inline_equation" + }, + { + "bbox": [ + 148, + 114, + 241, + 129 + ], + "score": 1.0, + "content": "depends only on d and", + "type": "text" + }, + { + "bbox": [ + 242, + 116, + 250, + 126 + ], + "score": 0.78, + "content": "C", + "type": "inline_equation" + }, + { + "bbox": [ + 250, + 114, + 254, + 129 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 2 + } + ], + "index": 2, + "bbox_fs": [ + 105, + 114, + 254, + 129 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 138, + 506, + 227 + ], + "lines": [ + { + "bbox": [ + 105, + 138, + 505, + 151 + ], + "spans": [ + { + "bbox": [ + 105, + 138, + 505, + 151 + ], + "score": 1.0, + "content": "Discussion. Theorem 4.1 shows that, for any model, the generalization gap depends largely on the", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 104, + 148, + 507, + 163 + ], + "spans": [ + { + "bbox": [ + 104, + 148, + 225, + 163 + ], + "score": 1.0, + "content": "model’s variation captured by", + "type": "text" + }, + { + "bbox": [ + 226, + 150, + 285, + 161 + ], + "score": 0.92, + "content": "\\mathcal { V } ^ { \\mathrm { s u p } } ( h , \\mathcal { E } _ { a v a i l } )", + "type": "inline_equation" + }, + { + "bbox": [ + 286, + 148, + 507, + 163 + ], + "score": 1.0, + "content": ". The result is irrelevant to the algorithm and provides a", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 160, + 506, + 173 + ], + "spans": [ + { + "bbox": [ + 105, + 160, + 275, + 173 + ], + "score": 1.0, + "content": "guarantee for the generalization gap from", + "type": "text" + }, + { + "bbox": [ + 275, + 161, + 300, + 172 + ], + "score": 0.91, + "content": "\\mathcal { E } _ { a v a i l }", + "type": "inline_equation" + }, + { + "bbox": [ + 300, + 160, + 311, + 173 + ], + "score": 1.0, + "content": "to", + "type": "text" + }, + { + "bbox": [ + 312, + 160, + 328, + 171 + ], + "score": 0.89, + "content": "{ \\mathcal { E } } _ { a l l }", + "type": "inline_equation" + }, + { + "bbox": [ + 328, + 160, + 448, + 173 + ], + "score": 1.0, + "content": ", so long as the learned model", + "type": "text" + }, + { + "bbox": [ + 449, + 161, + 456, + 172 + ], + "score": 0.84, + "content": "f", + "type": "inline_equation" + }, + { + "bbox": [ + 456, + 160, + 506, + 173 + ], + "score": 1.0, + "content": "is invariant,", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 104, + 170, + 507, + 185 + ], + "spans": [ + { + "bbox": [ + 104, + 170, + 124, + 185 + ], + "score": 1.0, + "content": "i.e.", + "type": "text" + }, + { + "bbox": [ + 124, + 171, + 183, + 183 + ], + "score": 0.92, + "content": "\\mathcal { V } ^ { \\mathrm { s u p } } ( h , \\mathcal { E } _ { a v a i l } )", + "type": "inline_equation" + }, + { + "bbox": [ + 184, + 170, + 252, + 185 + ], + "score": 1.0, + "content": "is small. When", + "type": "text" + }, + { + "bbox": [ + 252, + 172, + 268, + 183 + ], + "score": 0.91, + "content": "s ( \\cdot )", + "type": "inline_equation" + }, + { + "bbox": [ + 268, + 170, + 396, + 185 + ], + "score": 1.0, + "content": "is fixed, a model with smaller", + "type": "text" + }, + { + "bbox": [ + 396, + 171, + 455, + 183 + ], + "score": 0.91, + "content": "\\mathcal { V } ^ { \\mathrm { s u p } } ( h , \\mathcal { E } _ { a v a i l } )", + "type": "inline_equation" + }, + { + "bbox": [ + 456, + 170, + 507, + 185 + ], + "score": 1.0, + "content": "results in a", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 104, + 181, + 506, + 196 + ], + "spans": [ + { + "bbox": [ + 104, + 181, + 388, + 196 + ], + "score": 1.0, + "content": "smaller gap, which matches our understanding that invariant features in", + "type": "text" + }, + { + "bbox": [ + 388, + 183, + 413, + 194 + ], + "score": 0.91, + "content": "\\mathcal { E } _ { a v a i l }", + "type": "inline_equation" + }, + { + "bbox": [ + 413, + 181, + 506, + 196 + ], + "score": 1.0, + "content": "are somehow invariant", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 104, + 191, + 506, + 207 + ], + "spans": [ + { + "bbox": [ + 104, + 191, + 117, + 207 + ], + "score": 1.0, + "content": "in", + "type": "text" + }, + { + "bbox": [ + 117, + 193, + 133, + 204 + ], + "score": 0.92, + "content": "{ \\mathcal { E } } _ { a l l }", + "type": "inline_equation" + }, + { + "bbox": [ + 133, + 191, + 164, + 207 + ], + "score": 1.0, + "content": ". When", + "type": "text" + }, + { + "bbox": [ + 164, + 193, + 224, + 205 + ], + "score": 0.92, + "content": "\\mathcal { V } ^ { \\mathrm { s u p } } ( h , \\mathcal { E } _ { a v a i l } )", + "type": "inline_equation" + }, + { + "bbox": [ + 224, + 191, + 506, + 207 + ], + "score": 1.0, + "content": "is fixed, more difficult generalization will generate a larger expansion", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 203, + 504, + 217 + ], + "spans": [ + { + "bbox": [ + 105, + 203, + 475, + 217 + ], + "score": 1.0, + "content": "function, which leads to a larger gap. For the Gaussian class with bounded mean and variance,", + "type": "text" + }, + { + "bbox": [ + 475, + 204, + 504, + 215 + ], + "score": 0.89, + "content": "\\alpha \\gg d", + "type": "inline_equation" + } + ], + "index": 9 + }, + { + "bbox": [ + 106, + 215, + 267, + 227 + ], + "spans": [ + { + "bbox": [ + 106, + 215, + 267, + 227 + ], + "score": 1.0, + "content": "and the convergent rate is almost linear.", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 6.5, + "bbox_fs": [ + 104, + 138, + 507, + 227 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 231, + 505, + 287 + ], + "lines": [ + { + "bbox": [ + 106, + 231, + 506, + 244 + ], + "spans": [ + { + "bbox": [ + 106, + 231, + 248, + 244 + ], + "score": 1.0, + "content": "However, without any constraint to", + "type": "text" + }, + { + "bbox": [ + 248, + 234, + 254, + 243 + ], + "score": 0.79, + "content": "g", + "type": "inline_equation" + }, + { + "bbox": [ + 254, + 231, + 506, + 244 + ], + "score": 1.0, + "content": ", the convergent rate might be small. Theorem 4.2 then offers a", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 242, + 505, + 254 + ], + "spans": [ + { + "bbox": [ + 105, + 242, + 433, + 254 + ], + "score": 1.0, + "content": "generalization bound with a linear convergent rate under mild assumptions when", + "type": "text" + }, + { + "bbox": [ + 434, + 244, + 441, + 254 + ], + "score": 0.79, + "content": "g", + "type": "inline_equation" + }, + { + "bbox": [ + 441, + 242, + 505, + 254 + ], + "score": 1.0, + "content": "is linear, which", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 253, + 505, + 266 + ], + "spans": [ + { + "bbox": [ + 105, + 253, + 505, + 266 + ], + "score": 1.0, + "content": "is common in reality. It relaxes the concentration assumption (Formula 4) and asks only for the", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 264, + 505, + 276 + ], + "spans": [ + { + "bbox": [ + 106, + 264, + 505, + 276 + ], + "score": 1.0, + "content": "integrability of the density. The convergent rate is identical to the convergent rate of the expansion", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 275, + 369, + 287 + ], + "spans": [ + { + "bbox": [ + 106, + 275, + 198, + 287 + ], + "score": 1.0, + "content": "function, showing that", + "type": "text" + }, + { + "bbox": [ + 199, + 275, + 214, + 287 + ], + "score": 0.9, + "content": "s ( \\cdot )", + "type": "inline_equation" + }, + { + "bbox": [ + 215, + 275, + 369, + 287 + ], + "score": 1.0, + "content": "captures the generalization quite well.", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 13, + "bbox_fs": [ + 105, + 231, + 506, + 287 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 297, + 505, + 365 + ], + "lines": [ + { + "bbox": [ + 106, + 298, + 505, + 309 + ], + "spans": [ + { + "bbox": [ + 106, + 298, + 505, + 309 + ], + "score": 1.0, + "content": "Proof Sketch of Theorem 4.1. The proof of the main result, Theorem 4.1, is decomposed into the", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 308, + 506, + 321 + ], + "spans": [ + { + "bbox": [ + 105, + 308, + 249, + 321 + ], + "score": 1.0, + "content": "following steps. First, we transform", + "type": "text" + }, + { + "bbox": [ + 249, + 309, + 275, + 321 + ], + "score": 0.88, + "content": "\\operatorname { e r r } ( f )", + "type": "inline_equation" + }, + { + "bbox": [ + 275, + 308, + 506, + 321 + ], + "score": 1.0, + "content": "into the Total Variation of joint distributions of features in", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 318, + 506, + 332 + ], + "spans": [ + { + "bbox": [ + 105, + 318, + 506, + 332 + ], + "score": 1.0, + "content": "different domains (Step 1). To bound the Total Variation, it is sufficient to bound the distance of the", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 331, + 505, + 342 + ], + "spans": [ + { + "bbox": [ + 105, + 331, + 505, + 342 + ], + "score": 1.0, + "content": "corresponding Fourier transform, and further, it is equivalent to bound the Radon transform of joint", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 339, + 506, + 356 + ], + "spans": [ + { + "bbox": [ + 105, + 339, + 300, + 356 + ], + "score": 1.0, + "content": "distributions (Step 2). Eventually, we show that", + "type": "text" + }, + { + "bbox": [ + 300, + 341, + 373, + 354 + ], + "score": 0.92, + "content": "\\mathcal { V } ^ { \\mathrm { s u p } } ( \\beta ^ { \\top } h , \\mathcal { E } _ { a v a i l } )", + "type": "inline_equation" + }, + { + "bbox": [ + 373, + 339, + 506, + 356 + ], + "score": 1.0, + "content": "can be used to bound the Radon", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 353, + 285, + 365 + ], + "spans": [ + { + "bbox": [ + 106, + 353, + 285, + 365 + ], + "score": 1.0, + "content": "transform, which finishes the proof (Step 3).", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 18.5, + "bbox_fs": [ + 105, + 298, + 506, + 365 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 368, + 339, + 380 + ], + "lines": [ + { + "bbox": [ + 106, + 368, + 340, + 381 + ], + "spans": [ + { + "bbox": [ + 106, + 368, + 340, + 381 + ], + "score": 1.0, + "content": "Step 1. The OOD generalization error can be bounded as:", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 22, + "bbox_fs": [ + 106, + 368, + 340, + 381 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 142, + 384, + 449, + 416 + ], + "lines": [ + { + "bbox": [ + 142, + 384, + 449, + 416 + ], + "spans": [ + { + "bbox": [ + 142, + 384, + 449, + 416 + ], + "score": 0.93, + "content": "\\mathrm { e r r } ( f ) \\le \\operatorname* { s u p } _ { ( e , e ^ { \\prime } ) \\in ( \\mathcal { E } _ { a v a i l } , \\mathcal { E } _ { a l l } ) } \\frac { C } { K } \\sum _ { y \\in \\mathcal { Y } } \\int _ { h \\in \\mathbb { R } ^ { d } } \\big | p _ { h ^ { e } | Y ^ { e } } ( h | y ) - p _ { h ^ { e ^ { \\prime } } | Y ^ { e ^ { \\prime } } } ( h | y ) ) \\big | \\mathrm { d } h .", + "type": "interline_equation", + "image_path": "9c22961cf79642649ac7052d3371c49427c7a1ab896566ae2ddd688515f2d0dd.jpg" + } + ] + } + ], + "index": 24, + "virtual_lines": [ + { + "bbox": [ + 142, + 384, + 449, + 394.6666666666667 + ], + "spans": [], + "index": 23 + }, + { + "bbox": [ + 142, + 394.6666666666667, + 449, + 405.33333333333337 + ], + "spans": [], + "index": 24 + }, + { + "bbox": [ + 142, + 405.33333333333337, + 449, + 416.00000000000006 + ], + "spans": [], + "index": 25 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 425, + 381, + 437 + ], + "lines": [ + { + "bbox": [ + 106, + 425, + 381, + 438 + ], + "spans": [ + { + "bbox": [ + 106, + 425, + 381, + 438 + ], + "score": 1.0, + "content": "Step 2. According to the assumption (4), the dominant term in (7) is", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 26, + "bbox_fs": [ + 106, + 425, + 381, + 438 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 172, + 440, + 417, + 469 + ], + "lines": [ + { + "bbox": [ + 172, + 440, + 417, + 469 + ], + "spans": [ + { + "bbox": [ + 172, + 440, + 417, + 469 + ], + "score": 0.92, + "content": "\\int _ { | h | \\leq r _ { 1 } } \\Big | \\int _ { | t | \\leq r _ { 2 } } e ^ { - i \\langle h , t \\rangle } \\big ( \\hat { p } _ { h ^ { e } | Y ^ { e } } ( t | y ) - \\hat { p } _ { h ^ { e ^ { \\prime } } | Y ^ { e ^ { \\prime } } } ( t | y ) \\big ) \\big ) \\mathrm { d } t \\Big | \\mathrm { d } t ,", + "type": "interline_equation", + "image_path": "8d84abc42db5ac0bce20ba33a361ed49a07b3c2d298320a7da3642158de97201.jpg" + } + ] + } + ], + "index": 28, + "virtual_lines": [ + { + "bbox": [ + 172, + 440, + 417, + 449.6666666666667 + ], + "spans": [], + "index": 27 + }, + { + "bbox": [ + 172, + 449.6666666666667, + 417, + 459.33333333333337 + ], + "spans": [], + "index": 28 + }, + { + "bbox": [ + 172, + 459.33333333333337, + 417, + 469.00000000000006 + ], + "spans": [], + "index": 29 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 473, + 506, + 497 + ], + "lines": [ + { + "bbox": [ + 105, + 473, + 505, + 487 + ], + "spans": [ + { + "bbox": [ + 105, + 473, + 134, + 487 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 135, + 476, + 145, + 485 + ], + "score": 0.84, + "content": "r _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 145, + 473, + 165, + 487 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 166, + 476, + 176, + 485 + ], + "score": 0.83, + "content": "r _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 176, + 473, + 349, + 487 + ], + "score": 1.0, + "content": "are well-selected scalars that depend on", + "type": "text" + }, + { + "bbox": [ + 350, + 473, + 423, + 487 + ], + "score": 0.9, + "content": "s \\big ( \\mathcal { V } _ { \\rho } ^ { \\mathrm { s u p } } ( h , \\mathcal { E } _ { a v a i l } ) \\big )", + "type": "inline_equation" + }, + { + "bbox": [ + 423, + 473, + 505, + 487 + ], + "score": 1.0, + "content": ". By the Projection", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 484, + 419, + 498 + ], + "spans": [ + { + "bbox": [ + 105, + 484, + 419, + 498 + ], + "score": 1.0, + "content": "Theorem [31, 42] and the Fourier Inversion Formula, (8) is bounded above by", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 30.5, + "bbox_fs": [ + 105, + 473, + 505, + 498 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 204, + 501, + 386, + 528 + ], + "lines": [ + { + "bbox": [ + 204, + 501, + 386, + 528 + ], + "spans": [ + { + "bbox": [ + 204, + 501, + 386, + 528 + ], + "score": 0.94, + "content": "O ( r _ { 1 } ^ { d } r _ { 2 } ^ { d } ) \\times \\int _ { u \\in \\mathbb { R } } \\big | \\mathscr { R } _ { e ^ { \\prime } } ( \\beta , u ) - \\mathscr { R } _ { e } ( \\beta , u ) \\big | \\mathrm { d } u ,", + "type": "interline_equation", + "image_path": "49950396e3d8096d0baf981fde2f0be38a8c185cda445644133fe0a23d73c6a7.jpg" + } + ] + } + ], + "index": 32.5, + "virtual_lines": [ + { + "bbox": [ + 204, + 501, + 386, + 514.5 + ], + "spans": [], + "index": 32 + }, + { + "bbox": [ + 204, + 514.5, + 386, + 528.0 + ], + "spans": [], + "index": 33 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 532, + 327, + 545 + ], + "lines": [ + { + "bbox": [ + 105, + 529, + 327, + 546 + ], + "spans": [ + { + "bbox": [ + 105, + 529, + 133, + 546 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 133, + 532, + 171, + 544 + ], + "score": 0.93, + "content": "\\mathcal { R } _ { e } ( \\beta , u )", + "type": "inline_equation" + }, + { + "bbox": [ + 171, + 529, + 276, + 546 + ], + "score": 1.0, + "content": "is the Radon transform of", + "type": "text" + }, + { + "bbox": [ + 277, + 532, + 324, + 545 + ], + "score": 0.93, + "content": "p _ { h ^ { e } | Y ^ { e } } ( t | y )", + "type": "inline_equation" + }, + { + "bbox": [ + 324, + 529, + 327, + 546 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 34, + "bbox_fs": [ + 105, + 529, + 327, + 546 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 550, + 505, + 584 + ], + "lines": [ + { + "bbox": [ + 105, + 549, + 506, + 564 + ], + "spans": [ + { + "bbox": [ + 105, + 549, + 351, + 564 + ], + "score": 1.0, + "content": "Step 3. The right-hand side of Formula 8 can be bounded by", + "type": "text" + }, + { + "bbox": [ + 351, + 549, + 460, + 564 + ], + "score": 0.9, + "content": "O \\big ( r _ { 1 } ^ { d } r _ { 2 } ^ { d } s \\big ( \\mathcal { V } _ { \\rho } ^ { \\mathrm { s u p } } ( h , \\mathcal { E } _ { a v a i l } ) \\big ) \\big )", + "type": "inline_equation" + }, + { + "bbox": [ + 461, + 549, + 506, + 564 + ], + "score": 1.0, + "content": ". We finish", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 561, + 506, + 574 + ], + "spans": [ + { + "bbox": [ + 105, + 561, + 241, + 574 + ], + "score": 1.0, + "content": "the proof by selecting appropriate", + "type": "text" + }, + { + "bbox": [ + 241, + 563, + 252, + 573 + ], + "score": 0.85, + "content": "r _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 252, + 561, + 269, + 574 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 269, + 563, + 280, + 573 + ], + "score": 0.85, + "content": "r _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 280, + 561, + 506, + 574 + ], + "score": 1.0, + "content": "to balance the rate of the dominant term and other minor", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 572, + 395, + 585 + ], + "spans": [ + { + "bbox": [ + 105, + 572, + 395, + 585 + ], + "score": 1.0, + "content": "terms. For more details, please see Appendix 2 for the complete proofs.", + "type": "text" + } + ], + "index": 37 + } + ], + "index": 36, + "bbox_fs": [ + 105, + 549, + 506, + 585 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 588, + 277, + 600 + ], + "lines": [ + { + "bbox": [ + 105, + 586, + 274, + 603 + ], + "spans": [ + { + "bbox": [ + 105, + 586, + 249, + 603 + ], + "score": 1.0, + "content": "Now we turn to the lower bound of", + "type": "text" + }, + { + "bbox": [ + 249, + 588, + 274, + 601 + ], + "score": 0.89, + "content": "\\operatorname { e r r } ( f )", + "type": "inline_equation" + } + ], + "index": 38 + } + ], + "index": 38, + "bbox_fs": [ + 105, + 586, + 274, + 603 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 603, + 505, + 672 + ], + "lines": [ + { + "bbox": [ + 105, + 601, + 505, + 617 + ], + "spans": [ + { + "bbox": [ + 105, + 601, + 316, + 617 + ], + "score": 1.0, + "content": "Theorem 4.3 (Lower Bound). Consider 0-1 loss:", + "type": "text" + }, + { + "bbox": [ + 316, + 603, + 398, + 615 + ], + "score": 0.9, + "content": "\\ell ( \\hat { y } , y ) = \\mathbb { I } ( \\hat { y } \\neq y )", + "type": "inline_equation" + }, + { + "bbox": [ + 398, + 601, + 440, + 617 + ], + "score": 1.0, + "content": ". For any", + "type": "text" + }, + { + "bbox": [ + 441, + 604, + 468, + 614 + ], + "score": 0.9, + "content": "\\delta > 0", + "type": "inline_equation" + }, + { + "bbox": [ + 468, + 601, + 505, + 617 + ], + "score": 1.0, + "content": "and any", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 613, + 507, + 641 + ], + "spans": [ + { + "bbox": [ + 105, + 615, + 122, + 641 + ], + "score": 1.0, + "content": "exps.t.", + "type": "text" + }, + { + "bbox": [ + 123, + 628, + 240, + 639 + ], + "score": 0.88, + "content": "k x \\leq s ( x ) < + \\infty , x \\in [ 0 , t ]", + "type": "inline_equation" + }, + { + "bbox": [ + 240, + 615, + 405, + 630 + ], + "score": 0.79, + "content": "\\begin{array} { r } { s _ { + } ^ { \\prime } ( 0 ) \\triangleq \\operatorname* { l i m } _ { x \\to 0 ^ { + } } \\frac { s ( x ) - s ( 0 ) } { x } \\in ( 1 , + \\infty ) } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 316, + 613, + 507, + 630 + ], + "score": 1.0, + "content": "s(x)−s(0) ∈ (1, +∞); 2) exists k > 1, t > 0,", + "type": "text" + }, + { + "bbox": [ + 338, + 629, + 351, + 639 + ], + "score": 0.83, + "content": "C _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 384, + 629, + 407, + 638 + ], + "score": 0.45, + "content": "O O D", + "type": "inline_equation" + }, + { + "bbox": [ + 407, + 615, + 447, + 641 + ], + "score": 1.0, + "content": "generaliz", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 107, + 638, + 506, + 652 + ], + "spans": [ + { + "bbox": [ + 107, + 639, + 158, + 650 + ], + "score": 0.87, + "content": "( \\mathcal { E } _ { a v a i l } , \\mathcal { E } _ { a l l } )", + "type": "inline_equation" + }, + { + "bbox": [ + 158, + 638, + 187, + 652 + ], + "score": 1.0, + "content": "that is", + "type": "text" + }, + { + "bbox": [ + 187, + 639, + 219, + 651 + ], + "score": 0.9, + "content": "( s ( \\cdot ) , \\delta )", + "type": "inline_equation" + }, + { + "bbox": [ + 220, + 638, + 371, + 652 + ], + "score": 1.0, + "content": "-learnable under linear feature space", + "type": "text" + }, + { + "bbox": [ + 371, + 639, + 380, + 649 + ], + "score": 0.66, + "content": "\\Phi", + "type": "inline_equation" + }, + { + "bbox": [ + 380, + 638, + 443, + 652 + ], + "score": 1.0, + "content": "w.r.t symmetric", + "type": "text" + }, + { + "bbox": [ + 443, + 639, + 457, + 649 + ], + "score": 0.34, + "content": "K L", + "type": "inline_equation" + }, + { + "bbox": [ + 457, + 638, + 506, + 652 + ], + "score": 1.0, + "content": "-divergence", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 106, + 649, + 506, + 662 + ], + "spans": [ + { + "bbox": [ + 106, + 652, + 113, + 661 + ], + "score": 0.52, + "content": "\\rho", + "type": "inline_equation" + }, + { + "bbox": [ + 113, + 649, + 136, + 662 + ], + "score": 1.0, + "content": ", s.t.", + "type": "text" + }, + { + "bbox": [ + 136, + 649, + 187, + 662 + ], + "score": 0.89, + "content": "\\forall \\varepsilon \\ \\in \\ [ 0 , \\frac { t } { 2 } ] .", + "type": "inline_equation" + }, + { + "bbox": [ + 187, + 649, + 284, + 662 + ], + "score": 1.0, + "content": ", the optimal classifier", + "type": "text" + }, + { + "bbox": [ + 284, + 650, + 292, + 661 + ], + "score": 0.83, + "content": "f", + "type": "inline_equation" + }, + { + "bbox": [ + 292, + 649, + 336, + 662 + ], + "score": 1.0, + "content": "satisfying", + "type": "text" + }, + { + "bbox": [ + 337, + 650, + 420, + 662 + ], + "score": 0.92, + "content": "\\mathcal { V } ^ { s u p } ( h , \\mathcal { E } _ { a v a i l } ) = \\varepsilon", + "type": "inline_equation" + }, + { + "bbox": [ + 420, + 649, + 506, + 662 + ], + "score": 1.0, + "content": "will have the OOD", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 660, + 263, + 673 + ], + "spans": [ + { + "bbox": [ + 105, + 660, + 263, + 673 + ], + "score": 1.0, + "content": "generalization error lower bounded by", + "type": "text" + } + ], + "index": 43 + } + ], + "index": 41, + "bbox_fs": [ + 105, + 601, + 507, + 673 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 228, + 676, + 363, + 690 + ], + "lines": [ + { + "bbox": [ + 228, + 676, + 363, + 690 + ], + "spans": [ + { + "bbox": [ + 228, + 676, + 363, + 690 + ], + "score": 0.9, + "content": "\\mathrm { e r r } ( f ) \\geq C _ { 0 } \\cdot s ( \\mathcal { V } ^ { s u p } ( h , \\mathcal { E } _ { a v a i l } ) ) .", + "type": "interline_equation", + "image_path": "b71427ee15ff225ca02625e3250ad817c12b27937efe9a03a49e7ec1eee529f3.jpg" + } + ] + } + ], + "index": 44, + "virtual_lines": [ + { + "bbox": [ + 228, + 676, + 363, + 690 + ], + "spans": [], + "index": 44 + } + ] + }, + { + "type": "text", + "bbox": [ + 105, + 699, + 504, + 723 + ], + "lines": [ + { + "bbox": [ + 105, + 699, + 505, + 712 + ], + "spans": [ + { + "bbox": [ + 105, + 699, + 210, + 712 + ], + "score": 1.0, + "content": "Theorem 4.3 shows that", + "type": "text" + }, + { + "bbox": [ + 210, + 700, + 236, + 712 + ], + "score": 0.9, + "content": "\\operatorname { e r r } ( f )", + "type": "inline_equation" + }, + { + "bbox": [ + 237, + 699, + 325, + 712 + ], + "score": 1.0, + "content": "of optimal classifier", + "type": "text" + }, + { + "bbox": [ + 326, + 700, + 333, + 711 + ], + "score": 0.82, + "content": "f", + "type": "inline_equation" + }, + { + "bbox": [ + 333, + 699, + 505, + 712 + ], + "score": 1.0, + "content": "is lower bounded by its variation. Here", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 710, + 505, + 723 + ], + "spans": [ + { + "bbox": [ + 105, + 710, + 293, + 723 + ], + "score": 1.0, + "content": "“optimal” means the classifier that minimize", + "type": "text" + }, + { + "bbox": [ + 294, + 711, + 342, + 723 + ], + "score": 0.92, + "content": "\\mathcal { L } ( f , \\mathcal { E } _ { a v a i l } )", + "type": "inline_equation" + }, + { + "bbox": [ + 343, + 710, + 505, + 723 + ], + "score": 1.0, + "content": ". Altogether, the above three theorems", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 106, + 73, + 506, + 84 + ], + "spans": [ + { + "bbox": [ + 106, + 73, + 506, + 84 + ], + "score": 1.0, + "content": "offer a bidirectional control of OOD generalization error, showing that our formulation can offer a", + "type": "text", + "cross_page": true + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 83, + 505, + 96 + ], + "spans": [ + { + "bbox": [ + 106, + 83, + 505, + 96 + ], + "score": 1.0, + "content": "fine-grained description of most OOD generalization problem in a theoretical way. To pursue a good", + "type": "text", + "cross_page": true + } + ], + "index": 1 + }, + { + "bbox": [ + 106, + 95, + 506, + 108 + ], + "spans": [ + { + "bbox": [ + 106, + 95, + 462, + 108 + ], + "score": 1.0, + "content": "OOD performance, OOD algorithm should focus on improving predictive performance on", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 462, + 95, + 487, + 106 + ], + "score": 0.91, + "content": "\\mathcal { E } _ { a v a i l }", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 487, + 95, + 506, + 108 + ], + "score": 1.0, + "content": "and", + "type": "text", + "cross_page": true + } + ], + "index": 2 + }, + { + "bbox": [ + 104, + 104, + 506, + 120 + ], + "spans": [ + { + "bbox": [ + 104, + 104, + 205, + 120 + ], + "score": 1.0, + "content": "controlling the variation", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 205, + 106, + 265, + 118 + ], + "score": 0.92, + "content": "\\mathcal { V } ^ { \\mathrm { s u p } } ( h , \\mathcal { E } _ { a v a i l } )", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 265, + 104, + 506, + 120 + ], + "score": 1.0, + "content": "simultaneously. Note that this bound starts from population", + "type": "text", + "cross_page": true + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 116, + 505, + 128 + ], + "spans": [ + { + "bbox": [ + 105, + 116, + 505, + 128 + ], + "score": 1.0, + "content": "error, and we call for future works to combine our generalization bound and traditional bound from", + "type": "text", + "cross_page": true + } + ], + "index": 4 + }, + { + "bbox": [ + 106, + 127, + 465, + 140 + ], + "spans": [ + { + "bbox": [ + 106, + 127, + 465, + 140 + ], + "score": 1.0, + "content": "data samples to population error, giving a more complete characterization of the problem.", + "type": "text", + "cross_page": true + } + ], + "index": 5 + } + ], + "index": 45.5, + "bbox_fs": [ + 105, + 699, + 505, + 723 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 106, + 72, + 505, + 139 + ], + "lines": [ + { + "bbox": [ + 106, + 73, + 506, + 84 + ], + "spans": [ + { + "bbox": [ + 106, + 73, + 506, + 84 + ], + "score": 1.0, + "content": "offer a bidirectional control of OOD generalization error, showing that our formulation can offer a", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 83, + 505, + 96 + ], + "spans": [ + { + "bbox": [ + 106, + 83, + 505, + 96 + ], + "score": 1.0, + "content": "fine-grained description of most OOD generalization problem in a theoretical way. 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Note that this bound starts from population", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 116, + 505, + 128 + ], + "spans": [ + { + "bbox": [ + 105, + 116, + 505, + 128 + ], + "score": 1.0, + "content": "error, and we call for future works to combine our generalization bound and traditional bound from", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 106, + 127, + 465, + 140 + ], + "spans": [ + { + "bbox": [ + 106, + 127, + 465, + 140 + ], + "score": 1.0, + "content": "data samples to population error, giving a more complete characterization of the problem.", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 2.5 + }, + { + "type": "title", + "bbox": [ + 106, + 154, + 381, + 168 + ], + "lines": [ + { + "bbox": [ + 104, + 153, + 382, + 171 + ], + "spans": [ + { + "bbox": [ + 104, + 153, + 382, + 171 + ], + "score": 1.0, + "content": "5 Variation as a Factor of Model Selection Criterion", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 6 + }, + { + "type": "text", + "bbox": [ + 106, + 180, + 505, + 301 + ], + "lines": [ + { + "bbox": [ + 106, + 181, + 505, + 192 + ], + "spans": [ + { + "bbox": [ + 106, + 181, + 505, + 192 + ], + "score": 1.0, + "content": "As is pointed out in [21], model selection has a significant effect on domain generalization, and", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 191, + 506, + 204 + ], + "spans": [ + { + "bbox": [ + 105, + 191, + 506, + 204 + ], + "score": 1.0, + "content": "any OOD algorithm without a model selection criterion is not complete. [21] trained more than", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 201, + 505, + 214 + ], + "spans": [ + { + "bbox": [ + 105, + 201, + 505, + 214 + ], + "score": 1.0, + "content": "45,900 models with different algorithms, and results show that when traditional selection methods", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 213, + 506, + 226 + ], + "spans": [ + { + "bbox": [ + 105, + 213, + 506, + 226 + ], + "score": 1.0, + "content": "are applied, none of OOD algorithms can outperform ERM [58] by a significant margin. This result", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 224, + 506, + 236 + ], + "spans": [ + { + "bbox": [ + 105, + 224, + 506, + 236 + ], + "score": 1.0, + "content": "is not strange, since traditional selection methods focus mainly on (validation) accuracy, which is", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 235, + 506, + 249 + ], + "spans": [ + { + "bbox": [ + 105, + 235, + 506, + 249 + ], + "score": 1.0, + "content": "biased in OOD generalization [21, 63]. A very typical example is Colored MNIST [5], where the", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 246, + 505, + 259 + ], + "spans": [ + { + "bbox": [ + 106, + 246, + 505, + 259 + ], + "score": 1.0, + "content": "image is colored according to the label, but the relationship varies across domains. As explained in", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 257, + 505, + 270 + ], + "spans": [ + { + "bbox": [ + 105, + 257, + 459, + 270 + ], + "score": 1.0, + "content": "[5], ERM principle will only capture this spurious feature (color) and performs badly in", + "type": "text" + }, + { + "bbox": [ + 460, + 257, + 476, + 268 + ], + "score": 0.9, + "content": "{ \\mathcal { E } } _ { a l l }", + "type": "inline_equation" + }, + { + "bbox": [ + 476, + 257, + 505, + 270 + ], + "score": 1.0, + "content": ". Since", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 267, + 506, + 282 + ], + "spans": [ + { + "bbox": [ + 105, + 267, + 252, + 282 + ], + "score": 1.0, + "content": "ERM is exactly minimizing loss in", + "type": "text" + }, + { + "bbox": [ + 252, + 268, + 277, + 279 + ], + "score": 0.91, + "content": "\\mathcal { E } _ { a v a i l }", + "type": "inline_equation" + }, + { + "bbox": [ + 277, + 267, + 506, + 282 + ], + "score": 1.0, + "content": ", any model selection method using validation accuracy", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 278, + 506, + 291 + ], + "spans": [ + { + "bbox": [ + 105, + 278, + 506, + 291 + ], + "score": 1.0, + "content": "alone is likely to choose ERM rather than any other OOD algorithm [63]. Thus no algorithm will", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 290, + 313, + 302 + ], + "spans": [ + { + "bbox": [ + 105, + 290, + 313, + 302 + ], + "score": 1.0, + "content": "have a significant improvement compared to ERM.", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 12 + }, + { + "type": "text", + "bbox": [ + 106, + 306, + 505, + 382 + ], + "lines": [ + { + "bbox": [ + 104, + 305, + 506, + 319 + ], + "spans": [ + { + "bbox": [ + 104, + 305, + 506, + 319 + ], + "score": 1.0, + "content": "A natural question arises: what else can we use, in addition to accuracy? Theorem 4.1 points out that,", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 317, + 506, + 330 + ], + "spans": [ + { + "bbox": [ + 105, + 317, + 283, + 330 + ], + "score": 1.0, + "content": "learning feature with small variation across", + "type": "text" + }, + { + "bbox": [ + 284, + 317, + 308, + 328 + ], + "score": 0.9, + "content": "\\mathcal { E } _ { a v a i l }", + "type": "inline_equation" + }, + { + "bbox": [ + 309, + 317, + 506, + 330 + ], + "score": 1.0, + "content": "is important for decreasing OOD generalization", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 327, + 506, + 342 + ], + "spans": [ + { + "bbox": [ + 105, + 327, + 190, + 342 + ], + "score": 1.0, + "content": "error. Once a model", + "type": "text" + }, + { + "bbox": [ + 191, + 328, + 198, + 339 + ], + "score": 0.84, + "content": "f", + "type": "inline_equation" + }, + { + "bbox": [ + 198, + 327, + 268, + 342 + ], + "score": 1.0, + "content": "achieves a small", + "type": "text" + }, + { + "bbox": [ + 268, + 328, + 327, + 339 + ], + "score": 0.92, + "content": "\\mathcal { V } ^ { \\mathrm { s u p } } ( h , \\mathcal { E } _ { a v a i l } )", + "type": "inline_equation" + }, + { + "bbox": [ + 328, + 327, + 352, + 342 + ], + "score": 1.0, + "content": ", then", + "type": "text" + }, + { + "bbox": [ + 352, + 328, + 378, + 340 + ], + "score": 0.87, + "content": "\\operatorname { e r r } ( f )", + "type": "inline_equation" + }, + { + "bbox": [ + 378, + 327, + 506, + 342 + ], + "score": 1.0, + "content": "will be small. If the validation", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 339, + 506, + 352 + ], + "spans": [ + { + "bbox": [ + 105, + 339, + 506, + 352 + ], + "score": 1.0, + "content": "accuracy is also high, we shall know that the OOD accuracy will remain high. To this end, we propose", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 349, + 505, + 362 + ], + "spans": [ + { + "bbox": [ + 106, + 349, + 505, + 362 + ], + "score": 1.0, + "content": "our heuristic selection criterion (Algorithm 1). Instead of considering validation accuracy alone, we", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 360, + 506, + 373 + ], + "spans": [ + { + "bbox": [ + 105, + 360, + 506, + 373 + ], + "score": 1.0, + "content": "combine it with feature variation and select the model with high validation accuracy as well as low", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 372, + 147, + 383 + ], + "spans": [ + { + "bbox": [ + 106, + 372, + 147, + 383 + ], + "score": 1.0, + "content": "variation.", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 21 + }, + { + "type": "title", + "bbox": [ + 107, + 397, + 231, + 408 + ], + "lines": [ + { + "bbox": [ + 106, + 397, + 232, + 408 + ], + "spans": [ + { + "bbox": [ + 106, + 397, + 232, + 408 + ], + "score": 1.0, + "content": "Algorithm 1: Model Selection", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 25 + }, + { + "type": "text", + "bbox": [ + 106, + 410, + 496, + 521 + ], + "lines": 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f \\in { \\mathcal { M } } } ( \\operatorname { A c c } _ { f } - r _ { 0 } \\mathcal { V } _ { f } )", + "type": "inline_equation" + } + ], + "index": 34 + } + ], + "index": 30 + }, + { + "type": "text", + "bbox": [ + 107, + 538, + 505, + 585 + ], + "lines": [ + { + "bbox": [ + 106, + 538, + 505, + 551 + ], + "spans": [ + { + "bbox": [ + 106, + 538, + 505, + 551 + ], + "score": 1.0, + "content": "We briefly explain Algorithm 1 here. For each candidate model, we calculate its variation using the", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 104, + 549, + 507, + 565 + ], + "spans": [ + { + "bbox": [ + 104, + 549, + 267, + 565 + ], + "score": 1.0, + "content": "average of each feature’s variation, i.e.,", + "type": "text" + }, + { + "bbox": [ + 267, + 549, + 360, + 564 + ], + "score": 0.93, + "content": "\\begin{array} { r } { \\frac { 1 } { d } \\sum _ { i \\in [ d ] } \\mathcal { V } ( \\phi _ { i } , \\mathcal { X } _ { a v a i l } ) } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 360, + 549, + 507, + 565 + ], + "score": 1.0, + "content": ". When deriving the bounds, we use", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 106, + 561, + 505, + 576 + ], + "spans": [ + { + "bbox": [ + 106, + 562, + 125, + 573 + ], + "score": 0.82, + "content": "\\mathcal { V } ^ { \\mathrm { s u p } }", + "type": "inline_equation" + }, + { + "bbox": [ + 125, + 561, + 505, + 576 + ], + "score": 1.0, + "content": "instead of their average because we need to consider the worst case, i.e., the worst top model. 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Here", + "type": "text" + }, + { + "bbox": [ + 494, + 630, + 504, + 639 + ], + "score": 0.8, + "content": "r _ { 0 }", + "type": "inline_equation" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 638, + 506, + 651 + ], + "spans": [ + { + "bbox": [ + 105, + 638, + 386, + 651 + ], + "score": 1.0, + "content": "is a hyper-parameter representing the concrete relationship between", + "type": "text" + }, + { + "bbox": [ + 386, + 639, + 412, + 650 + ], + "score": 0.91, + "content": "\\operatorname { e r r } ( f )", + "type": "inline_equation" + }, + { + "bbox": [ + 412, + 638, + 431, + 651 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 432, + 639, + 444, + 651 + ], + "score": 0.88, + "content": "\\mathcal { V } _ { f }", + "type": "inline_equation" + }, + { + "bbox": [ + 444, + 638, + 506, + 651 + ], + "score": 1.0, + "content": ". Although we", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 649, + 506, + 662 + ], + "spans": [ + { + "bbox": [ + 105, + 649, + 506, + 662 + ], + "score": 1.0, + "content": "have already used one hyper-parameter to help select multiple hyper-parameter combinations, it", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 660, + 505, + 673 + ], + "spans": [ + { + "bbox": [ + 105, + 660, + 388, + 673 + ], + "score": 1.0, + "content": "is natural to ask whether we can further get rid of the selection of", + "type": "text" + }, + { + "bbox": [ + 389, + 662, + 399, + 671 + ], + "score": 0.86, + "content": "r _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 399, + 660, + 432, + 673 + ], + "score": 1.0, + "content": ". 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However, we can empirically estimate", + "type": "text" + }, + { + "bbox": [ + 469, + 684, + 479, + 693 + ], + "score": 0.84, + "content": "r _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 479, + 680, + 506, + 696 + ], + "score": 1.0, + "content": "using", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 106, + 693, + 505, + 712 + ], + "spans": [ + { + "bbox": [ + 106, + 693, + 176, + 712 + ], + "score": 0.93, + "content": "\\begin{array} { r } { r _ { 0 } = \\frac { \\mathrm { S t d } _ { f \\in \\hat { \\mathcal { M } } } \\mathrm { A c c } _ { f } } { \\mathrm { S t d } _ { f \\in \\hat { \\mathcal { M } } } \\mathcal { V } _ { f } } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 178, + 695, + 208, + 709 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 209, + 694, + 248, + 707 + ], + "score": 0.9, + "content": "\\hat { \\mathcal { M } } \\subset \\mathcal { M }", + "type": "inline_equation" + }, + { + "bbox": [ + 249, + 695, + 505, + 709 + ], + "score": 1.0, + "content": "is the model with not bad validation accuracy. We do not use", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 105, + 709, + 505, + 724 + ], + "spans": [ + { + "bbox": [ + 105, + 709, + 163, + 724 + ], + "score": 1.0, + "content": "the whole set", + "type": "text" + }, + { + "bbox": [ + 164, + 712, + 176, + 721 + ], + "score": 0.81, + "content": "\\mathcal { M }", + "type": "inline_equation" + }, + { + "bbox": [ + 177, + 709, + 505, + 724 + ], + "score": 1.0, + "content": "because some OOD algorithms will perform extremely bad when the penalty is", + "type": "text" + } + ], + "index": 48 + } + ], + "index": 44.5 + } + ], + "page_idx": 6, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 302, + 741, + 309, + 750 + ], + "lines": [ + { + "bbox": [ + 302, + 741, + 309, + 752 + ], + "spans": [ + { + "bbox": [ + 302, + 741, + 309, + 752 + ], + "score": 1.0, + "content": "7", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 106, + 72, + 505, + 139 + ], + "lines": [], + "index": 2.5, + "bbox_fs": [ + 104, + 73, + 506, + 140 + ], + "lines_deleted": true + }, + { + "type": "title", + "bbox": [ + 106, + 154, + 381, + 168 + ], + "lines": [ + { + "bbox": [ + 104, + 153, + 382, + 171 + ], + "spans": [ + { + "bbox": [ + 104, + 153, + 382, + 171 + ], + "score": 1.0, + "content": "5 Variation as a Factor of Model Selection Criterion", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 6 + }, + { + "type": "text", + "bbox": [ + 106, + 180, + 505, + 301 + ], + "lines": [ + { + "bbox": [ + 106, + 181, + 505, + 192 + ], + "spans": [ + { + "bbox": [ + 106, + 181, + 505, + 192 + ], + "score": 1.0, + "content": "As is pointed out in [21], model selection has a significant effect on domain generalization, and", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 191, + 506, + 204 + ], + "spans": [ + { + "bbox": [ + 105, + 191, + 506, + 204 + ], + "score": 1.0, + "content": "any OOD algorithm without a model selection criterion is not complete. [21] trained more than", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 201, + 505, + 214 + ], + "spans": [ + { + "bbox": [ + 105, + 201, + 505, + 214 + ], + "score": 1.0, + "content": "45,900 models with different algorithms, and results show that when traditional selection methods", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 213, + 506, + 226 + ], + "spans": [ + { + "bbox": [ + 105, + 213, + 506, + 226 + ], + "score": 1.0, + "content": "are applied, none of OOD algorithms can outperform ERM [58] by a significant margin. This result", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 224, + 506, + 236 + ], + "spans": [ + { + "bbox": [ + 105, + 224, + 506, + 236 + ], + "score": 1.0, + "content": "is not strange, since traditional selection methods focus mainly on (validation) accuracy, which is", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 235, + 506, + 249 + ], + "spans": [ + { + "bbox": [ + 105, + 235, + 506, + 249 + ], + "score": 1.0, + "content": "biased in OOD generalization [21, 63]. A very typical example is Colored MNIST [5], where the", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 246, + 505, + 259 + ], + "spans": [ + { + "bbox": [ + 106, + 246, + 505, + 259 + ], + "score": 1.0, + "content": "image is colored according to the label, but the relationship varies across domains. As explained in", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 257, + 505, + 270 + ], + "spans": [ + { + "bbox": [ + 105, + 257, + 459, + 270 + ], + "score": 1.0, + "content": "[5], ERM principle will only capture this spurious feature (color) and performs badly in", + "type": "text" + }, + { + "bbox": [ + 460, + 257, + 476, + 268 + ], + "score": 0.9, + "content": "{ \\mathcal { E } } _ { a l l }", + "type": "inline_equation" + }, + { + "bbox": [ + 476, + 257, + 505, + 270 + ], + "score": 1.0, + "content": ". Since", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 267, + 506, + 282 + ], + "spans": [ + { + "bbox": [ + 105, + 267, + 252, + 282 + ], + "score": 1.0, + "content": "ERM is exactly minimizing loss in", + "type": "text" + }, + { + "bbox": [ + 252, + 268, + 277, + 279 + ], + "score": 0.91, + "content": "\\mathcal { E } _ { a v a i l }", + "type": "inline_equation" + }, + { + "bbox": [ + 277, + 267, + 506, + 282 + ], + "score": 1.0, + "content": ", any model selection method using validation accuracy", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 278, + 506, + 291 + ], + "spans": [ + { + "bbox": [ + 105, + 278, + 506, + 291 + ], + "score": 1.0, + "content": "alone is likely to choose ERM rather than any other OOD algorithm [63]. Thus no algorithm will", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 290, + 313, + 302 + ], + "spans": [ + { + "bbox": [ + 105, + 290, + 313, + 302 + ], + "score": 1.0, + "content": "have a significant improvement compared to ERM.", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 12, + "bbox_fs": [ + 105, + 181, + 506, + 302 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 306, + 505, + 382 + ], + "lines": [ + { + "bbox": [ + 104, + 305, + 506, + 319 + ], + "spans": [ + { + "bbox": [ + 104, + 305, + 506, + 319 + ], + "score": 1.0, + "content": "A natural question arises: what else can we use, in addition to accuracy? Theorem 4.1 points out that,", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 317, + 506, + 330 + ], + "spans": [ + { + "bbox": [ + 105, + 317, + 283, + 330 + ], + "score": 1.0, + "content": "learning feature with small variation across", + "type": "text" + }, + { + "bbox": [ + 284, + 317, + 308, + 328 + ], + "score": 0.9, + "content": "\\mathcal { E } _ { a v a i l }", + "type": "inline_equation" + }, + { + "bbox": [ + 309, + 317, + 506, + 330 + ], + "score": 1.0, + "content": "is important for decreasing OOD generalization", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 327, + 506, + 342 + ], + "spans": [ + { + "bbox": [ + 105, + 327, + 190, + 342 + ], + "score": 1.0, + "content": "error. Once a model", + "type": "text" + }, + { + "bbox": [ + 191, + 328, + 198, + 339 + ], + "score": 0.84, + "content": "f", + "type": "inline_equation" + }, + { + "bbox": [ + 198, + 327, + 268, + 342 + ], + "score": 1.0, + "content": "achieves a small", + "type": "text" + }, + { + "bbox": [ + 268, + 328, + 327, + 339 + ], + "score": 0.92, + "content": "\\mathcal { V } ^ { \\mathrm { s u p } } ( h , \\mathcal { E } _ { a v a i l } )", + "type": "inline_equation" + }, + { + "bbox": [ + 328, + 327, + 352, + 342 + ], + "score": 1.0, + "content": ", then", + "type": "text" + }, + { + "bbox": [ + 352, + 328, + 378, + 340 + ], + "score": 0.87, + "content": "\\operatorname { e r r } ( f )", + "type": "inline_equation" + }, + { + "bbox": [ + 378, + 327, + 506, + 342 + ], + "score": 1.0, + "content": "will be small. If the validation", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 339, + 506, + 352 + ], + "spans": [ + { + "bbox": [ + 105, + 339, + 506, + 352 + ], + "score": 1.0, + "content": "accuracy is also high, we shall know that the OOD accuracy will remain high. To this end, we propose", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 349, + 505, + 362 + ], + "spans": [ + { + "bbox": [ + 106, + 349, + 505, + 362 + ], + "score": 1.0, + "content": "our heuristic selection criterion (Algorithm 1). Instead of considering validation accuracy alone, we", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 360, + 506, + 373 + ], + "spans": [ + { + "bbox": [ + 105, + 360, + 506, + 373 + ], + "score": 1.0, + "content": "combine it with feature variation and select the model with high validation accuracy as well as low", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 372, + 147, + 383 + ], + "spans": [ + { + "bbox": [ + 106, + 372, + 147, + 383 + ], + "score": 1.0, + "content": "variation.", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 21, + "bbox_fs": [ + 104, + 305, + 506, + 383 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 397, + 231, + 408 + ], + "lines": [ + { + "bbox": [ + 106, + 397, + 232, + 408 + ], + "spans": [ + { + "bbox": [ + 106, + 397, + 232, + 408 + ], + "score": 1.0, + "content": "Algorithm 1: Model Selection", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 25 + }, + { + "type": "list", + 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Since high validation means large", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 84, + 506, + 98 + ], + "spans": [ + { + "bbox": [ + 105, + 84, + 287, + 98 + ], + "score": 1.0, + "content": "informativeness in learned features, the use of", + "type": "text" + }, + { + "bbox": [ + 288, + 84, + 300, + 95 + ], + "score": 0.88, + "content": "\\hat { \\mathcal { M } }", + "type": "inline_equation" + }, + { + "bbox": [ + 300, + 84, + 506, + 98 + ], + "score": 1.0, + "content": "is an implicit application of informative assumption.", + "type": "text" + } + ], + "index": 1 + } + ], + "index": 0.5 + }, + { + "type": "text", + "bbox": [ + 107, + 101, + 505, + 135 + ], + "lines": [ + { + "bbox": [ + 105, + 101, + 506, + 115 + ], + "spans": [ + { + "bbox": [ + 105, + 101, + 506, + 115 + ], + "score": 1.0, + "content": "As shown in Section 6.1, our method can select models with higher OOD accuracy in various OOD", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 106, + 113, + 505, + 125 + ], + "spans": [ + { + "bbox": [ + 106, + 113, + 505, + 125 + ], + "score": 1.0, + "content": "datasets. We also explain in Appendix 3 why our method can outperform the traditional method in", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 106, + 123, + 506, + 136 + ], + "spans": [ + { + "bbox": [ + 106, + 123, + 506, + 136 + ], + "score": 1.0, + "content": "Color MNIST, where the dataset is hand-make and simple enough to calculate the expansion function.", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 3 + }, + { + "type": "title", + "bbox": [ + 107, + 149, + 191, + 163 + ], + "lines": [ + { + "bbox": [ + 104, + 147, + 193, + 166 + ], + "spans": [ + { + "bbox": [ + 104, + 147, + 193, + 166 + ], + "score": 1.0, + "content": "6 Experiments", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 5 + }, + { + "type": "text", + "bbox": [ + 107, + 173, + 505, + 250 + ], + "lines": [ + { + "bbox": [ + 106, + 173, + 505, + 186 + ], + "spans": [ + { + "bbox": [ + 106, + 173, + 505, + 186 + ], + "score": 1.0, + "content": "In this section, we conduct experiments to compare our model selection criterion (Section 5) with", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 183, + 506, + 196 + ], + "spans": [ + { + "bbox": [ + 105, + 183, + 506, + 196 + ], + "score": 1.0, + "content": "the baseline method5 [21]. Since both the variation and informativeness in Definition 3.1 are based", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 194, + 505, + 209 + ], + "spans": [ + { + "bbox": [ + 105, + 194, + 505, + 209 + ], + "score": 1.0, + "content": "on one-dimensional features, we can directly estimate these quantities feature-by-feature and design", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 207, + 505, + 218 + ], + "spans": [ + { + "bbox": [ + 106, + 207, + 505, + 218 + ], + "score": 1.0, + "content": "model selection method based on them. 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We train models on different datasets, different", + "type": "text" + }, + { + "bbox": [ + 435, + 294, + 460, + 305 + ], + "score": 0.9, + "content": "\\mathcal { E } _ { a v a i l }", + "type": "inline_equation" + }, + { + "bbox": [ + 460, + 294, + 506, + 307 + ], + "score": 1.0, + "content": ", and select", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 304, + 506, + 317 + ], + "spans": [ + { + "bbox": [ + 105, + 304, + 506, + 317 + ], + "score": 1.0, + "content": "models according to a different selection criteria. We then compare the OOD accuracy of selected", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 316, + 140, + 327 + ], + "spans": [ + { + "bbox": [ + 106, + 316, + 140, + 327 + ], + "score": 1.0, + "content": "models.", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 15.5 + }, + { + "type": "text", + "bbox": [ + 106, + 337, + 506, + 463 + ], + "lines": [ + { + "bbox": [ + 105, + 337, + 506, + 350 + ], + "spans": [ + { + "bbox": [ + 105, + 337, + 506, + 350 + ], + "score": 1.0, + "content": "Settings We train our model on three benchmark OOD datasets (PACS [34], OfficeHome [59],", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 348, + 506, + 363 + ], + "spans": [ + { + "bbox": [ + 105, + 348, + 319, + 363 + ], + "score": 1.0, + "content": "VLCS [57]) and consider all possible selections of", + "type": "text" + }, + { + "bbox": [ + 320, + 349, + 371, + 361 + ], + "score": 0.9, + "content": "( \\mathcal { E } _ { a v a i l } , \\mathcal { E } _ { a l l } )", + "type": "inline_equation" + }, + { + "bbox": [ + 372, + 348, + 506, + 363 + ], + "score": 1.0, + "content": ". 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After training, we employ", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 392, + 506, + 406 + ], + "spans": [ + { + "bbox": [ + 105, + 392, + 506, + 406 + ], + "score": 1.0, + "content": "different selection methods and compare the OOD accuracy of the selected models. As stated in", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 403, + 505, + 417 + ], + "spans": [ + { + "bbox": [ + 106, + 405, + 283, + 417 + ], + "score": 1.0, + "content": "Section 5, we use the standard deviation of", + "type": "text" + }, + { + "bbox": [ + 284, + 405, + 292, + 415 + ], + "score": 0.79, + "content": "\\nu", + "type": "inline_equation" + }, + { + "bbox": [ + 293, + 405, + 402, + 417 + ], + "score": 1.0, + "content": "and validation accuracy in", + "type": "text" + }, + { + "bbox": [ + 402, + 403, + 416, + 415 + ], + "score": 0.86, + "content": "\\hat { \\mathcal { M } }", + "type": "inline_equation" + }, + { + "bbox": [ + 416, + 405, + 463, + 417 + ], + "score": 1.0, + "content": "to estimate", + "type": "text" + }, + { + "bbox": [ + 463, + 407, + 474, + 416 + ], + "score": 0.84, + "content": "r _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 474, + 405, + 505, + 417 + ], + "score": 1.0, + "content": ", where", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 416, + 506, + 432 + ], + "spans": [ + { + "bbox": [ + 106, + 416, + 295, + 432 + ], + "score": 0.9, + "content": "{ \\hat { \\mathcal { M } } } = \\{ f \\in { \\mathcal { M } } : \\operatorname { A c c } _ { f } \\geq \\operatorname* { m a x } _ { \\hat { f } } \\operatorname { A c c } _ { \\hat { f } } - 0 . 1 \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 296, + 416, + 388, + 432 + ], + "score": 1.0, + "content": ". Note that calculating", + "type": "text" + }, + { + "bbox": [ + 388, + 417, + 443, + 430 + ], + "score": 0.92, + "content": "\\mathcal { V } ( \\phi _ { i } , \\mathcal { X } _ { a v a i l } )", + "type": "inline_equation" + }, + { + "bbox": [ + 444, + 416, + 506, + 432 + ], + "score": 1.0, + "content": "takes calculus", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 430, + 506, + 443 + ], + "spans": [ + { + "bbox": [ + 106, + 430, + 506, + 443 + ], + "score": 1.0, + "content": "many times, so we design a parallel GPU kernel density estimation to speed up the whole process a", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 440, + 507, + 455 + ], + "spans": [ + { + "bbox": [ + 105, + 440, + 507, + 455 + ], + "score": 1.0, + "content": "hundred times and manage to finish one model in seconds. For more details about the experiments,", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 453, + 173, + 464 + ], + "spans": [ + { + "bbox": [ + 105, + 453, + 173, + 464 + ], + "score": 1.0, + "content": "see Appendix 4.", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 23 + }, + { + "type": "table", + "bbox": [ + 123, + 510, + 488, + 614 + ], + "blocks": [ + { + "type": "table_caption", + "bbox": [ + 106, + 478, + 504, + 502 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 105, + 477, + 505, + 492 + ], + "spans": [ + { + "bbox": [ + 105, + 477, + 505, + 492 + ], + "score": 1.0, + "content": "Table 1: Model Selection Result. “Env” denotes the unseen domain during training. “Val” denotes", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 488, + 350, + 505 + ], + "spans": [ + { + "bbox": [ + 105, + 488, + 350, + 505 + ], + "score": 1.0, + "content": "the OOD accuracy of model selected by validation accuracy.", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 29.5 + }, + { + "type": "table_body", + "bbox": [ + 123, + 510, + 488, + 614 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 123, + 510, + 488, + 614 + ], + "spans": [ + { + "bbox": [ + 123, + 510, + 488, + 614 + ], + "score": 0.984, + "html": "
PACSEnvACPSavgacc inc
Val85.20%80.42%96.17%77.86%84.91%1
Ours88.72%81.74%96.83%79.00%86.57%1.66%↑
OfficeHomeEnvACPRavgacc inc
Val61.85%55.56%74.72%76.25%67.09%-
Ours65.76%55.07%75.20%76.31%68.09%1.00%↑
VLCSEnvCLSVavgacc inc
Val97.46%64.83%69.50%670.97%75.69%1
Ours97.81%66.98%69.50%70.97%76.32%0.63%↑
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For each environment setup, we select", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 639, + 507, + 653 + ], + "spans": [ + { + "bbox": [ + 105, + 639, + 507, + 653 + ], + "score": 1.0, + "content": "the best model according to Algorithm 1 and validation accuracy. The results show that on all datasets,", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 650, + 507, + 664 + ], + "spans": [ + { + "bbox": [ + 105, + 650, + 507, + 664 + ], + "score": 1.0, + "content": "our selection criterion significantly outperforms the validation accuracy in average OOD accuracy.", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 659, + 507, + 676 + ], + "spans": [ + { + "bbox": [ + 105, + 659, + 507, + 676 + ], + "score": 1.0, + "content": "For a more detailed comparison, our method improves the OOD accuracy in most of the 12 setups.", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 672, + 501, + 686 + ], + "spans": [ + { + "bbox": [ + 105, + 672, + 501, + 686 + ], + "score": 1.0, + "content": "Our experiments demonstrate that our criterion can help select models with higher OOD accuracy.", + "type": "text" + } + ], + "index": 38 + } + ], + "index": 36 + } + ], + "page_idx": 7, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 106, + 690, + 505, + 722 + ], + "lines": [ + { + "bbox": [ + 118, + 689, + 505, + 703 + ], + "spans": [ + { + "bbox": [ + 118, + 689, + 505, + 703 + ], + "score": 1.0, + "content": "5Our experiments is conducted in DomainBed: https://github.com/facebookresearch/DomainBed.", + "type": "text" + } + ] + }, + { + "bbox": [ + 118, + 699, + 506, + 713 + ], + "spans": [ + { + "bbox": [ + 118, + 699, + 506, + 713 + ], + "score": 1.0, + "content": "6Notice that some OOD accuracy are the same in the two methods since the same model is selected. 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PACSEnvACPSavgacc inc
Val85.20%80.42%96.17%77.86%84.91%1
Ours88.72%81.74%96.83%79.00%86.57%1.66%↑
OfficeHomeEnvACPRavgacc inc
Val61.85%55.56%74.72%76.25%67.09%-
Ours65.76%55.07%75.20%76.31%68.09%1.00%↑
VLCSEnvCLSVavgacc inc
Val97.46%64.83%69.50%670.97%75.69%1
Ours97.81%66.98%69.50%70.97%76.32%0.63%↑
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There are approximately 2 million points in", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 106, + 215, + 505, + 227 + ], + "spans": [ + { + "bbox": [ + 106, + 215, + 505, + 227 + ], + "score": 1.0, + "content": "each image, with each point representing a feature, and its color represents its informativeness. The", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 225, + 505, + 238 + ], + "spans": [ + { + "bbox": [ + 106, + 225, + 402, + 238 + ], + "score": 1.0, + "content": "solid red line stands for the expansion function under the corresponding", + "type": "text" + }, + { + "bbox": [ + 402, + 225, + 408, + 235 + ], + "score": 0.51, + "content": "\\delta", + "type": "inline_equation" + }, + { + "bbox": [ + 409, + 225, + 440, + 238 + ], + "score": 1.0, + "content": ". 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When", + "type": "text" + }, + { + "bbox": [ + 255, + 236, + 279, + 246 + ], + "score": 0.9, + "content": "\\delta = 0", + "type": "inline_equation" + }, + { + "bbox": [ + 280, + 236, + 463, + 248 + ], + "score": 1.0, + "content": ", no expansion function can make it learnable.", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 5 + } + ], + "index": 3.0 + }, + { + "type": "text", + "bbox": [ + 107, + 259, + 505, + 326 + ], + "lines": [ + { + "bbox": [ + 105, + 259, + 505, + 272 + ], + "spans": [ + { + "bbox": [ + 105, + 259, + 505, + 272 + ], + "score": 1.0, + "content": "One may wonder if the expansion function really exists and what it will look like for a real-world", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 270, + 505, + 283 + ], + "spans": [ + { + "bbox": [ + 105, + 270, + 505, + 283 + ], + "score": 1.0, + "content": "OOD generalization task. In this section, we consider the OOD dataset Office-Home [59]. We", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 281, + 506, + 294 + ], + "spans": [ + { + "bbox": [ + 105, + 281, + 249, + 294 + ], + "score": 1.0, + "content": "explicitly plot millions of features’", + "type": "text" + }, + { + "bbox": [ + 249, + 281, + 302, + 294 + ], + "score": 0.92, + "content": "\\mathcal { V } _ { \\rho } \\big ( \\phi , \\mathcal { E } _ { a v a i l } \\big )", + "type": "inline_equation" + }, + { + "bbox": [ + 303, + 281, + 321, + 294 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 321, + 281, + 366, + 294 + ], + "score": 0.93, + "content": "\\mathcal { V } _ { \\rho } \\big ( \\phi , \\mathcal { E } _ { a l l } \\big )", + "type": "inline_equation" + }, + { + "bbox": [ + 366, + 281, + 450, + 294 + ], + "score": 1.0, + "content": "with Total Variation", + "type": "text" + }, + { + "bbox": [ + 450, + 283, + 456, + 293 + ], + "score": 0.8, + "content": "\\rho", + "type": "inline_equation" + }, + { + "bbox": [ + 457, + 281, + 506, + 294 + ], + "score": 1.0, + "content": "to see what", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 293, + 505, + 304 + ], + "spans": [ + { + "bbox": [ + 106, + 293, + 505, + 304 + ], + "score": 1.0, + "content": "the expansion function is like in this task. We take the architecture as ResNet-50 [23], and we trained", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 302, + 505, + 316 + ], + "spans": [ + { + "bbox": [ + 105, + 302, + 505, + 316 + ], + "score": 1.0, + "content": "thousands of models with more than five algorithms, obtaining about 2 million features. The results", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 313, + 170, + 326 + ], + "spans": [ + { + "bbox": [ + 105, + 313, + 170, + 326 + ], + "score": 1.0, + "content": "are in Figure 1.", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 10.5 + }, + { + "type": "text", + "bbox": [ + 107, + 336, + 505, + 392 + ], + "lines": [ + { + "bbox": [ + 105, + 334, + 504, + 351 + ], + "spans": [ + { + "bbox": [ + 105, + 334, + 160, + 351 + ], + "score": 1.0, + "content": "Existence of", + "type": "text" + }, + { + "bbox": [ + 160, + 337, + 176, + 349 + ], + "score": 0.9, + "content": "s ( \\cdot )", + "type": "inline_equation" + }, + { + "bbox": [ + 177, + 334, + 214, + 351 + ], + "score": 1.0, + "content": ". When", + "type": "text" + }, + { + "bbox": [ + 214, + 337, + 238, + 347 + ], + "score": 0.89, + "content": "\\delta = 0", + "type": "inline_equation" + }, + { + "bbox": [ + 238, + 334, + 478, + 351 + ], + "score": 1.0, + "content": ", some non-informative features are nearly 0-invariant across", + "type": "text" + }, + { + "bbox": [ + 479, + 337, + 504, + 348 + ], + "score": 0.86, + "content": "\\mathcal { E } _ { a v a i l }", + "type": "inline_equation" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 348, + 506, + 360 + ], + "spans": [ + { + "bbox": [ + 105, + 348, + 198, + 360 + ], + "score": 1.0, + "content": "but are varying across", + "type": "text" + }, + { + "bbox": [ + 198, + 348, + 214, + 359 + ], + "score": 0.89, + "content": "{ \\mathcal { E } } _ { a l l }", + "type": "inline_equation" + }, + { + "bbox": [ + 214, + 348, + 506, + 360 + ], + "score": 1.0, + "content": ", so no expansion function can make this task learnable, i.e., this task is", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 357, + 505, + 371 + ], + "spans": [ + { + "bbox": [ + 105, + 357, + 129, + 371 + ], + "score": 1.0, + "content": "NOT", + "type": "text" + }, + { + "bbox": [ + 130, + 358, + 162, + 371 + ], + "score": 0.9, + "content": "( s ( \\cdot ) , 0 )", + "type": "inline_equation" + }, + { + "bbox": [ + 163, + 357, + 303, + 371 + ], + "score": 1.0, + "content": "for any expansion function. But as", + "type": "text" + }, + { + "bbox": [ + 304, + 359, + 310, + 369 + ], + "score": 0.77, + "content": "\\delta", + "type": "inline_equation" + }, + { + "bbox": [ + 311, + 357, + 505, + 371 + ], + "score": 1.0, + "content": "increases, only informative features are left, and", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 370, + 505, + 383 + ], + "spans": [ + { + "bbox": [ + 105, + 370, + 220, + 383 + ], + "score": 1.0, + "content": "now we can find appropriate", + "type": "text" + }, + { + "bbox": [ + 221, + 370, + 236, + 382 + ], + "score": 0.92, + "content": "s ( \\cdot )", + "type": "inline_equation" + }, + { + "bbox": [ + 237, + 370, + 488, + 383 + ], + "score": 1.0, + "content": "to make it learnable. We can clearly realize from the figure that", + "type": "text" + }, + { + "bbox": [ + 489, + 370, + 505, + 382 + ], + "score": 0.89, + "content": "s ( \\cdot )", + "type": "inline_equation" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 380, + 205, + 393 + ], + "spans": [ + { + "bbox": [ + 105, + 380, + 164, + 393 + ], + "score": 1.0, + "content": "do exist when", + "type": "text" + }, + { + "bbox": [ + 164, + 381, + 201, + 392 + ], + "score": 0.91, + "content": "\\delta \\geq 0 . 1 5", + "type": "inline_equation" + }, + { + "bbox": [ + 201, + 380, + 205, + 393 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 16 + }, + { + "type": "text", + "bbox": [ + 106, + 403, + 505, + 513 + ], + "lines": [ + { + "bbox": [ + 106, + 403, + 504, + 415 + ], + "spans": [ + { + "bbox": [ + 106, + 403, + 187, + 415 + ], + "score": 1.0, + "content": "Trade-off between", + "type": "text" + }, + { + "bbox": [ + 187, + 403, + 204, + 415 + ], + "score": 0.9, + "content": "s ( \\cdot )", + "type": "inline_equation" + }, + { + "bbox": [ + 204, + 403, + 225, + 415 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 225, + 403, + 231, + 413 + ], + "score": 0.61, + "content": "\\delta", + "type": "inline_equation" + }, + { + "bbox": [ + 231, + 403, + 426, + 415 + ], + "score": 1.0, + "content": ". The second phenomenon is that the slope of", + "type": "text" + }, + { + "bbox": [ + 426, + 403, + 443, + 415 + ], + "score": 0.9, + "content": "s ( \\cdot )", + "type": "inline_equation" + }, + { + "bbox": [ + 443, + 403, + 497, + 415 + ], + "score": 1.0, + "content": "decreases as", + "type": "text" + }, + { + "bbox": [ + 498, + 403, + 504, + 413 + ], + "score": 0.66, + "content": "\\delta", + "type": "inline_equation" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 414, + 505, + 426 + ], + "spans": [ + { + "bbox": [ + 106, + 415, + 269, + 426 + ], + "score": 1.0, + "content": "increases, showing a trade-off between", + "type": "text" + }, + { + "bbox": [ + 270, + 414, + 286, + 426 + ], + "score": 0.89, + "content": "s ( \\cdot )", + "type": "inline_equation" + }, + { + "bbox": [ + 286, + 415, + 305, + 426 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 305, + 415, + 311, + 424 + ], + "score": 0.63, + "content": "\\delta", + "type": "inline_equation" + }, + { + "bbox": [ + 312, + 415, + 505, + 426 + ], + "score": 1.0, + "content": ". Although this trade-off comes naturally from", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 425, + 505, + 438 + ], + "spans": [ + { + "bbox": [ + 106, + 425, + 429, + 438 + ], + "score": 1.0, + "content": "the definition of learnability, it has a deep meaning. As is shown in Section 4,", + "type": "text" + }, + { + "bbox": [ + 430, + 425, + 456, + 437 + ], + "score": 0.82, + "content": "\\operatorname { e r r } ( f )", + "type": "inline_equation" + }, + { + "bbox": [ + 456, + 425, + 505, + 438 + ], + "score": 1.0, + "content": "is bounded", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 436, + 505, + 449 + ], + "spans": [ + { + "bbox": [ + 106, + 436, + 119, + 449 + ], + "score": 1.0, + "content": "by", + "type": "text" + }, + { + "bbox": [ + 120, + 436, + 153, + 448 + ], + "score": 0.92, + "content": "O ( s ( \\varepsilon ) )", + "type": "inline_equation" + }, + { + "bbox": [ + 153, + 436, + 182, + 449 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 183, + 438, + 189, + 446 + ], + "score": 0.77, + "content": "\\varepsilon", + "type": "inline_equation" + }, + { + "bbox": [ + 190, + 436, + 505, + 449 + ], + "score": 1.0, + "content": "is the variation of the model. To make the bound tighter, a natural idea is to", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 447, + 506, + 460 + ], + "spans": [ + { + "bbox": [ + 106, + 447, + 169, + 460 + ], + "score": 1.0, + "content": "choose a flatter", + "type": "text" + }, + { + "bbox": [ + 169, + 447, + 185, + 459 + ], + "score": 0.89, + "content": "s ( \\cdot )", + "type": "inline_equation" + }, + { + "bbox": [ + 186, + 447, + 263, + 460 + ], + "score": 1.0, + "content": ". However, a flatter", + "type": "text" + }, + { + "bbox": [ + 263, + 447, + 279, + 459 + ], + "score": 0.89, + "content": "s ( \\cdot )", + "type": "inline_equation" + }, + { + "bbox": [ + 279, + 447, + 373, + 460 + ], + "score": 1.0, + "content": "corresponds to a larger", + "type": "text" + }, + { + "bbox": [ + 374, + 448, + 380, + 457 + ], + "score": 0.59, + "content": "\\delta", + "type": "inline_equation" + }, + { + "bbox": [ + 380, + 447, + 506, + 460 + ], + "score": 1.0, + "content": ". Typically, learning a model to", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 458, + 505, + 470 + ], + "spans": [ + { + "bbox": [ + 105, + 458, + 505, + 470 + ], + "score": 1.0, + "content": "meet this higher informativeness requirement is more difficult, and it is possible that the algorithm", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 469, + 505, + 480 + ], + "spans": [ + { + "bbox": [ + 106, + 469, + 505, + 480 + ], + "score": 1.0, + "content": "achieves this by capturing more domain-specific features, which will therefore increase the variation", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 479, + 506, + 492 + ], + "spans": [ + { + "bbox": [ + 105, + 479, + 160, + 492 + ], + "score": 1.0, + "content": "of the model,", + "type": "text" + }, + { + "bbox": [ + 161, + 482, + 167, + 489 + ], + "score": 0.65, + "content": "\\varepsilon", + "type": "inline_equation" + }, + { + "bbox": [ + 167, + 479, + 313, + 492 + ], + "score": 1.0, + "content": ". As a result, we are not sure whether", + "type": "text" + }, + { + "bbox": [ + 313, + 479, + 331, + 492 + ], + "score": 0.9, + "content": "s ( \\varepsilon )", + "type": "inline_equation" + }, + { + "bbox": [ + 331, + 479, + 506, + 492 + ], + "score": 1.0, + "content": "will increase or decrease. We believe this is", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 491, + 505, + 503 + ], + "spans": [ + { + "bbox": [ + 106, + 491, + 505, + 503 + ], + "score": 1.0, + "content": "also the essence of model selection: i.e., to trade-off between the variation and informativeness of a", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 502, + 255, + 513 + ], + "spans": [ + { + "bbox": [ + 106, + 502, + 255, + 513 + ], + "score": 1.0, + "content": "model, which is done in Formula 11.", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 23.5 + }, + { + "type": "title", + "bbox": [ + 107, + 527, + 232, + 541 + ], + "lines": [ + { + "bbox": [ + 105, + 526, + 234, + 543 + ], + "spans": [ + { + "bbox": [ + 105, + 526, + 234, + 543 + ], + "score": 1.0, + "content": "7 More Related Works", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 29 + }, + { + "type": "text", + "bbox": [ + 106, + 552, + 505, + 651 + ], + "lines": [ + { + "bbox": [ + 106, + 552, + 506, + 565 + ], + "spans": [ + { + "bbox": [ + 106, + 552, + 506, + 565 + ], + "score": 1.0, + "content": "Domain generalization [12, 39], or OOD generalization, has drawn much attention recently [21, 30].", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 563, + 506, + 576 + ], + "spans": [ + { + "bbox": [ + 106, + 563, + 506, + 576 + ], + "score": 1.0, + "content": "The goal is to learn a model from several training domains and expect good performance on unseen", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 574, + 506, + 587 + ], + "spans": [ + { + "bbox": [ + 105, + 574, + 506, + 587 + ], + "score": 1.0, + "content": "test domains. [60, 64] offer a comprehensive survey. A popular solution is to extract domain-invariant", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 585, + 506, + 597 + ], + "spans": [ + { + "bbox": [ + 106, + 585, + 506, + 597 + ], + "score": 1.0, + "content": "feature representation. [45] and [49] proved that when the model is linear, the invariance under", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 597, + 506, + 608 + ], + "spans": [ + { + "bbox": [ + 106, + 597, + 506, + 608 + ], + "score": 1.0, + "content": "training domains can help discover invariant features on test domains. [5] introduces the invariant", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 607, + 505, + 619 + ], + "spans": [ + { + "bbox": [ + 105, + 607, + 505, + 619 + ], + "score": 1.0, + "content": "prediction into neural networks and proposes a practical objective function. After that, a lot of works", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 618, + 504, + 630 + ], + "spans": [ + { + "bbox": [ + 106, + 618, + 504, + 630 + ], + "score": 1.0, + "content": "arise from the view of causal discovery, distributional robustness and conditional independence", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 106, + 628, + 506, + 642 + ], + "spans": [ + { + "bbox": [ + 106, + 628, + 506, + 642 + ], + "score": 1.0, + "content": "[1, 7, 16, 15, 26, 32, 33, 43, 51, 61]. On the other hand, some works point out the weakness of", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 106, + 640, + 456, + 652 + ], + "spans": [ + { + "bbox": [ + 106, + 640, + 456, + 652 + ], + "score": 1.0, + "content": "existing methods from the theoretical and experimental perspectives [2, 21, 29, 41, 50].", + "type": "text" + } + ], + "index": 38 + } + ], + "index": 34 + }, + { + "type": "text", + "bbox": [ + 107, + 656, + 505, + 722 + ], + "lines": [ + { + "bbox": [ + 106, + 656, + 505, + 668 + ], + "spans": [ + { + "bbox": [ + 106, + 656, + 505, + 668 + ], + "score": 1.0, + "content": "The OOD generalization requires restrictions on how the target domains may differ. A straightforward", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 667, + 506, + 680 + ], + "spans": [ + { + "bbox": [ + 105, + 667, + 506, + 680 + ], + "score": 1.0, + "content": "approach is to define a set of test domains around the training domain using some distribution distance", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 106, + 678, + 505, + 689 + ], + "spans": [ + { + "bbox": [ + 106, + 678, + 505, + 689 + ], + "score": 1.0, + "content": "measure [6, 13, 19, 25, 51, 53, 54, 61]. Another feasible route is the causal framework which is robust", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 106, + 689, + 506, + 700 + ], + "spans": [ + { + "bbox": [ + 106, + 689, + 506, + 700 + ], + "score": 1.0, + "content": "to the test distributions caused by interventions[44, 46] on variables, e.g., [5, 24, 36, 37, 40, 47, 49, 52].", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 106, + 700, + 506, + 712 + ], + "spans": [ + { + "bbox": [ + 106, + 700, + 506, + 712 + ], + "score": 1.0, + "content": "The principle of these methods is that a causal model is invariant and can achieve the minimal worst-", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 106, + 711, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 106, + 711, + 505, + 722 + ], + "score": 1.0, + "content": "case risk [4, 22, 44, 49]. 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The", + "type": "text" + }, + { + "bbox": [ + 478, + 194, + 484, + 203 + ], + "score": 0.46, + "content": "\\mathbf { X } ^ { } -", + "type": "inline_equation" + }, + { + "bbox": [ + 485, + 192, + 505, + 205 + ], + "score": 1.0, + "content": "-axis", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 202, + 506, + 218 + ], + "spans": [ + { + "bbox": [ + 105, + 202, + 149, + 218 + ], + "score": 1.0, + "content": "stands for", + "type": "text" + }, + { + "bbox": [ + 150, + 203, + 199, + 215 + ], + "score": 0.93, + "content": "\\mathcal { V } ( \\phi , \\mathcal { E } _ { a v a i l } )", + "type": "inline_equation" + }, + { + "bbox": [ + 200, + 202, + 276, + 218 + ], + "score": 1.0, + "content": "and the y-axis for", + "type": "text" + }, + { + "bbox": [ + 277, + 203, + 317, + 216 + ], + "score": 0.93, + "content": "\\mathcal { V } ( \\phi , \\mathcal { E } _ { a l l } )", + "type": "inline_equation" + }, + { + "bbox": [ + 318, + 202, + 506, + 218 + ], + "score": 1.0, + "content": ". There are approximately 2 million points in", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 106, + 215, + 505, + 227 + ], + "spans": [ + { + "bbox": [ + 106, + 215, + 505, + 227 + ], + "score": 1.0, + "content": "each image, with each point representing a feature, and its color represents its informativeness. The", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 225, + 505, + 238 + ], + "spans": [ + { + "bbox": [ + 106, + 225, + 402, + 238 + ], + "score": 1.0, + "content": "solid red line stands for the expansion function under the corresponding", + "type": "text" + }, + { + "bbox": [ + 402, + 225, + 408, + 235 + ], + "score": 0.51, + "content": "\\delta", + "type": "inline_equation" + }, + { + "bbox": [ + 409, + 225, + 440, + 238 + ], + "score": 1.0, + "content": ". When", + "type": "text" + }, + { + "bbox": [ + 440, + 225, + 447, + 235 + ], + "score": 0.68, + "content": "\\delta", + "type": "inline_equation" + }, + { + "bbox": [ + 447, + 225, + 505, + 238 + ], + "score": 1.0, + "content": "increases, the", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 236, + 463, + 248 + ], + "spans": [ + { + "bbox": [ + 106, + 236, + 255, + 248 + ], + "score": 1.0, + "content": "expansion function decreases. When", + "type": "text" + }, + { + "bbox": [ + 255, + 236, + 279, + 246 + ], + "score": 0.9, + "content": "\\delta = 0", + "type": "inline_equation" + }, + { + "bbox": [ + 280, + 236, + 463, + 248 + ], + "score": 1.0, + "content": ", no expansion function can make it learnable.", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 5 + } + ], + "index": 3.0 + }, + { + "type": "text", + "bbox": [ + 107, + 259, + 505, + 326 + ], + "lines": [ + { + "bbox": [ + 105, + 259, + 505, + 272 + ], + "spans": [ + { + "bbox": [ + 105, + 259, + 505, + 272 + ], + "score": 1.0, + "content": "One may wonder if the expansion function really exists and what it will look like for a real-world", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 270, + 505, + 283 + ], + "spans": [ + { + "bbox": [ + 105, + 270, + 505, + 283 + ], + "score": 1.0, + "content": "OOD generalization task. In this section, we consider the OOD dataset Office-Home [59]. We", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 281, + 506, + 294 + ], + "spans": [ + { + "bbox": [ + 105, + 281, + 249, + 294 + ], + "score": 1.0, + "content": "explicitly plot millions of features’", + "type": "text" + }, + { + "bbox": [ + 249, + 281, + 302, + 294 + ], + "score": 0.92, + "content": "\\mathcal { V } _ { \\rho } \\big ( \\phi , \\mathcal { E } _ { a v a i l } \\big )", + "type": "inline_equation" + }, + { + "bbox": [ + 303, + 281, + 321, + 294 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 321, + 281, + 366, + 294 + ], + "score": 0.93, + "content": "\\mathcal { V } _ { \\rho } \\big ( \\phi , \\mathcal { E } _ { a l l } \\big )", + "type": "inline_equation" + }, + { + "bbox": [ + 366, + 281, + 450, + 294 + ], + "score": 1.0, + "content": "with Total Variation", + "type": "text" + }, + { + "bbox": [ + 450, + 283, + 456, + 293 + ], + "score": 0.8, + "content": "\\rho", + "type": "inline_equation" + }, + { + "bbox": [ + 457, + 281, + 506, + 294 + ], + "score": 1.0, + "content": "to see what", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 293, + 505, + 304 + ], + "spans": [ + { + "bbox": [ + 106, + 293, + 505, + 304 + ], + "score": 1.0, + "content": "the expansion function is like in this task. We take the architecture as ResNet-50 [23], and we trained", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 302, + 505, + 316 + ], + "spans": [ + { + "bbox": [ + 105, + 302, + 505, + 316 + ], + "score": 1.0, + "content": "thousands of models with more than five algorithms, obtaining about 2 million features. The results", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 313, + 170, + 326 + ], + "spans": [ + { + "bbox": [ + 105, + 313, + 170, + 326 + ], + "score": 1.0, + "content": "are in Figure 1.", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 10.5, + "bbox_fs": [ + 105, + 259, + 506, + 326 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 336, + 505, + 392 + ], + "lines": [ + { + "bbox": [ + 105, + 334, + 504, + 351 + ], + "spans": [ + { + "bbox": [ + 105, + 334, + 160, + 351 + ], + "score": 1.0, + "content": "Existence of", + "type": "text" + }, + { + "bbox": [ + 160, + 337, + 176, + 349 + ], + "score": 0.9, + "content": "s ( \\cdot )", + "type": "inline_equation" + }, + { + "bbox": [ + 177, + 334, + 214, + 351 + ], + "score": 1.0, + "content": ". When", + "type": "text" + }, + { + "bbox": [ + 214, + 337, + 238, + 347 + ], + "score": 0.89, + "content": "\\delta = 0", + "type": "inline_equation" + }, + { + "bbox": [ + 238, + 334, + 478, + 351 + ], + "score": 1.0, + "content": ", some non-informative features are nearly 0-invariant across", + "type": "text" + }, + { + "bbox": [ + 479, + 337, + 504, + 348 + ], + "score": 0.86, + "content": "\\mathcal { E } _ { a v a i l }", + "type": "inline_equation" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 348, + 506, + 360 + ], + "spans": [ + { + "bbox": [ + 105, + 348, + 198, + 360 + ], + "score": 1.0, + "content": "but are varying across", + "type": "text" + }, + { + "bbox": [ + 198, + 348, + 214, + 359 + ], + "score": 0.89, + "content": "{ \\mathcal { E } } _ { a l l }", + "type": "inline_equation" + }, + { + "bbox": [ + 214, + 348, + 506, + 360 + ], + "score": 1.0, + "content": ", so no expansion function can make this task learnable, i.e., this task is", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 357, + 505, + 371 + ], + "spans": [ + { + "bbox": [ + 105, + 357, + 129, + 371 + ], + "score": 1.0, + "content": "NOT", + "type": "text" + }, + { + "bbox": [ + 130, + 358, + 162, + 371 + ], + "score": 0.9, + "content": "( s ( \\cdot ) , 0 )", + "type": "inline_equation" + }, + { + "bbox": [ + 163, + 357, + 303, + 371 + ], + "score": 1.0, + "content": "for any expansion function. But as", + "type": "text" + }, + { + "bbox": [ + 304, + 359, + 310, + 369 + ], + "score": 0.77, + "content": "\\delta", + "type": "inline_equation" + }, + { + "bbox": [ + 311, + 357, + 505, + 371 + ], + "score": 1.0, + "content": "increases, only informative features are left, and", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 370, + 505, + 383 + ], + "spans": [ + { + "bbox": [ + 105, + 370, + 220, + 383 + ], + "score": 1.0, + "content": "now we can find appropriate", + "type": "text" + }, + { + "bbox": [ + 221, + 370, + 236, + 382 + ], + "score": 0.92, + "content": "s ( \\cdot )", + "type": "inline_equation" + }, + { + "bbox": [ + 237, + 370, + 488, + 383 + ], + "score": 1.0, + "content": "to make it learnable. We can clearly realize from the figure that", + "type": "text" + }, + { + "bbox": [ + 489, + 370, + 505, + 382 + ], + "score": 0.89, + "content": "s ( \\cdot )", + "type": "inline_equation" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 380, + 205, + 393 + ], + "spans": [ + { + "bbox": [ + 105, + 380, + 164, + 393 + ], + "score": 1.0, + "content": "do exist when", + "type": "text" + }, + { + "bbox": [ + 164, + 381, + 201, + 392 + ], + "score": 0.91, + "content": "\\delta \\geq 0 . 1 5", + "type": "inline_equation" + }, + { + "bbox": [ + 201, + 380, + 205, + 393 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 16, + "bbox_fs": [ + 105, + 334, + 506, + 393 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 403, + 505, + 513 + ], + "lines": [ + { + "bbox": [ + 106, + 403, + 504, + 415 + ], + "spans": [ + { + "bbox": [ + 106, + 403, + 187, + 415 + ], + "score": 1.0, + "content": "Trade-off between", + "type": "text" + }, + { + "bbox": [ + 187, + 403, + 204, + 415 + ], + "score": 0.9, + "content": "s ( \\cdot )", + "type": "inline_equation" + }, + { + "bbox": [ + 204, + 403, + 225, + 415 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 225, + 403, + 231, + 413 + ], + "score": 0.61, + "content": "\\delta", + "type": "inline_equation" + }, + { + "bbox": [ + 231, + 403, + 426, + 415 + ], + "score": 1.0, + "content": ". The second phenomenon is that the slope of", + "type": "text" + }, + { + "bbox": [ + 426, + 403, + 443, + 415 + ], + "score": 0.9, + "content": "s ( \\cdot )", + "type": "inline_equation" + }, + { + "bbox": [ + 443, + 403, + 497, + 415 + ], + "score": 1.0, + "content": "decreases as", + "type": "text" + }, + { + "bbox": [ + 498, + 403, + 504, + 413 + ], + "score": 0.66, + "content": "\\delta", + "type": "inline_equation" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 414, + 505, + 426 + ], + "spans": [ + { + "bbox": [ + 106, + 415, + 269, + 426 + ], + "score": 1.0, + "content": "increases, showing a trade-off between", + "type": "text" + }, + { + "bbox": [ + 270, + 414, + 286, + 426 + ], + "score": 0.89, + "content": "s ( \\cdot )", + "type": "inline_equation" + }, + { + "bbox": [ + 286, + 415, + 305, + 426 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 305, + 415, + 311, + 424 + ], + "score": 0.63, + "content": "\\delta", + "type": "inline_equation" + }, + { + "bbox": [ + 312, + 415, + 505, + 426 + ], + "score": 1.0, + "content": ". Although this trade-off comes naturally from", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 425, + 505, + 438 + ], + "spans": [ + { + "bbox": [ + 106, + 425, + 429, + 438 + ], + "score": 1.0, + "content": "the definition of learnability, it has a deep meaning. As is shown in Section 4,", + "type": "text" + }, + { + "bbox": [ + 430, + 425, + 456, + 437 + ], + "score": 0.82, + "content": "\\operatorname { e r r } ( f )", + "type": "inline_equation" + }, + { + "bbox": [ + 456, + 425, + 505, + 438 + ], + "score": 1.0, + "content": "is bounded", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 436, + 505, + 449 + ], + "spans": [ + { + "bbox": [ + 106, + 436, + 119, + 449 + ], + "score": 1.0, + "content": "by", + "type": "text" + }, + { + "bbox": [ + 120, + 436, + 153, + 448 + ], + "score": 0.92, + "content": "O ( s ( \\varepsilon ) )", + "type": "inline_equation" + }, + { + "bbox": [ + 153, + 436, + 182, + 449 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 183, + 438, + 189, + 446 + ], + "score": 0.77, + "content": "\\varepsilon", + "type": "inline_equation" + }, + { + "bbox": [ + 190, + 436, + 505, + 449 + ], + "score": 1.0, + "content": "is the variation of the model. To make the bound tighter, a natural idea is to", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 447, + 506, + 460 + ], + "spans": [ + { + "bbox": [ + 106, + 447, + 169, + 460 + ], + "score": 1.0, + "content": "choose a flatter", + "type": "text" + }, + { + "bbox": [ + 169, + 447, + 185, + 459 + ], + "score": 0.89, + "content": "s ( \\cdot )", + "type": "inline_equation" + }, + { + "bbox": [ + 186, + 447, + 263, + 460 + ], + "score": 1.0, + "content": ". However, a flatter", + "type": "text" + }, + { + "bbox": [ + 263, + 447, + 279, + 459 + ], + "score": 0.89, + "content": "s ( \\cdot )", + "type": "inline_equation" + }, + { + "bbox": [ + 279, + 447, + 373, + 460 + ], + "score": 1.0, + "content": "corresponds to a larger", + "type": "text" + }, + { + "bbox": [ + 374, + 448, + 380, + 457 + ], + "score": 0.59, + "content": "\\delta", + "type": "inline_equation" + }, + { + "bbox": [ + 380, + 447, + 506, + 460 + ], + "score": 1.0, + "content": ". Typically, learning a model to", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 458, + 505, + 470 + ], + "spans": [ + { + "bbox": [ + 105, + 458, + 505, + 470 + ], + "score": 1.0, + "content": "meet this higher informativeness requirement is more difficult, and it is possible that the algorithm", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 469, + 505, + 480 + ], + "spans": [ + { + "bbox": [ + 106, + 469, + 505, + 480 + ], + "score": 1.0, + "content": "achieves this by capturing more domain-specific features, which will therefore increase the variation", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 479, + 506, + 492 + ], + "spans": [ + { + "bbox": [ + 105, + 479, + 160, + 492 + ], + "score": 1.0, + "content": "of the model,", + "type": "text" + }, + { + "bbox": [ + 161, + 482, + 167, + 489 + ], + "score": 0.65, + "content": "\\varepsilon", + "type": "inline_equation" + }, + { + "bbox": [ + 167, + 479, + 313, + 492 + ], + "score": 1.0, + "content": ". As a result, we are not sure whether", + "type": "text" + }, + { + "bbox": [ + 313, + 479, + 331, + 492 + ], + "score": 0.9, + "content": "s ( \\varepsilon )", + "type": "inline_equation" + }, + { + "bbox": [ + 331, + 479, + 506, + 492 + ], + "score": 1.0, + "content": "will increase or decrease. We believe this is", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 491, + 505, + 503 + ], + "spans": [ + { + "bbox": [ + 106, + 491, + 505, + 503 + ], + "score": 1.0, + "content": "also the essence of model selection: i.e., to trade-off between the variation and informativeness of a", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 502, + 255, + 513 + ], + "spans": [ + { + "bbox": [ + 106, + 502, + 255, + 513 + ], + "score": 1.0, + "content": "model, which is done in Formula 11.", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 23.5, + "bbox_fs": [ + 105, + 403, + 506, + 513 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 527, + 232, + 541 + ], + "lines": [ + { + "bbox": [ + 105, + 526, + 234, + 543 + ], + "spans": [ + { + "bbox": [ + 105, + 526, + 234, + 543 + ], + "score": 1.0, + "content": "7 More Related Works", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 29 + }, + { + "type": "text", + "bbox": [ + 106, + 552, + 505, + 651 + ], + "lines": [ + { + "bbox": [ + 106, + 552, + 506, + 565 + ], + "spans": [ + { + "bbox": [ + 106, + 552, + 506, + 565 + ], + "score": 1.0, + "content": "Domain generalization [12, 39], or OOD generalization, has drawn much attention recently [21, 30].", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 563, + 506, + 576 + ], + "spans": [ + { + "bbox": [ + 106, + 563, + 506, + 576 + ], + "score": 1.0, + "content": "The goal is to learn a model from several training domains and expect good performance on unseen", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 574, + 506, + 587 + ], + "spans": [ + { + "bbox": [ + 105, + 574, + 506, + 587 + ], + "score": 1.0, + "content": "test domains. [60, 64] offer a comprehensive survey. A popular solution is to extract domain-invariant", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 585, + 506, + 597 + ], + "spans": [ + { + "bbox": [ + 106, + 585, + 506, + 597 + ], + "score": 1.0, + "content": "feature representation. [45] and [49] proved that when the model is linear, the invariance under", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 597, + 506, + 608 + ], + "spans": [ + { + "bbox": [ + 106, + 597, + 506, + 608 + ], + "score": 1.0, + "content": "training domains can help discover invariant features on test domains. [5] introduces the invariant", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 607, + 505, + 619 + ], + "spans": [ + { + "bbox": [ + 105, + 607, + 505, + 619 + ], + "score": 1.0, + "content": "prediction into neural networks and proposes a practical objective function. After that, a lot of works", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 618, + 504, + 630 + ], + "spans": [ + { + "bbox": [ + 106, + 618, + 504, + 630 + ], + "score": 1.0, + "content": "arise from the view of causal discovery, distributional robustness and conditional independence", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 106, + 628, + 506, + 642 + ], + "spans": [ + { + "bbox": [ + 106, + 628, + 506, + 642 + ], + "score": 1.0, + "content": "[1, 7, 16, 15, 26, 32, 33, 43, 51, 61]. On the other hand, some works point out the weakness of", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 106, + 640, + 456, + 652 + ], + "spans": [ + { + "bbox": [ + 106, + 640, + 456, + 652 + ], + "score": 1.0, + "content": "existing methods from the theoretical and experimental perspectives [2, 21, 29, 41, 50].", + "type": "text" + } + ], + "index": 38 + } + ], + "index": 34, + "bbox_fs": [ + 105, + 552, + 506, + 652 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 656, + 505, + 722 + ], + "lines": [ + { + "bbox": [ + 106, + 656, + 505, + 668 + ], + "spans": [ + { + "bbox": [ + 106, + 656, + 505, + 668 + ], + "score": 1.0, + "content": "The OOD generalization requires restrictions on how the target domains may differ. A straightforward", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 667, + 506, + 680 + ], + "spans": [ + { + "bbox": [ + 105, + 667, + 506, + 680 + ], + "score": 1.0, + "content": "approach is to define a set of test domains around the training domain using some distribution distance", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 106, + 678, + 505, + 689 + ], + "spans": [ + { + "bbox": [ + 106, + 678, + 505, + 689 + ], + "score": 1.0, + "content": "measure [6, 13, 19, 25, 51, 53, 54, 61]. 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[N/A]", + "type": "text" + } + ], + "index": 47, + "is_list_end_line": true + } + ], + "index": 44.5, + "bbox_fs": [ + 106, + 646, + 506, + 718 + ] + } + ] + } + ], + "_backend": "pipeline", + "_version_name": "2.2.2" +} \ No newline at end of file diff --git a/parse/train/BJOFETxR-/BJOFETxR-.md b/parse/train/BJOFETxR-/BJOFETxR-.md new file mode 100644 index 0000000000000000000000000000000000000000..fe055f5a1665a6a125adce520446bba652faad4a --- /dev/null +++ b/parse/train/BJOFETxR-/BJOFETxR-.md @@ -0,0 +1,320 @@ +# LEARNING TO REPRESENT PROGRAMS WITH GRAPHS + +Miltiadis Allamanis +Microsoft Research +Cambridge, UK +miallama@microsoft.com +Marc Brockschmidt +Microsoft Research +Cambridge, UK +mabrocks@microsoft.com + +Mahmoud Khademi∗ Simon Fraser University Burnaby, BC, Canada mkhademi@sfu.ca + +# ABSTRACT + +Learning tasks on source code (i.e., formal languages) have been considered recently, but most work has tried to transfer natural language methods and does not capitalize on the unique opportunities offered by code’s known sematics. For example, long-range dependencies induced by using the same variable or function in distant locations are often not considered. We propose to use graphs to represent both the syntactic and semantic structure of code and use graph-based deep learning methods to learn to reason over program structures. + +In this work, we present how to construct graphs from source code and how to scale Gated Graph Neural Networks training to such large graphs. We evaluate our method on two tasks: VARNAMING, in which a network attempts to predict the name of a variable given its usage, and VARMISUSE, in which the network learns to reason about selecting the correct variable that should be used at a given program location. Our comparison to methods that use less structured program representations shows the advantages of modeling known structure, and suggests that our models learn to infer meaningful names and to solve the VARMISUSE task in many cases. Additionally, our testing showed that VARMISUSE identifies a number of bugs in mature open-source projects. + +# 1 INTRODUCTION + +The advent of large repositories of source code as well as scalable machine learning methods naturally leads to the idea of “big code”, i.e., largely unsupervised methods that support software engineers by generalizing from existing source code (Allamanis et al., 2017). Currently, existing deep learning models of source code capture its shallow, textual structure, e.g. as a sequence of tokens (Hindle et al., 2012; Raychev et al., 2014; Allamanis et al., 2016), as parse trees (Maddison & Tarlow, 2014; Bielik et al., 2016), or as a flat dependency networks of variables (Raychev et al., 2015). Such models miss out on the opportunity to capitalize on the rich and well-defined semantics of source code. In this work, we take a step to alleviate this by including two additional signal sources in source code: data flow and type hierarchies. We do this by encoding programs as graphs, in which edges represent syntactic relationships (e.g. “token before/after”) as well as semantic relationships (“variable last used/written here”, “formal parameter for argument is called stream”, etc.). Our key insight is that exposing these semantics explicitly as structured input to a machine learning model lessens the requirements on amounts of training data, model capacity and training regime and allows us to solve tasks that are beyond the current state of the art. + +We explore two tasks to illustrate the advantages of exposing more semantic structure of programs. First, we consider the VARNAMING task (Allamanis et al., 2014; Raychev et al., 2015), in which given some source code, the “correct” variable name is inferred as a sequence of subtokens. This requires some understanding of how a variable is used, i.e., requires reasoning about lines of code far apart in the source file. Secondly, we introduce the variable misuse prediction task (VARMISUSE), in which the network aims to infer which variable should be used in a program location. To illustrate the task, Figure 1 shows a slightly simplified snippet of a bug our model detected in a popular open-source project. Specifically, instead of the variable clazz, variable first should have been used in the yellow highlighted slot. Existing static analysis methods cannot detect such issues, even though a software engineer would easily identify this as an error from experience. + +![](images/1bc113e2a4c0fb9e0e51adeae6b01b6e7ddf6b162f6335e7e776d43b92e616d2.jpg) +Figure 1: A snippet of a detected bug in RavenDB an open-source C# project. The code has been slightly simplified. Our model detects correctly that the variable used in the highlighted (yellow) slot is incorrect. Instead, first should have been placed at the slot. We reported this problem which was fixed in PR 4138. + +To achieve high accuracy on these tasks, we need to learn representations of program semantics. For both tasks, we need to learn the semantic role of a variable (e.g., “is it a counter?”, “is it a filename?”). Additionally, for VARMISUSE, learning variable usage semantics (e.g., “a filename is needed here”) is required. This “fill the blank element” task is related to methods for learning distributed representations of natural language words, such as Word2Vec (Mikolov et al., 2013) and GLoVe (Pennington et al., 2014). However, we can learn from a much richer structure such as data flow information. This work is a step towards learning program representations, and we expect them to be valuable in a wide range of other tasks, such as code completion (“this is the variable you are looking for”) and more advanced bug finding (“you should lock before using this object”). + +To summarize, our contributions are: (i) We define the VARMISUSE task as a challenge for machine learning modeling of source code, that requires to learn (some) semantics of programs (cf. section 3). (ii) We present deep learning models for solving the VARNAMING and VARMISUSE tasks by modeling the code’s graph structure and learning program representations over those graphs (cf. section 4). (iii) We evaluate our models on a large dataset of 2.9 million lines of real-world source code, showing that our best model achieves $3 2 . 9 \%$ accuracy on the VARNAMING task and $8 5 . 5 \%$ accuracy on the VARMISUSE task, beating simpler baselines (cf. section 5). (iv) We document practical relevance of VARMISUSE by summarizing some bugs that we found in mature open-source software projects $( c f .$ subsection 5.3). Our implementation of graph neural networks (on a simpler task) can be found at https://github.com/Microsoft/gated-graph-neural-network-samples and the dataset can be found at https://aka.ms/iclr18-prog-graphs-dataset. + +# 2 RELATED WORK + +Our work builds upon the recent field of using machine learning for source code artifacts (Allamanis et al., 2017). For example, Hindle et al. (2012); Bhoopchand et al. (2016) model the code as a sequence of tokens, while Maddison & Tarlow (2014); Raychev et al. (2016) model the syntax tree structure of code. All works on language models of code find that predicting variable and method identifiers is one of biggest challenges in the task. + +Closest to our work is the work of Allamanis et al. (2015) who learn distributed representations of variables using all their usages to predict their names. However, they do not use data flow information and we are not aware of any model that does so. Raychev et al. (2015) and Bichsel et al. (2016) use conditional random fields to model a variety of relationships between variables, AST elements and types to predict variable names and types (resp. to deobfuscate Android apps), but without considering the flow of data explicitly. In these works, all variable usages are deterministically known beforehand (as the code is complete and remains unmodified), as in Allamanis et al. (2014; 2015). + +Our work is remotely related to work on program synthesis using sketches (Solar-Lezama, 2008) and automated code transplantation (Barr et al., 2015). However, these approaches require a set of specifications (e.g. input-output examples, test suites) to complete the gaps, rather than statistics learned from big code. These approaches can be thought as complementary to ours, since we learn to statistically complete the gaps without any need for specifications, by learning common variable usage patterns from code. + +Neural networks on graphs (Gori et al., 2005; Li et al., 2015; Defferrard et al., 2016; Kipf & Welling, 2016; Gilmer et al., 2017) adapt a variety of deep learning methods to graph-structured input. They have been used in a series of applications, such as link prediction and classification (Grover & Leskovec, 2016) and semantic role labeling in NLP (Marcheggiani & Titov, 2017). Somewhat related to source code is the work of Wang et al. (2017) who learn graph-based representations of mathematical formulas for premise selection in theorem proving. + +# 3 THE VARMISUSE TASK + +Detecting variable misuses in code is a task that requires understanding and reasoning about program semantics. To successfully tackle the task one needs to infer the role and function of the program elements and understand how they relate. For example, given a program such as Fig. 1, the task is to automatically detect that the marked use of $\mathtt { C 1 a z z }$ is a mistake and that first should be used instead. While this task resembles standard code completion, it differs significantly in its scope and purpose, by considering only variable identifiers and a mostly complete program. + +Task Description We view a source code file as a sequence of tokens $t _ { 0 } \ldots t _ { N } = \mathcal { T }$ , in which some tokens $t _ { \lambda _ { 0 } } , t _ { \lambda _ { 1 } } \ldots$ are variables. Furthermore, let $\mathbb { V } _ { t } \subset \mathbb { V }$ refer to the set of all type-correct variables in scope at the location of $t$ , i.e., those variables that can be used at $t$ without raising a compiler error. We call a token $t o k _ { \lambda }$ where we want to predict the correct variable usage a slot. We define a separate task for each slot $t _ { \lambda }$ : Given $t _ { 0 } \ldots t _ { \lambda - 1 }$ and $t _ { \lambda + 1 } , \dots , t _ { N }$ , correctly select $t _ { \lambda }$ from $\mathbb { V } _ { t _ { \lambda } }$ . For training and evaluation purposes, a correct solution is one that simply matches the ground truth, but note that in practice, several possible assignments could be considered correct (i.e., when several variables refer to the same value in memory). + +# 4 MODEL: PROGRAMS AS GRAPHS + +In this section, we discuss how to transform program source code into program graphs and learn representations over them. These program graphs not only encode the program text but also the semantic information that can be obtained using standard compiler tools. + +Gated Graph Neural Networks Our work builds on Gated Graph Neural Networks (Li et al., 2015) (GGNN) and we summarize them here. A graph $\mathcal { G } = ( \nu , \pmb { \varepsilon } , \pmb { X } )$ is composed of a set of nodes $\nu$ , node features $\boldsymbol { X }$ , and a list of directed edge sets $\pmb { \mathcal { E } } = ( \mathcal { E } _ { 1 } , \ldots , \mathcal { E } _ { K } )$ where $K$ is the number of edge types. We annotate each $v \in \mathcal V$ with a real-valued vector $\pmb { x } ^ { ( v ) } \in \mathbb { R } ^ { D }$ representing the features of the node (e.g., the embedding of a string label of that node). + +We associate every node $v$ with a state vector $\mathbf { \Omega } _ { h } ( v )$ , initialized from the node label $\pmb { x } ^ { ( v ) }$ . The sizes of the state vector and feature vector are typically the same, but we can use larger state vectors through padding of node features. To propagate information throughout the graph, “messages” of type $k$ are sent from each $v$ to its neighbors, where each message is computed from its current state vector as m(vk $m _ { k } ^ { ( v ) } = f _ { k } ( { \pmb h } ^ { ( v ) } )$ . Here, $f _ { k }$ can be an arbitrary function; we choose a linear layer in our case. By computing messages for all graph edges at the same time, all states can be updated at the same time. In particular, a new state for a node $v$ is computed by aggregating all incoming messages as $\tilde { m } ^ { ( v ) } = g ( \{ m _ { k } ^ { ( u ) } \ |$ there is an edge of type $k$ from $u$ to $v \}$ ). $g$ is an aggregation function, which we implement as elementwise summation. Given the aggregated message $\tilde { m } ^ { ( v ) }$ and the current state vector $\mathbf { \Omega } _ { h } ( v )$ of node $v$ , the state of the next time step $\boldsymbol { h ^ { \prime } } ^ { ( v ) }$ is computed as $\pmb { h } ^ { \prime ( v ) } = \mathbf { G } \mathbf { R } \mathbf { U } ( \tilde { \pmb { m } } ^ { ( v ) } , \pmb { h } ^ { ( v ) } )$ , where GRU is the recurrent cell function of gated recurrent unit (GRU) (Cho et al., 2014). The + +(a) Simplified syntax graph for line 2 of Fig. 1, where blue rounded boxes are syntax nodes, black rectangular boxes syntax tokens, blue edges Child edges and double black edges NextToken edges. + +![](images/f0957bde3bf55bfd9fc41e58429404103dadf9736f4a693bec1cc3c471bc40e7.jpg) +Figure 2: Examples of graph edges used in program representation. + +![](images/83a96cffbd0f2a1c8368982c7c4bae4e01f2122dfee415273de350b2c11528e5.jpg) + +(b) Data flow edges for $\begin{array} { r l r } { ( \overline { { \bf x } } ) ^ { 1 } , \overline { { \boldsymbol { \mathbf { y } } } } } & { { } = } & { \bf F o o ( \begin{array} { l } { ) } \ ; } \end{array} \end{array}$ while $\left. \overline { { \mathbf { x } } } \right. ^ { 3 } \ > \ \mathsf { 0 } .$ ) $\underline { { \nabla } } \mathbf { \Sigma } ^ { 4 } \mathbf { \Sigma } = \underline { { \nabla } } \mathbf { \bar { x } } \mathbf { \Sigma } ^ { 5 } \mathbf { \Sigma } ^ { \subset } \mathbf { \Sigma } \boxed { \nabla } \mathbf { \bar { y } } \mathbf { \Sigma } ^ { 6 }$ (indices added for clarity), with red dotted LastUse edges, green dashed LastWrite edges and dashdotted purple ComputedFrom edges. + +dynamics defined by the above equations are repeated for a fixed number of time steps. Then, we use the state vectors from the last time step as the node representations.1 + +Program Graphs We represent program source code as graphs and use different edge types to model syntactic and semantic relationships between different tokens. The backbone of a program graph is the program’s abstract syntax tree (AST), consisting of syntax nodes (corresponding to nonterminals in the programming language’s grammar) and syntax tokens (corresponding to terminals). We label syntax nodes with the name of the nonterminal from the program’s grammar, whereas syntax tokens are labeled with the string that they represent. We use Child edges to connect nodes according to the AST. As this does not induce an order on children of a syntax node, we additionally add NextToken edges connecting each syntax token to its successor. An example of this is shown in Fig. 2a. + +To capture the flow of control and data through a program, we add additional edges connecting different uses and updates of syntax tokens corresponding to variables. For such a token $v$ , let $\mathcal { D } ^ { R } ( v )$ be the set of syntax tokens at which the variable could have been used last. This set may contain several nodes (for example, when using a variable after a conditional in which it was used in both branches), and even syntax tokens that follow in the program code (in the case of loops). Similarly, let $\mathcal { D } ^ { W } ( v )$ be the set of syntax tokens at which the variable was last written to. Using these, we add LastRead (resp. LastWrite) edges connecting $v$ to all elements of $\mathcal { D } ^ { R } ( v )$ (resp. $\tilde { \mathcal { D } } ^ { W } ( v ) )$ . Additionally, whenever we observe an assignment $v = e x p r$ , we connect $v$ to all variable tokens occurring in expr using ComputedFrom edges. An example of such semantic edges is shown in Fig. 2b. + +We extend the graph to chain all uses of the same variable using LastLexicalUse edges (independent of data flow, i.e., in if (...) { ... v ...} else { ... v ...}, we link the two occurrences of $v$ ). We also connect return tokens to the method declaration using ReturnsTo edges (this creates a “shortcut” to its name and type). Inspired by Rice et al. (2017), we connect arguments in method calls to the formal parameters that they are matched to with FormalArgName edges, i.e., if we observe a call Foo(bar) and a method declaration Foo(InputStream stream), we connect the bar token to the stream token. Finally, we connect every token corresponding to a variable to enclosing guard expressions that use the variable with GuardedBy and GuardedByNegation edges. For example, in if ( $\mathrm { ~ \bf ~ \cdot ~ x ~ } > \mathrm { ~ \bf ~ y ~ }$ ) { ... x ...} else $\{ \ \begin{array} { r l } { \mathrm { ~ \texttt ~ { ~ . ~ . ~ } ~ } \underline { { \mathrm { ~ y ~ } } } \mathrm { ~ . ~ . ~ . ~ } \} } \end{array}$ , we add a GuardedBy edge from $\underline { { \boldsymbol { \mathrm { X } } } }$ (resp. a GuardedByNegation edge from y) to the AST node corresponding to $\mathrm { ~ ~ { ~ x ~ } ~ } > \mathrm { ~ ~ { ~ y ~ } ~ }$ . + +Finally, for all types of edges we introduce their respective backwards edges (transposing the adjacency matrix), doubling the number of edges and edge types. Backwards edges help with propagating information faster across the GGNN and make the model more expressive. + +Leveraging Variable Type Information We assume a statically typed language and that the source code can be compiled, and thus each variable has a (known) type $\tau ( v )$ . To use it, we define a learnable embedding function $\mathbf { r } ( \tau )$ for known types and additionally define an “UNKTYPE” for all unknown/unrepresented types. We also leverage the rich type hierarchy that is available in many object-oriented languages. For this, we map a variable’s type $\tau ( v )$ to the set of its supertypes, i.e. $\tau ^ { * } \bar { ( } v ) = \{ \tau : \tau ( v ) \quad$ implements type $\tau \} \cup \{ \bar { \tau } ( v ) \}$ . We then compute the type representation $\mathbf { r } ^ { * } ( v )$ of a variable $v$ as the element-wise maximum of $\{ \mathbf { r } ( \tau ) : \tau \in \tau ^ { * } ( v ) \}$ . We chose the maximum here, as it is a natural pooling operation for representing partial ordering relations (such as type lattices). Using all types in $\tau ^ { * } ( v )$ allows us to generalize to unseen types that implement common supertypes or interfaces. For example, List ${ \tt < K > }$ has multiple concrete types (e.g. List, List). Nevertheless, these types implement a common interface (IList) and share common characteristics. During training, we randomly select a non-empty subset of $\tau ^ { * } ( v )$ which ensures training of all known types in the lattice. This acts both like a dropout mechanism and allows us to learn a good representation for all types in the type lattice. + +Initial Node Representation To compute the initial node state, we combine information from the textual representation of the token and its type. Concretely, we split the name of a node representing a token into subtokens (e.g. classTypes will be split into two subtokens class and types) on camelCase and pascal_case. We then average the embeddings of all subtokens to retrieve an embedding for the node name. Finally, we concatenate the learned type representation $\mathbf { r } ^ { * } ( v )$ , computed as discussed earlier, with the node name representation, and pass it through a linear layer to obtain the initial representations for each node in the graph. + +Programs Graphs for VARNAMING Given a program and an existing variable $v$ , we build a program graph as discussed above and then replace the variable name in all corresponding variable tokens by a special ${ < } S \mathrm { L O T } >$ token. To predict a name, we use the initial node labels computed as the concatenation of learnable token embeddings and type embeddings as discussed above, run GGNN propagation for 8 time steps2 and then compute a variable usage representation by averaging the representations for all ${ < } S \mathrm { L O T } >$ tokens. This representation is then used as the initial state of a one-layer GRU, which predicts the target name as a sequence of subtokens (e.g., the name inputStreamBuffer is treated as the sequence [input, stream, buffer]). We train this graph2seq architecture using a maximum likelihood objective. In section 5, we report the accuracy for predicting the exact name and the F1 score for predicting its subtokens. + +Program Graphs for VARMISUSE To model VARMISUSE with program graphs we need to modify the graph. First, to compute a context representation $\mathbf { } c ( t )$ for a slot $t$ where we want to predict the used variable, we insert a new node $v _ { < S \tt L O T > }$ at the position of $t$ , corresponding to a “hole” at this point, and connect it to the remaining graph using all applicable edges that do not depend on the chosen variable at the slot (i.e., everything but LastUse, LastWrite, LastLexicalUse, and GuardedBy edges). Then, to compute the usage representation $\mathbf { u } ( t , v )$ of each candidate variable $v$ at the target slot, we insert a “candidate” node $v _ { t , v }$ for all $v$ in $\mathbb { V } _ { t }$ , and connect it to the graph by inserting the LastUse, LastWrite and LastLexicalUse edges that would be used if the variable were to be used at this slot. Each of these candidate nodes represents the speculative placement of the variable within the scope. + +Using the initial node representations, concatenated with an extra bit that is set to one for the candidate nodes $v _ { t , v }$ , we run GGNN propagation for 8 time steps.2 The context and usage representation are then the final node states of the nodes, i.e., $\pmb { c } ( t ) = \pmb { h } ^ { ( v _ { < \mathrm { S L O T } > } ) }$ and $\mathbf { u } ( t , v ) = h ^ { ( v _ { t , v } ) }$ . Finally, the correct variable usage at the location is computed as arg $\textstyle \operatorname* { m a x } _ { v } W [ c ( t ) , \mathbf { u } ( t , v ) ]$ where $W$ is a linear layer that uses the concatenation of $\mathbf { } c ( t )$ and $\mathbf { u } ( t , v )$ . We train using a max-margin objective. + +# 4.1 IMPLEMENTATION + +Using GGNNs for sets of large, diverse graphs requires some engineering effort, as efficient batching is hard in the presence of diverse shapes. An important observation is that large graphs are normally very sparse, and thus a representation of edges as an adjacency list would usually be advantageous to reduce memory consumption. In our case, this can be easily implemented using a sparse tensor representation, allowing large batch sizes that exploit the parallelism of modern GPUs efficiently. A second key insight is to represent a batch of graphs as one large graph with many disconnected components. This just requires appropriate pre-processing to make node identities unique. As this makes batch construction somewhat CPU-intensive, we found it useful to prepare minibatches on a separate thread. Our TensorFlow (Abadi et al., 2016) implementation scales to 55 graphs per second during training and 219 graphs per second during test-time using a single NVidia GeForce GTX Titan X with graphs having on average 2,228 (median 936) nodes and 8,350 (median 3,274) edges and 8 GGNN unrolling iterations, all 20 edge types (forward and backward edges for 10 original edge types) and the size of the hidden layer set to 64. The number of types of edges in the GGNN contributes proportionally to the running time. For example, a GGNN run for our ablation study using only the two most common edge types (NextToken, Child) achieves 105 graphs/second during training and 419 graphs/second at test time with the same hyperparameters. Our (generic) implementation of GGNNs is available at https://github.com/Microsoft/ gated-graph-neural-network-samples, using a simpler demonstration task. + +# 5 EVALUATION + +Dataset We collected a dataset for the VARMISUSE task from open source $C ^ { \# }$ projects on GitHub. To select projects, we picked the top-starred (non-fork) projects in GitHub. We then filtered out projects that we could not (easily) compile in full using Roslyn3, as we require a compilation to extract precise type information for the code (including those types present in external libraries). Our final dataset contains 29 projects from a diverse set of domains (compilers, databases, . . . ) with about 2.9 million non-empty lines of code. A full table is shown in Appendix D. + +For the task of detecting variable misuses, we collect data from all projects by selecting all variable usage locations, filtering out variable declarations, where at least one other type-compatible replacement variable is in scope. The task is then to infer the correct variable that originally existed in that location. Thus, by construction there is at least one type-correct replacement variable, i.e. picking it would not raise an error during type checking. In our test datasets, at each slot there are on average 3.8 type-correct alternative variables (median 3, $\sigma = 2 . 6$ ). + +From our dataset, we selected two projects as our development set. From the rest of the projects, we selected three projects for UNSEENPROJTEST to allow testing on projects with completely unknown structure and types. We split the remaining 23 projects into train/validation/test sets in the proportion 60-10-30, splitting along files (i.e., all examples from one source file are in the same set). We call the test set obtained like this SEENPROJTEST. + +Baselines For VARMISUSE, we consider two bidirectional RNN-based baselines. The local model (LOC) is a simple two-layer bidirectional GRU run over the tokens before and after the target location. For this baseline, $\mathbf { } c ( t )$ is set to the slot representation computed by the RNN, and the usage context of each variable $\mathbf { u } ( t , v )$ is the embedding of the name and type of the variable, computed in the same way as the initial node labels in the GGNN. This baseline allows us to evaluate how important the usage context information is for this task. The flat dataflow model (AVGBIRNN) is an extension to LOC, where the usage representation $\mathbf { u } ( t , v )$ is computed using another two-layer bidirectional RNN run over the tokens before/after each usage, and then averaging over the computed representations at the variable token $v$ . The local context, $\mathbf { } c ( t )$ , is identical to LOC. AVGBIRNN is a significantly stronger baseline that already takes some structural information into account, as the averaging over all variables usages helps with long-range dependencies. Both models pick the variable that maximizes $\boldsymbol { c } ( t ) ^ { T } \mathbf { u } ( t , v )$ . + +For VARNAMING, we replace LOC by AVGLBL, which uses a log-bilinear model for 4 left and 4 right context tokens of each variable usage, and then averages over these context representations (this corresponds to the model in Allamanis et al. (2015)). We also test AVGBIRNN on VARNAMING, which essentially replaces the log-bilinear context model by a bidirectional RNN. + +Table 1: Evaluation of models. SEENPROJTEST refers to the test set containing projects that have files in the training set, UNSEENPROJTEST refers to projects that have no files in the training data. Results averaged over two runs. + +
SEENPROJTESTUNSEENPROJTEST
LocAVGLBLAVGBIRNNGGNNLocAVGLBLAVGBIRNNGGNN
VARMISUSE
Accuracy (%)50.0二73.785.528.960.278.2
PR AUC0.7880.9410.9800.6110.8950.958
VARNAMING Accuracy (%)36.142.9
53.622.723.444.0
F1 (%)44.050.165.830.632.062.0
+ +Table 2: Ablation study for the GGNN model on SEENPROJTEST for the two tasks. + +
Ablation DescriptionAccuracy (%) VARMISUSE
VARNAMING
Standard Model (reported in Table 1)85.553.6
Only NextToken, Child, LastUse_LastWrite edges80.631.2
Only semantic edges (all but NextToken, Child) Only syntax edges (NextToken, Child)78.4 55.352.9 34.3
Node Labels: Tokens instead of subtokens
Node Labels:Disabled85.6 84.334.5
31.8
+ +# 5.1 QUANTITATIVE EVALUATION + +Table 1 shows the evaluation results of the models for both tasks.4 As LOC captures very little information, it performs relatively badly. AVGLBL and AVGBIRNN, which capture information from many variable usage sites, but do not explicitly encode the rich structure of the problem, still lag behind the GGNN by a wide margin. The performance difference is larger for VARMISUSE, since the structure and the semantics of code are far more important within this setting. + +Generalization to new projects Generalizing across a diverse set of source code projects with different domains is an important challenge in machine learning. We repeat the evaluation using the UNSEENPROJTEST set stemming from projects that have no files in the training set. The right side of Table 1 shows that our models still achieve good performance, although it is slightly lower compared to SEENPROJTEST. This is expected since the type lattice is mostly unknown in UNSEENPROJTEST. + +We believe that the dominant problem in applying a trained model to an unknown project (i.e., domain) is the fact that its type hierarchy is unknown and the used vocabulary (e.g. in variables, method and class names, etc.) can differ substantially. + +Ablation Study To study the effect of some of the design choices for our models, we have run some additional experiments and show their results in Table 2. First, we varied the edges used in the program graph. We find that restricting the model to syntactic information has a large impact on performance on both tasks, whereas restricting it to semantic edges seems to mostly impact performance on VARMISUSE. Similarly, the ComputedFrom, FormalArgName and ReturnsTo edges give a small boost on VARMISUSE, but greatly improve performance on VARNAMING. As evidenced by the experiments with the node label representation, syntax node and token names seem to matter little for VARMISUSE, but naturally have a great impact on VARNAMING. + +# 5.2 QUALITATIVE EVALUATION + +Figure 3 illustrates the predictions that GGNN makes on a sample test snippet. The snippet recursively searches for the global directives file by gradually descending into the root folder. Reasoning about the correct variable usages is hard, even for humans, but the GGNN correctly predicts the variable usages at all locations except two (slot 1 and 8). As a software engineer is writing the code, it is imaginable that she may make a mistake misusing one variable in the place of another. Since all variables are string variables, no type errors will be raised. As the probabilities in Fig. 3 suggest most potential variable misuses can be flagged by the model yielding valuable warnings to software engineers. Additional samples with comments can be found in Appendix B. + +![](images/cd8c9c73a863608276a7923a4347ae44724d33325aed2f76dd26669048968bd2.jpg) +Figure 3: VARMISUSE predictions on slots within a snippet of the SEENPROJTEST set for the ServiceStack project. Additional visualizations are available in Appendix B. The underlined tokens are the correct tokens. The model has to select among a number of string variables at each slot, where all of them represent some kind of path. The GGNN accurately predicts the correct variable usage in 11 out of the 13 slots reasoning about the complex ways the variables interact among them. + +![](images/1a46d813f5e2a8d4ca1585320818302c03d99347f9237a6ad178e6da4a0a1573.jpg) +Figure 4: A bug found (yellow) in RavenDB open-source project. The code unnecessarily ensures that the buffer is of size length rather than size (which our model predicts as the correct variable here). + +Furthermore, Appendix C shows samples of pairs of code snippets that share similar representations as computed by the cosine similarity of the usage representation $\mathbf { u } ( t , v )$ of GGNN. The reader can notice that the network learns to group variable usages that share semantic similarities together. For example, checking for null before the use of a variable yields similar distributed representations across code segments (Sample 1 in Appendix C). + +# 5.3 DISCOVERED VARIABLE MISUSE BUGS + +We have used our VARMISUSE model to identify likely locations of bugs in RavenDB (a document database) and Roslyn (Microsoft’s $C ^ { \# }$ compiler framework). For this, we manually reviewed a sample of the top 500 locations in both projects where our model was most confident about a choosing a variable differing from the ground truth, and found three bugs in each of the projects. + +Figs. 1,4,5 show the issues discovered in RavenDB. The bug in Fig. 1 was possibly caused by copy-pasting, and cannot be easily caught by traditional methods. A compiler will not warn about unused variables (since first is used) and virtually nobody would write a test testing another test. Fig. 4 shows an issue that, although not critical, can lead to increased memory consumption. Fig. 5 shows another issue arising from a non-informative error message. We privately reported three additional bugs to the Roslyn developers, who have fixed the issues in the meantime (cf. https://github.com/dotnet/roslyn/pull/23437). One of the reported bugs could cause a crash in Visual Studio when using certain Roslyn features. + +![](images/336e106c897c8cfa8eb64fc3ca64fbd06fdd03a060af8c210b6cbadf7e1fcc42.jpg) +Figure 5: A bug found (yellow) in the RavenDB open-source project. Although backupFilename is found to be invalid by IsValidBackup, the user is notified that backupLocation is invalid instead. + +Finding these issues in widely released and tested code suggests that our model can be useful during the software development process, complementing classic program analysis tools. For example, one usage scenario would be to guide the code reviewing process to locations a VARMISUSE model has identified as unusual, or use it as a prior to focus testing or expensive code analysis efforts. + +# 6 DISCUSSION & CONCLUSIONS + +Although source code is well understood and studied within other disciplines such as programming language research, it is a relatively new domain for deep learning. It presents novel opportunities compared to textual or perceptual data, as its (local) semantics are well-defined and rich additional information can be extracted using well-known, efficient program analyses. On the other hand, integrating this wealth of structured information poses an interesting challenge. Our VARMISUSE task exposes these opportunities, going beyond simpler tasks such as code completion. We consider it as a first proxy for the core challenge of learning the meaning of source code, as it requires to probabilistically refine standard information included in type systems. + +# REFERENCES + +Martín Abadi, Ashish Agarwal, Paul Barham, Eugene Brevdo, Zhifeng Chen, Craig Citro, Greg S Corrado, Andy Davis, Jeffrey Dean, Matthieu Devin, et al. Tensorflow: Large-scale machine learning on heterogeneous distributed systems. arXiv preprint arXiv:1603.04467, 2016. +Miltiadis Allamanis, Earl T Barr, Christian Bird, and Charles Sutton. Learning natural coding conventions. In Foundations of Software Engineering (FSE), 2014. +Miltiadis Allamanis, Earl T Barr, Christian Bird, and Charles Sutton. Suggesting accurate method and class names. In Foundations of Software Engineering (FSE), 2015. +Miltiadis Allamanis, Hao Peng, and Charles Sutton. A convolutional attention network for extreme summarization of source code. In International Conference on Machine Learning (ICML), pp. 2091–2100, 2016. +Miltiadis Allamanis, Earl T Barr, Premkumar Devanbu, and Charles Sutton. A survey of machine learning for big code and naturalness. arXiv preprint arXiv:1709.06182, 2017. +Earl T Barr, Mark Harman, Yue Jia, Alexandru Marginean, and Justyna Petke. Automated software transplantation. In International Symposium on Software Testing and Analysis (ISSTA), 2015. +Al Bessey, Ken Block, Ben Chelf, Andy Chou, Bryan Fulton, Seth Hallem, Charles Henri-Gros, Asya Kamsky, Scott McPeak, and Dawson Engler. A few billion lines of code later: using static analysis to find bugs in the real world. Communications of the ACM, 53(2):66–75, 2010. +Avishkar Bhoopchand, Tim Rocktäschel, Earl Barr, and Sebastian Riedel. Learning Python code suggestion with a sparse pointer network. arXiv preprint arXiv:1611.08307, 2016. + +Benjamin Bichsel, Veselin Raychev, Petar Tsankov, and Martin Vechev. Statistical deobfuscation of android applications. In Conference on Computer and Communications Security (CCS), 2016. + +Pavol Bielik, Veselin Raychev, and Martin Vechev. PHOG: probabilistic model for code. In International Conference on Machine Learning (ICML), 2016. + +Kyunghyun Cho, Bart van Merriënboer, Dzmitry Bahdanau, and Yoshua Bengio. On the properties of neural machine translation: Encoder–decoder approaches. Syntax, Semantics and Structure in Statistical Translation, 2014. + +Michaël Defferrard, Xavier Bresson, and Pierre Vandergheynst. Convolutional neural networks on graphs with fast localized spectral filtering. In Neural Information Processing Systems (NIPS), pp. 3844–3852, 2016. + +Justin Gilmer, Samuel S. Schoenholz, Patrick F. Riley, Oriol Vinyals, and George E. Dahl. Neural message passing for quantum chemistry. arXiv preprint arXiv:1704.01212, 2017. + +Marco Gori, Gabriele Monfardini, and Franco Scarselli. A new model for learning in graph domains. In IEEE International Joint Conference Neural Networks (IJCNN). IEEE, 2005. + +Aditya Grover and Jure Leskovec. node2vec: Scalable feature learning for networks. In International Conference on Knowledge Discovery and Data Mining (SIGKDD), pp. 855–864. ACM, 2016. + +Abram Hindle, Earl T Barr, Zhendong Su, Mark Gabel, and Premkumar Devanbu. On the naturalness of software. In International Conference on Software Engineering (ICSE), 2012. + +Thomas N Kipf and Max Welling. Semi-supervised classification with graph convolutional networks. arXiv preprint arXiv:1609.02907, 2016. + +Yujia Li, Daniel Tarlow, Marc Brockschmidt, and Richard Zemel. Gated graph sequence neural networks. In International Conference on Learning Representations (ICLR), 2015. + +Chris J Maddison and Daniel Tarlow. Structured generative models of natural source code. In International Conference on Machine Learning (ICML), 2014. + +Diego Marcheggiani and Ivan Titov. Encoding sentences with graph convolutional networks for semantic role labeling. In ACL, 2017. + +Tomas Mikolov, Ilya Sutskever, Kai Chen, Greg S Corrado, and Jeff Dean. Distributed representations of words and phrases and their compositionality. In Neural Information Processing Systems (NIPS), 2013. + +Jeffrey Pennington, Richard Socher, and Christopher D Manning. GloVe: Global vectors for word representation. In EMNLP, 2014. + +Veselin Raychev, Martin Vechev, and Eran Yahav. Code completion with statistical language models. In Programming Languages Design and Implementation (PLDI), pp. 419–428, 2014. + +Veselin Raychev, Martin Vechev, and Andreas Krause. Predicting program properties from Big Code. In Principles of Programming Languages (POPL), 2015. + +Veselin Raychev, Pavol Bielik, and Martin Vechev. Probabilistic model for code with decision trees. In Object-Oriented Programming, Systems, Languages, and Applications (OOPSLA), 2016. + +Andrew Rice, Edward Aftandilian, Ciera Jaspan, Emily Johnston, Michael Pradel, and Yulissa Arroyo-Paredes. Detecting argument selection defects. Proceedings of the ACM on Programming Languages, 1(OOPSLA):104, 2017. + +Michael Schlichtkrull, Thomas N. Kipf, Peter Bloem, Rianne van den Berg, Ivan Titov, and Max Welling. Modeling relational data with graph convolutional network. arXiv preprint arXiv:1703.06103, 2017. + +Armando Solar-Lezama. Program synthesis by sketching. University of California, Berkeley, 2008. + +Mingzhe Wang, Yihe Tang, Jian Wang, and Jia Deng. Premise selection for theorem proving by deep graph embedding. In Advances in Neural Information Processing Systems, pp. 2783–2793, 2017. + +![](images/8d4734049cdf665102b0203edbfe7d102c214d1ca0e59c1e940ccf68db6434c3.jpg) +Figure 6: Precision-Recall and ROC curves for the GGNN model on VARMISUSE. Note that the $y$ axis starts from $50 \%$ . + +Table 3: Performance of GGNN model on VARMISUSE per number of type-correct, in-scope candidate variables. Here we compute the performance of the full GGNN model that uses subtokens. + +
# of candidates2346or78+
Accuracy On SEENPROJTEST (%)91.684.581.878.675.177.5
Accuracy On UNSEENPROJTEST (%)85.777.175.769.071.562.4
+ +# A PERFORMANCE CURVES + +Figure 6 shows the ROC and precision-recall curves for the GGNN model. As the reader may observe, setting a false positive rate to $10 \%$ we get a true positive rate5 of $73 \%$ for the SEENPROJTEST and $69 \%$ for the unseen test. This suggests that this model can be practically used at a high precision setting with acceptable performance. + +# B VARMISUSE PREDICTION SAMPLES + +Below we list a set of samples from our SEENPROJTEST projects with comments about the model performance. Code comments and formatting may have been altered for typesetting reasons. The ground truth choice is underlined. + +# Sample 1 + +![](images/abfe6409dabc3b628a3cb9baa2fdc14d3f39bb8b79f6efc499c55f175cb161d8.jpg) + +. The model correctly predicts all variables in the loop. + +# Sample 2 + +![](images/67d1357095f3d80bff9433916647f24a7195c8a63c4dcc5ead0f08be17f5bdaf.jpg) + +#1 name: $86 \%$ , DIR_PATH: $14 \%$ #2 path: $90 \%$ , name: $8 \%$ , DIR_PATH: $2 \%$ #3 path: $76 \%$ , name: $16 \%$ , DIR_PATH: $8 \%$ + +$\triangleright$ String variables are not confused their semantic role is inferred correctly. + +# Sample 3 + +[global::System.Diagnostics.DebuggerNonUserCodeAttribute] +public void MergeFrom(pb::CodedInputStream input) { uint tag; while ((tag $=$ input.ReadTag()) $\ ! = ~ 0$ ) { switch(tag) { default: input.SkipLastField(); break; case 10: { #1 .AddEntriesFrom(input, _repeated_payload_codec); break; } } +} + +#1 Payload: $66 \%$ , payload_: $44 \%$ + +$\triangleright$ The model is commonly confused by aliases, i.e. variables that point to the same location in memory. +In this sample, either choice would have yielded identical behavior. + +# Sample 4 + +![](images/a45880cd7de37c697125e92c3c541033a38ec7e20081fe1d6ce9b55998761b1c.jpg) + +#1 gate: $9 9 \%$ , _observers: $1 \%$ #2 _isDisposed: $90 \%$ , _isStopped: $8 \%$ , HasObservers: $2 \%$ + +. The ReturnsTo edge can help predict variables that otherwise would have been impossible. + +# Sample 5 + +![](images/382d0a5b3bd9d9fd228858bebcaa0e6dfce97d4d8f24faa63d9010e590be558b.jpg) + +#1 error: $93 \%$ , _exception: $7 \%$ +#2 error: $98 \%$ , _exception: $2 \%$ +#3 _gate: $100 \%$ , _observers: $0 \%$ +#4 isStopped: $86 \%$ , _isDisposed: $13 \%$ , HasObservers: $1 \%$ +#5 isStopped: $91 \%$ , _isDisposed: $9 \%$ , HasObservers: $0 \%$ +#6 _exception: $100 \%$ , error: $0 \%$ +#7 error: $98 \%$ , _exception: $2 \%$ +#8 _exception: $9 9 \%$ , error: $1 \%$ + +$\triangleright$ The model predicts the correct variables from all slots apart from the last. Reasoning about the last one, requires interprocedural understanding of the code across the class file. + +# Sample 6 + +private bool BecomingCommand(object message) if (ReceiveCommand #1 return true; if #2 .ToString() $= =$ #3 #4 .Tell #5 else return false; return true; +} + +#1 message: $100 \%$ , Response: $0 \%$ , Message: $0 \%$ #2 message: $100 \%$ , Response: $0 \%$ , Message: $0 \%$ #3 Response: $91 \%$ , Message: $9 \%$ #4 Probe: $98 \%$ , AskedForDelete: $2 \%$ #5 Response: $98 \%$ , Message: $2 \%$ + +. The model predicts correctly all usages except from the one in slot #3. Reasoning about this snippet requires additional semantic information about the intent of the code. + +# Sample 7 + +var response $=$ ResultsFilter(typeof(TResponse), #1 #2 , request); + +#1 httpMethod: $9 9 \%$ , absoluteUrl: $1 \%$ , UserName: $0 \%$ , UserAgent: $0 \%$ #2 absoluteUrl: $9 9 \%$ , httpMethod: $1 \%$ , UserName: $0 \%$ , UserAgent: $0 \%$ + +$\triangleright$ The model knows about selecting the correct string parameters because it matches them to the formal parameter names. + +# Sample 8 + +if #1 $> =$ #2 ) throw new InvalidOperationException(Strings_Core.FAILED_CLOCK_MONITORING) + +#1 n: $100 \%$ , MAXERROR: $0 \%$ , SYNC_MAXRETRIES: $0 \%$ #2 MAXERROR: $62 \%$ , SYNC_MAXRETRIES: $22 \%$ , n: $16 \%$ + +$\triangleright$ It is hard for the model to reason about conditionals, especially with rare constants as in slot #2. + +# C NEAREST NEIGHBOR OF GGNN USAGE REPRESENTATIONS + +Here we show pairs of nearest neighbors based on the cosine similarity of the learned representations $\mathbf { u } ( t , v )$ . Each slot $t$ is marked in dark blue and all usages of $v$ are marked in yellow (i.e. variableName ). This is a set of hand-picked examples showing good and bad examples. A brief description follows after each pair. + +# Sample 1 + +![](images/3af9a60fadeb14347e4f874e0c1dc803122c0a76e9fdc856fd08d2c98658db62.jpg) + +$\triangleright$ Slots that are checked for null-ness have similar representations. + +# Sample 2 + +![](images/d1ada4fdbb0e1a6e130ce0b48d0b02926d9a7f74e9bd9bde7dc55ae52c59ef8f.jpg) + +$\triangleright$ Slots that follow similar API protocols have similar representations. Note that the function HasAddress is a local function, seen only in the testset. + +# Sample 3 + +![](images/2138340e86ff73ed6c4633db684eb824cdcf8b0c1f5f5434c9e65b177116c2f6.jpg) + +$\triangleright$ Adding elements to a collection-like object yields similar representations. + +# D DATASET + +The collected dataset and its characteristics are listed in Table 4. The full dataset as a set of projects and its parsed JSON will become available online. + +Table 4: Projects in our dataset. Ordered alphabetically. kLOC measures the number of non-empty lines of C# code. Projects marked with Devwere used as a development set. Projects marked with †were in the test-only dataset. The rest of the projects were split into train-validation-test. The dataset contains in total about 2.9MLOC. + +
NameGit SHAkLOCsSlotsVarsDescription
Akka.NET719335a124051.3k51.2kActor-based Concurrent &Distributed Framework
AutoMapper2ca7c2b5463.7k10.7kObject-to-Object Mapping Library
BenchmarkDotNet1670ca34285.1k6.1kBenchmarking Library
BotBuilder190117c3446.4k8.7kSDK for Building Bots
choco93985688363.8k5.2kWindows Package Manager
commandline†09677b16111.1k2.3kCommand Line Parser
CommonMark.NETDevf3d54530142.6k1.4kMarkdown Parser
Dapper931c700d183.3k4.7kObject Mapper Library
EntityFrameworkfa0b7ec826333.4k39.3kObject-Relational Mapper
Hangfireffc4912f333.6k6.1kBackground Job Processing Library
Humanizertcclla77e272.4k4.4kString Manipulation and Formatting
Lean†f574bfd719026.4k28.3kAlgorithmic Trading Engine
Nancy72elf614707.5k15.7HTTP Service Framework
Newtonsoft.Json6057d9b812314.9k16.1k JSON Library
Ninject7006297f130.7k2.1kCode Injection Library
NLog643e326a758.3k11.0kLogging Library
Opserver51b032e7243.7k4.5kMonitoring System
OptiKey7d35c718346.1k3.9kAssistive On-Screen Keyboard
orleans Pollye0d6a150 0afdbc32300 3230.7k 3.8k35.6k 9.1kDistributed Virtual Actor Model
Resilience & Transient Fault Handling Library
quartznetb33e6f86499.6k9.8kScheduler
ravendbDev5523092264778.0k82.7kDocument Database
RestSharp70de357b204.0k4.5kREST and HTTP API Client Library
Rx.NET2d146fe518014.0k21.9kReactive Language Extensions
scriptcsf3cc8bcb182.7k4.3kC# Text Editor
ServiceStack6d59da7523138.0k46.2kWeb Framework
ShareX718dd71112522.3k18.1kSharing Application
SignalRfa88089e536.5k10.5kPush Notification Framework
Woxcdaf6272132.0k2.1kApplication Launcher
+ +For this work, we released a large portion of the data, with the exception of projects with a GPL license. The data can be found at https://aka.ms/iclr18-prog-graphs-dataset. Since we are excluding some projects from the data, below we report the results, averaged over three runs, on the published dataset: + +
Accuracy (%)PR AUC
SEENPROJTEST84.00.976
UNSEENPROJTEST74.10.934
\ No newline at end of file diff --git a/parse/train/BJOFETxR-/BJOFETxR-_content_list.json b/parse/train/BJOFETxR-/BJOFETxR-_content_list.json new file mode 100644 index 0000000000000000000000000000000000000000..447c1f2d53c824e820b30c0ac0e7482e4c3cc500 --- /dev/null +++ b/parse/train/BJOFETxR-/BJOFETxR-_content_list.json @@ -0,0 +1,1690 @@ +[ + { + "type": "text", + "text": "LEARNING TO REPRESENT PROGRAMS WITH GRAPHS ", + "text_level": 1, + "bbox": [ + 173, + 98, + 816, + 121 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Miltiadis Allamanis \nMicrosoft Research \nCambridge, UK \nmiallama@microsoft.com \nMarc Brockschmidt \nMicrosoft Research \nCambridge, UK \nmabrocks@microsoft.com ", + "bbox": [ + 183, + 145, + 401, + 200 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "", + "bbox": [ + 596, + 145, + 813, + 200 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Mahmoud Khademi∗ Simon Fraser University Burnaby, BC, Canada mkhademi@sfu.ca ", + "bbox": [ + 183, + 222, + 344, + 277 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "ABSTRACT ", + "text_level": 1, + "bbox": [ + 454, + 314, + 544, + 329 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Learning tasks on source code (i.e., formal languages) have been considered recently, but most work has tried to transfer natural language methods and does not capitalize on the unique opportunities offered by code’s known sematics. For example, long-range dependencies induced by using the same variable or function in distant locations are often not considered. We propose to use graphs to represent both the syntactic and semantic structure of code and use graph-based deep learning methods to learn to reason over program structures. ", + "bbox": [ + 233, + 347, + 764, + 443 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "In this work, we present how to construct graphs from source code and how to scale Gated Graph Neural Networks training to such large graphs. We evaluate our method on two tasks: VARNAMING, in which a network attempts to predict the name of a variable given its usage, and VARMISUSE, in which the network learns to reason about selecting the correct variable that should be used at a given program location. Our comparison to methods that use less structured program representations shows the advantages of modeling known structure, and suggests that our models learn to infer meaningful names and to solve the VARMISUSE task in many cases. Additionally, our testing showed that VARMISUSE identifies a number of bugs in mature open-source projects. ", + "bbox": [ + 233, + 446, + 764, + 584 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "1 INTRODUCTION ", + "text_level": 1, + "bbox": [ + 176, + 612, + 334, + 627 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "The advent of large repositories of source code as well as scalable machine learning methods naturally leads to the idea of “big code”, i.e., largely unsupervised methods that support software engineers by generalizing from existing source code (Allamanis et al., 2017). Currently, existing deep learning models of source code capture its shallow, textual structure, e.g. as a sequence of tokens (Hindle et al., 2012; Raychev et al., 2014; Allamanis et al., 2016), as parse trees (Maddison & Tarlow, 2014; Bielik et al., 2016), or as a flat dependency networks of variables (Raychev et al., 2015). Such models miss out on the opportunity to capitalize on the rich and well-defined semantics of source code. In this work, we take a step to alleviate this by including two additional signal sources in source code: data flow and type hierarchies. We do this by encoding programs as graphs, in which edges represent syntactic relationships (e.g. “token before/after”) as well as semantic relationships (“variable last used/written here”, “formal parameter for argument is called stream”, etc.). Our key insight is that exposing these semantics explicitly as structured input to a machine learning model lessens the requirements on amounts of training data, model capacity and training regime and allows us to solve tasks that are beyond the current state of the art. ", + "bbox": [ + 174, + 643, + 825, + 838 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "We explore two tasks to illustrate the advantages of exposing more semantic structure of programs. First, we consider the VARNAMING task (Allamanis et al., 2014; Raychev et al., 2015), in which given some source code, the “correct” variable name is inferred as a sequence of subtokens. This requires some understanding of how a variable is used, i.e., requires reasoning about lines of code far apart in the source file. Secondly, we introduce the variable misuse prediction task (VARMISUSE), in which the network aims to infer which variable should be used in a program location. To illustrate the task, Figure 1 shows a slightly simplified snippet of a bug our model detected in a popular open-source project. Specifically, instead of the variable clazz, variable first should have been used in the yellow highlighted slot. Existing static analysis methods cannot detect such issues, even though a software engineer would easily identify this as an error from experience. ", + "bbox": [ + 174, + 845, + 825, + 901 + ], + "page_idx": 0 + }, + { + "type": "image", + "img_path": "images/1bc113e2a4c0fb9e0e51adeae6b01b6e7ddf6b162f6335e7e776d43b92e616d2.jpg", + "image_caption": [ + "Figure 1: A snippet of a detected bug in RavenDB an open-source C# project. The code has been slightly simplified. Our model detects correctly that the variable used in the highlighted (yellow) slot is incorrect. Instead, first should have been placed at the slot. We reported this problem which was fixed in PR 4138. " + ], + "image_footnote": [], + "bbox": [ + 169, + 102, + 825, + 214 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "", + "bbox": [ + 174, + 304, + 825, + 388 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "To achieve high accuracy on these tasks, we need to learn representations of program semantics. For both tasks, we need to learn the semantic role of a variable (e.g., “is it a counter?”, “is it a filename?”). Additionally, for VARMISUSE, learning variable usage semantics (e.g., “a filename is needed here”) is required. This “fill the blank element” task is related to methods for learning distributed representations of natural language words, such as Word2Vec (Mikolov et al., 2013) and GLoVe (Pennington et al., 2014). However, we can learn from a much richer structure such as data flow information. This work is a step towards learning program representations, and we expect them to be valuable in a wide range of other tasks, such as code completion (“this is the variable you are looking for”) and more advanced bug finding (“you should lock before using this object”). ", + "bbox": [ + 174, + 395, + 825, + 521 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "To summarize, our contributions are: (i) We define the VARMISUSE task as a challenge for machine learning modeling of source code, that requires to learn (some) semantics of programs (cf. section 3). (ii) We present deep learning models for solving the VARNAMING and VARMISUSE tasks by modeling the code’s graph structure and learning program representations over those graphs (cf. section 4). (iii) We evaluate our models on a large dataset of 2.9 million lines of real-world source code, showing that our best model achieves $3 2 . 9 \\%$ accuracy on the VARNAMING task and $8 5 . 5 \\%$ accuracy on the VARMISUSE task, beating simpler baselines (cf. section 5). (iv) We document practical relevance of VARMISUSE by summarizing some bugs that we found in mature open-source software projects $( c f .$ subsection 5.3). Our implementation of graph neural networks (on a simpler task) can be found at https://github.com/Microsoft/gated-graph-neural-network-samples and the dataset can be found at https://aka.ms/iclr18-prog-graphs-dataset. ", + "bbox": [ + 173, + 526, + 826, + 680 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "2 RELATED WORK ", + "text_level": 1, + "bbox": [ + 176, + 710, + 344, + 728 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "Our work builds upon the recent field of using machine learning for source code artifacts (Allamanis et al., 2017). For example, Hindle et al. (2012); Bhoopchand et al. (2016) model the code as a sequence of tokens, while Maddison & Tarlow (2014); Raychev et al. (2016) model the syntax tree structure of code. All works on language models of code find that predicting variable and method identifiers is one of biggest challenges in the task. ", + "bbox": [ + 174, + 750, + 825, + 819 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "Closest to our work is the work of Allamanis et al. (2015) who learn distributed representations of variables using all their usages to predict their names. However, they do not use data flow information and we are not aware of any model that does so. Raychev et al. (2015) and Bichsel et al. (2016) use conditional random fields to model a variety of relationships between variables, AST elements and types to predict variable names and types (resp. to deobfuscate Android apps), but without considering the flow of data explicitly. In these works, all variable usages are deterministically known beforehand (as the code is complete and remains unmodified), as in Allamanis et al. (2014; 2015). ", + "bbox": [ + 174, + 825, + 825, + 924 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "Our work is remotely related to work on program synthesis using sketches (Solar-Lezama, 2008) and automated code transplantation (Barr et al., 2015). However, these approaches require a set of specifications (e.g. input-output examples, test suites) to complete the gaps, rather than statistics learned from big code. These approaches can be thought as complementary to ours, since we learn to statistically complete the gaps without any need for specifications, by learning common variable usage patterns from code. ", + "bbox": [ + 174, + 103, + 825, + 186 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Neural networks on graphs (Gori et al., 2005; Li et al., 2015; Defferrard et al., 2016; Kipf & Welling, 2016; Gilmer et al., 2017) adapt a variety of deep learning methods to graph-structured input. They have been used in a series of applications, such as link prediction and classification (Grover & Leskovec, 2016) and semantic role labeling in NLP (Marcheggiani & Titov, 2017). Somewhat related to source code is the work of Wang et al. (2017) who learn graph-based representations of mathematical formulas for premise selection in theorem proving. ", + "bbox": [ + 174, + 194, + 825, + 279 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "3 THE VARMISUSE TASK ", + "text_level": 1, + "bbox": [ + 176, + 304, + 400, + 320 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Detecting variable misuses in code is a task that requires understanding and reasoning about program semantics. To successfully tackle the task one needs to infer the role and function of the program elements and understand how they relate. For example, given a program such as Fig. 1, the task is to automatically detect that the marked use of $\\mathtt { C 1 a z z }$ is a mistake and that first should be used instead. While this task resembles standard code completion, it differs significantly in its scope and purpose, by considering only variable identifiers and a mostly complete program. ", + "bbox": [ + 174, + 339, + 825, + 422 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Task Description We view a source code file as a sequence of tokens $t _ { 0 } \\ldots t _ { N } = \\mathcal { T }$ , in which some tokens $t _ { \\lambda _ { 0 } } , t _ { \\lambda _ { 1 } } \\ldots$ are variables. Furthermore, let $\\mathbb { V } _ { t } \\subset \\mathbb { V }$ refer to the set of all type-correct variables in scope at the location of $t$ , i.e., those variables that can be used at $t$ without raising a compiler error. We call a token $t o k _ { \\lambda }$ where we want to predict the correct variable usage a slot. We define a separate task for each slot $t _ { \\lambda }$ : Given $t _ { 0 } \\ldots t _ { \\lambda - 1 }$ and $t _ { \\lambda + 1 } , \\dots , t _ { N }$ , correctly select $t _ { \\lambda }$ from $\\mathbb { V } _ { t _ { \\lambda } }$ . For training and evaluation purposes, a correct solution is one that simply matches the ground truth, but note that in practice, several possible assignments could be considered correct (i.e., when several variables refer to the same value in memory). ", + "bbox": [ + 174, + 444, + 825, + 556 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "4 MODEL: PROGRAMS AS GRAPHS ", + "text_level": 1, + "bbox": [ + 174, + 582, + 478, + 598 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "In this section, we discuss how to transform program source code into program graphs and learn representations over them. These program graphs not only encode the program text but also the semantic information that can be obtained using standard compiler tools. ", + "bbox": [ + 174, + 617, + 823, + 659 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Gated Graph Neural Networks Our work builds on Gated Graph Neural Networks (Li et al., 2015) (GGNN) and we summarize them here. A graph $\\mathcal { G } = ( \\nu , \\pmb { \\varepsilon } , \\pmb { X } )$ is composed of a set of nodes $\\nu$ , node features $\\boldsymbol { X }$ , and a list of directed edge sets $\\pmb { \\mathcal { E } } = ( \\mathcal { E } _ { 1 } , \\ldots , \\mathcal { E } _ { K } )$ where $K$ is the number of edge types. We annotate each $v \\in \\mathcal V$ with a real-valued vector $\\pmb { x } ^ { ( v ) } \\in \\mathbb { R } ^ { D }$ representing the features of the node (e.g., the embedding of a string label of that node). ", + "bbox": [ + 174, + 680, + 825, + 752 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "We associate every node $v$ with a state vector $\\mathbf { \\Omega } _ { h } ( v )$ , initialized from the node label $\\pmb { x } ^ { ( v ) }$ . The sizes of the state vector and feature vector are typically the same, but we can use larger state vectors through padding of node features. To propagate information throughout the graph, “messages” of type $k$ are sent from each $v$ to its neighbors, where each message is computed from its current state vector as m(vk $m _ { k } ^ { ( v ) } = f _ { k } ( { \\pmb h } ^ { ( v ) } )$ . Here, $f _ { k }$ can be an arbitrary function; we choose a linear layer in our case. By computing messages for all graph edges at the same time, all states can be updated at the same time. In particular, a new state for a node $v$ is computed by aggregating all incoming messages as $\\tilde { m } ^ { ( v ) } = g ( \\{ m _ { k } ^ { ( u ) } \\ |$ there is an edge of type $k$ from $u$ to $v \\}$ ). $g$ is an aggregation function, which we implement as elementwise summation. Given the aggregated message $\\tilde { m } ^ { ( v ) }$ and the current state vector $\\mathbf { \\Omega } _ { h } ( v )$ of node $v$ , the state of the next time step $\\boldsymbol { h ^ { \\prime } } ^ { ( v ) }$ is computed as $\\pmb { h } ^ { \\prime ( v ) } = \\mathbf { G } \\mathbf { R } \\mathbf { U } ( \\tilde { \\pmb { m } } ^ { ( v ) } , \\pmb { h } ^ { ( v ) } )$ , where GRU is the recurrent cell function of gated recurrent unit (GRU) (Cho et al., 2014). The ", + "bbox": [ + 174, + 758, + 825, + 924 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "(a) Simplified syntax graph for line 2 of Fig. 1, where blue rounded boxes are syntax nodes, black rectangular boxes syntax tokens, blue edges Child edges and double black edges NextToken edges. ", + "bbox": [ + 173, + 219, + 482, + 270 + ], + "page_idx": 3 + }, + { + "type": "image", + "img_path": "images/f0957bde3bf55bfd9fc41e58429404103dadf9736f4a693bec1cc3c471bc40e7.jpg", + "image_caption": [ + "Figure 2: Examples of graph edges used in program representation. " + ], + "image_footnote": [], + "bbox": [ + 196, + 114, + 459, + 215 + ], + "page_idx": 3 + }, + { + "type": "image", + "img_path": "images/83a96cffbd0f2a1c8368982c7c4bae4e01f2122dfee415273de350b2c11528e5.jpg", + "image_caption": [], + "image_footnote": [], + "bbox": [ + 522, + 102, + 771, + 198 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "(b) Data flow edges for $\\begin{array} { r l r } { ( \\overline { { \\bf x } } ) ^ { 1 } , \\overline { { \\boldsymbol { \\mathbf { y } } } } } & { { } = } & { \\bf F o o ( \\begin{array} { l } { ) } \\ ; } \\end{array} \\end{array}$ while $\\left. \\overline { { \\mathbf { x } } } \\right. ^ { 3 } \\ > \\ \\mathsf { 0 } .$ ) $\\underline { { \\nabla } } \\mathbf { \\Sigma } ^ { 4 } \\mathbf { \\Sigma } = \\underline { { \\nabla } } \\mathbf { \\bar { x } } \\mathbf { \\Sigma } ^ { 5 } \\mathbf { \\Sigma } ^ { \\subset } \\mathbf { \\Sigma } \\boxed { \\nabla } \\mathbf { \\bar { y } } \\mathbf { \\Sigma } ^ { 6 }$ (indices added for clarity), with red dotted LastUse edges, green dashed LastWrite edges and dashdotted purple ComputedFrom edges. ", + "bbox": [ + 516, + 205, + 825, + 270 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "dynamics defined by the above equations are repeated for a fixed number of time steps. Then, we use the state vectors from the last time step as the node representations.1 ", + "bbox": [ + 173, + 314, + 821, + 342 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Program Graphs We represent program source code as graphs and use different edge types to model syntactic and semantic relationships between different tokens. The backbone of a program graph is the program’s abstract syntax tree (AST), consisting of syntax nodes (corresponding to nonterminals in the programming language’s grammar) and syntax tokens (corresponding to terminals). We label syntax nodes with the name of the nonterminal from the program’s grammar, whereas syntax tokens are labeled with the string that they represent. We use Child edges to connect nodes according to the AST. As this does not induce an order on children of a syntax node, we additionally add NextToken edges connecting each syntax token to its successor. An example of this is shown in Fig. 2a. ", + "bbox": [ + 173, + 366, + 825, + 492 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "To capture the flow of control and data through a program, we add additional edges connecting different uses and updates of syntax tokens corresponding to variables. For such a token $v$ , let $\\mathcal { D } ^ { R } ( v )$ be the set of syntax tokens at which the variable could have been used last. This set may contain several nodes (for example, when using a variable after a conditional in which it was used in both branches), and even syntax tokens that follow in the program code (in the case of loops). Similarly, let $\\mathcal { D } ^ { W } ( v )$ be the set of syntax tokens at which the variable was last written to. Using these, we add LastRead (resp. LastWrite) edges connecting $v$ to all elements of $\\mathcal { D } ^ { R } ( v )$ (resp. $\\tilde { \\mathcal { D } } ^ { W } ( v ) )$ . Additionally, whenever we observe an assignment $v = e x p r$ , we connect $v$ to all variable tokens occurring in expr using ComputedFrom edges. An example of such semantic edges is shown in Fig. 2b. ", + "bbox": [ + 173, + 498, + 825, + 637 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "We extend the graph to chain all uses of the same variable using LastLexicalUse edges (independent of data flow, i.e., in if (...) { ... v ...} else { ... v ...}, we link the two occurrences of $v$ ). We also connect return tokens to the method declaration using ReturnsTo edges (this creates a “shortcut” to its name and type). Inspired by Rice et al. (2017), we connect arguments in method calls to the formal parameters that they are matched to with FormalArgName edges, i.e., if we observe a call Foo(bar) and a method declaration Foo(InputStream stream), we connect the bar token to the stream token. Finally, we connect every token corresponding to a variable to enclosing guard expressions that use the variable with GuardedBy and GuardedByNegation edges. For example, in if ( $\\mathrm { ~ \\bf ~ \\cdot ~ x ~ } > \\mathrm { ~ \\bf ~ y ~ }$ ) { ... x ...} else $\\{ \\ \\begin{array} { r l } { \\mathrm { ~ \\texttt ~ { ~ . ~ . ~ } ~ } \\underline { { \\mathrm { ~ y ~ } } } \\mathrm { ~ . ~ . ~ . ~ } \\} } \\end{array}$ , we add a GuardedBy edge from $\\underline { { \\boldsymbol { \\mathrm { X } } } }$ (resp. a GuardedByNegation edge from y) to the AST node corresponding to $\\mathrm { ~ ~ { ~ x ~ } ~ } > \\mathrm { ~ ~ { ~ y ~ } ~ }$ . ", + "bbox": [ + 173, + 643, + 826, + 797 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Finally, for all types of edges we introduce their respective backwards edges (transposing the adjacency matrix), doubling the number of edges and edge types. Backwards edges help with propagating information faster across the GGNN and make the model more expressive. ", + "bbox": [ + 176, + 804, + 823, + 847 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Leveraging Variable Type Information We assume a statically typed language and that the source code can be compiled, and thus each variable has a (known) type $\\tau ( v )$ . To use it, we define a learnable embedding function $\\mathbf { r } ( \\tau )$ for known types and additionally define an “UNKTYPE” for all unknown/unrepresented types. We also leverage the rich type hierarchy that is available in many object-oriented languages. For this, we map a variable’s type $\\tau ( v )$ to the set of its supertypes, i.e. $\\tau ^ { * } \\bar { ( } v ) = \\{ \\tau : \\tau ( v ) \\quad$ implements type $\\tau \\} \\cup \\{ \\bar { \\tau } ( v ) \\}$ . We then compute the type representation $\\mathbf { r } ^ { * } ( v )$ of a variable $v$ as the element-wise maximum of $\\{ \\mathbf { r } ( \\tau ) : \\tau \\in \\tau ^ { * } ( v ) \\}$ . We chose the maximum here, as it is a natural pooling operation for representing partial ordering relations (such as type lattices). Using all types in $\\tau ^ { * } ( v )$ allows us to generalize to unseen types that implement common supertypes or interfaces. For example, List ${ \\tt < K > }$ has multiple concrete types (e.g. List, List). Nevertheless, these types implement a common interface (IList) and share common characteristics. During training, we randomly select a non-empty subset of $\\tau ^ { * } ( v )$ which ensures training of all known types in the lattice. This acts both like a dropout mechanism and allows us to learn a good representation for all types in the type lattice. ", + "bbox": [ + 174, + 103, + 825, + 297 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Initial Node Representation To compute the initial node state, we combine information from the textual representation of the token and its type. Concretely, we split the name of a node representing a token into subtokens (e.g. classTypes will be split into two subtokens class and types) on camelCase and pascal_case. We then average the embeddings of all subtokens to retrieve an embedding for the node name. Finally, we concatenate the learned type representation $\\mathbf { r } ^ { * } ( v )$ , computed as discussed earlier, with the node name representation, and pass it through a linear layer to obtain the initial representations for each node in the graph. ", + "bbox": [ + 174, + 314, + 825, + 411 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Programs Graphs for VARNAMING Given a program and an existing variable $v$ , we build a program graph as discussed above and then replace the variable name in all corresponding variable tokens by a special ${ < } S \\mathrm { L O T } >$ token. To predict a name, we use the initial node labels computed as the concatenation of learnable token embeddings and type embeddings as discussed above, run GGNN propagation for 8 time steps2 and then compute a variable usage representation by averaging the representations for all ${ < } S \\mathrm { L O T } >$ tokens. This representation is then used as the initial state of a one-layer GRU, which predicts the target name as a sequence of subtokens (e.g., the name inputStreamBuffer is treated as the sequence [input, stream, buffer]). We train this graph2seq architecture using a maximum likelihood objective. In section 5, we report the accuracy for predicting the exact name and the F1 score for predicting its subtokens. ", + "bbox": [ + 174, + 426, + 825, + 565 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Program Graphs for VARMISUSE To model VARMISUSE with program graphs we need to modify the graph. First, to compute a context representation $\\mathbf { } c ( t )$ for a slot $t$ where we want to predict the used variable, we insert a new node $v _ { < S \\tt L O T > }$ at the position of $t$ , corresponding to a “hole” at this point, and connect it to the remaining graph using all applicable edges that do not depend on the chosen variable at the slot (i.e., everything but LastUse, LastWrite, LastLexicalUse, and GuardedBy edges). Then, to compute the usage representation $\\mathbf { u } ( t , v )$ of each candidate variable $v$ at the target slot, we insert a “candidate” node $v _ { t , v }$ for all $v$ in $\\mathbb { V } _ { t }$ , and connect it to the graph by inserting the LastUse, LastWrite and LastLexicalUse edges that would be used if the variable were to be used at this slot. Each of these candidate nodes represents the speculative placement of the variable within the scope. ", + "bbox": [ + 174, + 582, + 825, + 720 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Using the initial node representations, concatenated with an extra bit that is set to one for the candidate nodes $v _ { t , v }$ , we run GGNN propagation for 8 time steps.2 The context and usage representation are then the final node states of the nodes, i.e., $\\pmb { c } ( t ) = \\pmb { h } ^ { ( v _ { < \\mathrm { S L O T } > } ) }$ and $\\mathbf { u } ( t , v ) = h ^ { ( v _ { t , v } ) }$ . Finally, the correct variable usage at the location is computed as arg $\\textstyle \\operatorname* { m a x } _ { v } W [ c ( t ) , \\mathbf { u } ( t , v ) ]$ where $W$ is a linear layer that uses the concatenation of $\\mathbf { } c ( t )$ and $\\mathbf { u } ( t , v )$ . We train using a max-margin objective. ", + "bbox": [ + 174, + 728, + 825, + 800 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "4.1 IMPLEMENTATION ", + "text_level": 1, + "bbox": [ + 174, + 818, + 341, + 832 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Using GGNNs for sets of large, diverse graphs requires some engineering effort, as efficient batching is hard in the presence of diverse shapes. An important observation is that large graphs are normally very sparse, and thus a representation of edges as an adjacency list would usually be advantageous to reduce memory consumption. In our case, this can be easily implemented using a sparse tensor representation, allowing large batch sizes that exploit the parallelism of modern GPUs efficiently. A second key insight is to represent a batch of graphs as one large graph with many disconnected components. This just requires appropriate pre-processing to make node identities unique. As this makes batch construction somewhat CPU-intensive, we found it useful to prepare minibatches on a separate thread. Our TensorFlow (Abadi et al., 2016) implementation scales to 55 graphs per second during training and 219 graphs per second during test-time using a single NVidia GeForce GTX Titan X with graphs having on average 2,228 (median 936) nodes and 8,350 (median 3,274) edges and 8 GGNN unrolling iterations, all 20 edge types (forward and backward edges for 10 original edge types) and the size of the hidden layer set to 64. The number of types of edges in the GGNN contributes proportionally to the running time. For example, a GGNN run for our ablation study using only the two most common edge types (NextToken, Child) achieves 105 graphs/second during training and 419 graphs/second at test time with the same hyperparameters. Our (generic) implementation of GGNNs is available at https://github.com/Microsoft/ gated-graph-neural-network-samples, using a simpler demonstration task. ", + "bbox": [ + 176, + 843, + 825, + 900 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "", + "bbox": [ + 174, + 103, + 825, + 299 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "5 EVALUATION ", + "text_level": 1, + "bbox": [ + 176, + 348, + 313, + 364 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Dataset We collected a dataset for the VARMISUSE task from open source $C ^ { \\# }$ projects on GitHub. To select projects, we picked the top-starred (non-fork) projects in GitHub. We then filtered out projects that we could not (easily) compile in full using Roslyn3, as we require a compilation to extract precise type information for the code (including those types present in external libraries). Our final dataset contains 29 projects from a diverse set of domains (compilers, databases, . . . ) with about 2.9 million non-empty lines of code. A full table is shown in Appendix D. ", + "bbox": [ + 174, + 397, + 825, + 481 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "For the task of detecting variable misuses, we collect data from all projects by selecting all variable usage locations, filtering out variable declarations, where at least one other type-compatible replacement variable is in scope. The task is then to infer the correct variable that originally existed in that location. Thus, by construction there is at least one type-correct replacement variable, i.e. picking it would not raise an error during type checking. In our test datasets, at each slot there are on average 3.8 type-correct alternative variables (median 3, $\\sigma = 2 . 6$ ). ", + "bbox": [ + 174, + 488, + 825, + 571 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "From our dataset, we selected two projects as our development set. From the rest of the projects, we selected three projects for UNSEENPROJTEST to allow testing on projects with completely unknown structure and types. We split the remaining 23 projects into train/validation/test sets in the proportion 60-10-30, splitting along files (i.e., all examples from one source file are in the same set). We call the test set obtained like this SEENPROJTEST. ", + "bbox": [ + 174, + 579, + 825, + 648 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Baselines For VARMISUSE, we consider two bidirectional RNN-based baselines. The local model (LOC) is a simple two-layer bidirectional GRU run over the tokens before and after the target location. For this baseline, $\\mathbf { } c ( t )$ is set to the slot representation computed by the RNN, and the usage context of each variable $\\mathbf { u } ( t , v )$ is the embedding of the name and type of the variable, computed in the same way as the initial node labels in the GGNN. This baseline allows us to evaluate how important the usage context information is for this task. The flat dataflow model (AVGBIRNN) is an extension to LOC, where the usage representation $\\mathbf { u } ( t , v )$ is computed using another two-layer bidirectional RNN run over the tokens before/after each usage, and then averaging over the computed representations at the variable token $v$ . The local context, $\\mathbf { } c ( t )$ , is identical to LOC. AVGBIRNN is a significantly stronger baseline that already takes some structural information into account, as the averaging over all variables usages helps with long-range dependencies. Both models pick the variable that maximizes $\\boldsymbol { c } ( t ) ^ { T } \\mathbf { u } ( t , v )$ . ", + "bbox": [ + 173, + 694, + 825, + 861 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "For VARNAMING, we replace LOC by AVGLBL, which uses a log-bilinear model for 4 left and 4 right context tokens of each variable usage, and then averages over these context representations (this corresponds to the model in Allamanis et al. (2015)). We also test AVGBIRNN on VARNAMING, which essentially replaces the log-bilinear context model by a bidirectional RNN. ", + "bbox": [ + 174, + 867, + 825, + 924 + ], + "page_idx": 5 + }, + { + "type": "table", + "img_path": "images/b50ee87c697642af7c15b71b6fc64db9212a957b4661dfb2395c47f5be19c8b6.jpg", + "table_caption": [ + "Table 1: Evaluation of models. SEENPROJTEST refers to the test set containing projects that have files in the training set, UNSEENPROJTEST refers to projects that have no files in the training data. Results averaged over two runs. " + ], + "table_footnote": [], + "table_body": "
SEENPROJTESTUNSEENPROJTEST
LocAVGLBLAVGBIRNNGGNNLocAVGLBLAVGBIRNNGGNN
VARMISUSE
Accuracy (%)50.0二73.785.528.960.278.2
PR AUC0.7880.9410.9800.6110.8950.958
VARNAMING Accuracy (%)36.142.9
53.622.723.444.0
F1 (%)44.050.165.830.632.062.0
", + "bbox": [ + 173, + 143, + 823, + 252 + ], + "page_idx": 6 + }, + { + "type": "table", + "img_path": "images/0af154e3d64763ffe10fe3cdcde26c9644db9e9a108844942ce96db280c1e44b.jpg", + "table_caption": [ + "Table 2: Ablation study for the GGNN model on SEENPROJTEST for the two tasks. " + ], + "table_footnote": [], + "table_body": "
Ablation DescriptionAccuracy (%) VARMISUSE
VARNAMING
Standard Model (reported in Table 1)85.553.6
Only NextToken, Child, LastUse_LastWrite edges80.631.2
Only semantic edges (all but NextToken, Child) Only syntax edges (NextToken, Child)78.4 55.352.9 34.3
Node Labels: Tokens instead of subtokens
Node Labels:Disabled85.6 84.334.5
31.8
", + "bbox": [ + 241, + 282, + 756, + 412 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "5.1 QUANTITATIVE EVALUATION ", + "text_level": 1, + "bbox": [ + 176, + 438, + 416, + 453 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "Table 1 shows the evaluation results of the models for both tasks.4 As LOC captures very little information, it performs relatively badly. AVGLBL and AVGBIRNN, which capture information from many variable usage sites, but do not explicitly encode the rich structure of the problem, still lag behind the GGNN by a wide margin. The performance difference is larger for VARMISUSE, since the structure and the semantics of code are far more important within this setting. ", + "bbox": [ + 174, + 464, + 825, + 535 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "Generalization to new projects Generalizing across a diverse set of source code projects with different domains is an important challenge in machine learning. We repeat the evaluation using the UNSEENPROJTEST set stemming from projects that have no files in the training set. The right side of Table 1 shows that our models still achieve good performance, although it is slightly lower compared to SEENPROJTEST. This is expected since the type lattice is mostly unknown in UNSEENPROJTEST. ", + "bbox": [ + 174, + 551, + 825, + 621 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "We believe that the dominant problem in applying a trained model to an unknown project (i.e., domain) is the fact that its type hierarchy is unknown and the used vocabulary (e.g. in variables, method and class names, etc.) can differ substantially. ", + "bbox": [ + 176, + 627, + 823, + 670 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "Ablation Study To study the effect of some of the design choices for our models, we have run some additional experiments and show their results in Table 2. First, we varied the edges used in the program graph. We find that restricting the model to syntactic information has a large impact on performance on both tasks, whereas restricting it to semantic edges seems to mostly impact performance on VARMISUSE. Similarly, the ComputedFrom, FormalArgName and ReturnsTo edges give a small boost on VARMISUSE, but greatly improve performance on VARNAMING. As evidenced by the experiments with the node label representation, syntax node and token names seem to matter little for VARMISUSE, but naturally have a great impact on VARNAMING. ", + "bbox": [ + 174, + 685, + 825, + 797 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "5.2 QUALITATIVE EVALUATION ", + "text_level": 1, + "bbox": [ + 176, + 815, + 405, + 830 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "Figure 3 illustrates the predictions that GGNN makes on a sample test snippet. The snippet recursively searches for the global directives file by gradually descending into the root folder. Reasoning about the correct variable usages is hard, even for humans, but the GGNN correctly predicts the variable usages at all locations except two (slot 1 and 8). As a software engineer is writing the code, it is imaginable that she may make a mistake misusing one variable in the place of another. Since all variables are string variables, no type errors will be raised. As the probabilities in Fig. 3 suggest most potential variable misuses can be flagged by the model yielding valuable warnings to software engineers. Additional samples with comments can be found in Appendix B. ", + "bbox": [ + 176, + 842, + 823, + 885 + ], + "page_idx": 6 + }, + { + "type": "image", + "img_path": "images/cd8c9c73a863608276a7923a4347ae44724d33325aed2f76dd26669048968bd2.jpg", + "image_caption": [ + "Figure 3: VARMISUSE predictions on slots within a snippet of the SEENPROJTEST set for the ServiceStack project. Additional visualizations are available in Appendix B. The underlined tokens are the correct tokens. The model has to select among a number of string variables at each slot, where all of them represent some kind of path. The GGNN accurately predicts the correct variable usage in 11 out of the 13 slots reasoning about the complex ways the variables interact among them. " + ], + "image_footnote": [], + "bbox": [ + 169, + 106, + 828, + 392 + ], + "page_idx": 7 + }, + { + "type": "image", + "img_path": "images/1a46d813f5e2a8d4ca1585320818302c03d99347f9237a6ad178e6da4a0a1573.jpg", + "image_caption": [ + "Figure 4: A bug found (yellow) in RavenDB open-source project. The code unnecessarily ensures that the buffer is of size length rather than size (which our model predicts as the correct variable here). " + ], + "image_footnote": [], + "bbox": [ + 169, + 497, + 818, + 560 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "", + "bbox": [ + 174, + 637, + 825, + 708 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "Furthermore, Appendix C shows samples of pairs of code snippets that share similar representations as computed by the cosine similarity of the usage representation $\\mathbf { u } ( t , v )$ of GGNN. The reader can notice that the network learns to group variable usages that share semantic similarities together. For example, checking for null before the use of a variable yields similar distributed representations across code segments (Sample 1 in Appendix C). ", + "bbox": [ + 174, + 715, + 825, + 785 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "5.3 DISCOVERED VARIABLE MISUSE BUGS ", + "text_level": 1, + "bbox": [ + 176, + 804, + 486, + 819 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "We have used our VARMISUSE model to identify likely locations of bugs in RavenDB (a document database) and Roslyn (Microsoft’s $C ^ { \\# }$ compiler framework). For this, we manually reviewed a sample of the top 500 locations in both projects where our model was most confident about a choosing a variable differing from the ground truth, and found three bugs in each of the projects. ", + "bbox": [ + 174, + 830, + 825, + 888 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "Figs. 1,4,5 show the issues discovered in RavenDB. The bug in Fig. 1 was possibly caused by copy-pasting, and cannot be easily caught by traditional methods. A compiler will not warn about unused variables (since first is used) and virtually nobody would write a test testing another test. Fig. 4 shows an issue that, although not critical, can lead to increased memory consumption. Fig. 5 shows another issue arising from a non-informative error message. We privately reported three additional bugs to the Roslyn developers, who have fixed the issues in the meantime (cf. https://github.com/dotnet/roslyn/pull/23437). One of the reported bugs could cause a crash in Visual Studio when using certain Roslyn features. ", + "bbox": [ + 174, + 895, + 823, + 924 + ], + "page_idx": 7 + }, + { + "type": "image", + "img_path": "images/336e106c897c8cfa8eb64fc3ca64fbd06fdd03a060af8c210b6cbadf7e1fcc42.jpg", + "image_caption": [ + "Figure 5: A bug found (yellow) in the RavenDB open-source project. Although backupFilename is found to be invalid by IsValidBackup, the user is notified that backupLocation is invalid instead. " + ], + "image_footnote": [], + "bbox": [ + 171, + 102, + 826, + 170 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "", + "bbox": [ + 173, + 242, + 826, + 327 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "Finding these issues in widely released and tested code suggests that our model can be useful during the software development process, complementing classic program analysis tools. For example, one usage scenario would be to guide the code reviewing process to locations a VARMISUSE model has identified as unusual, or use it as a prior to focus testing or expensive code analysis efforts. ", + "bbox": [ + 174, + 334, + 825, + 390 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "6 DISCUSSION & CONCLUSIONS ", + "text_level": 1, + "bbox": [ + 176, + 409, + 459, + 425 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "Although source code is well understood and studied within other disciplines such as programming language research, it is a relatively new domain for deep learning. It presents novel opportunities compared to textual or perceptual data, as its (local) semantics are well-defined and rich additional information can be extracted using well-known, efficient program analyses. On the other hand, integrating this wealth of structured information poses an interesting challenge. Our VARMISUSE task exposes these opportunities, going beyond simpler tasks such as code completion. 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", + "bbox": [ + 171, + 869, + 823, + 886 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "Mingzhe Wang, Yihe Tang, Jian Wang, and Jia Deng. Premise selection for theorem proving by deep graph embedding. In Advances in Neural Information Processing Systems, pp. 2783–2793, 2017. ", + "bbox": [ + 171, + 892, + 823, + 921 + ], + "page_idx": 9 + }, + { + "type": "image", + "img_path": "images/8d4734049cdf665102b0203edbfe7d102c214d1ca0e59c1e940ccf68db6434c3.jpg", + "image_caption": [ + "Figure 6: Precision-Recall and ROC curves for the GGNN model on VARMISUSE. Note that the $y$ axis starts from $50 \\%$ . " + ], + "image_footnote": [], + "bbox": [ + 171, + 99, + 823, + 270 + ], + "page_idx": 10 + }, + { + "type": "table", + "img_path": "images/dd6c7d57e2baa84336760f0c78a1b6a4a7c222370ffeb1c9b7940bd6dc5e4789.jpg", + "table_caption": [ + "Table 3: Performance of GGNN model on VARMISUSE per number of type-correct, in-scope candidate variables. Here we compute the performance of the full GGNN model that uses subtokens. " + ], + "table_footnote": [], + "table_body": "
# of candidates2346or78+
Accuracy On SEENPROJTEST (%)91.684.581.878.675.177.5
Accuracy On UNSEENPROJTEST (%)85.777.175.769.071.562.4
", + "bbox": [ + 218, + 366, + 779, + 425 + ], + "page_idx": 10 + }, + { + "type": "text", + "text": "A PERFORMANCE CURVES ", + "text_level": 1, + "bbox": [ + 178, + 450, + 413, + 467 + ], + "page_idx": 10 + }, + { + "type": "text", + "text": "Figure 6 shows the ROC and precision-recall curves for the GGNN model. As the reader may observe, setting a false positive rate to $10 \\%$ we get a true positive rate5 of $73 \\%$ for the SEENPROJTEST and $69 \\%$ for the unseen test. This suggests that this model can be practically used at a high precision setting with acceptable performance. ", + "bbox": [ + 173, + 483, + 826, + 540 + ], + "page_idx": 10 + }, + { + "type": "text", + "text": "B VARMISUSE PREDICTION SAMPLES", + "text_level": 1, + "bbox": [ + 174, + 563, + 506, + 579 + ], + "page_idx": 10 + }, + { + "type": "text", + "text": "Below we list a set of samples from our SEENPROJTEST projects with comments about the model performance. Code comments and formatting may have been altered for typesetting reasons. The ground truth choice is underlined. ", + "bbox": [ + 173, + 595, + 826, + 638 + ], + "page_idx": 10 + }, + { + "type": "text", + "text": "Sample 1 ", + "text_level": 1, + "bbox": [ + 173, + 643, + 240, + 659 + ], + "page_idx": 10 + }, + { + "type": "image", + "img_path": "images/abfe6409dabc3b628a3cb9baa2fdc14d3f39bb8b79f6efc499c55f175cb161d8.jpg", + "image_caption": [], + "image_footnote": [], + "bbox": [ + 171, + 664, + 831, + 858 + ], + "page_idx": 10 + }, + { + "type": "text", + "text": ". The model correctly predicts all variables in the loop. ", + "bbox": [ + 173, + 868, + 537, + 883 + ], + "page_idx": 10 + }, + { + "type": "text", + "text": "Sample 2 ", + "text_level": 1, + "bbox": [ + 173, + 99, + 241, + 114 + ], + "page_idx": 11 + }, + { + "type": "image", + "img_path": "images/67d1357095f3d80bff9433916647f24a7195c8a63c4dcc5ead0f08be17f5bdaf.jpg", + "image_caption": [], + "image_footnote": [], + "bbox": [ + 171, + 121, + 830, + 174 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": "#1 name: $86 \\%$ , DIR_PATH: $14 \\%$ #2 path: $90 \\%$ , name: $8 \\%$ , DIR_PATH: $2 \\%$ #3 path: $76 \\%$ , name: $16 \\%$ , DIR_PATH: $8 \\%$ ", + "bbox": [ + 174, + 183, + 513, + 234 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": "$\\triangleright$ String variables are not confused their semantic role is inferred correctly. ", + "bbox": [ + 173, + 246, + 663, + 261 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": "Sample 3 ", + "text_level": 1, + "bbox": [ + 173, + 266, + 240, + 280 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": "[global::System.Diagnostics.DebuggerNonUserCodeAttribute] \npublic void MergeFrom(pb::CodedInputStream input) { uint tag; while ((tag $=$ input.ReadTag()) $\\ ! = ~ 0$ ) { switch(tag) { default: input.SkipLastField(); break; case 10: { #1 .AddEntriesFrom(input, _repeated_payload_codec); break; } } \n} ", + "bbox": [ + 171, + 289, + 707, + 484 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": "#1 Payload: $66 \\%$ , payload_: $44 \\%$ ", + "bbox": [ + 186, + 493, + 464, + 511 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": "$\\triangleright$ The model is commonly confused by aliases, i.e. variables that point to the same location in memory. \nIn this sample, either choice would have yielded identical behavior. ", + "bbox": [ + 173, + 522, + 825, + 551 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": "Sample 4 ", + "text_level": 1, + "bbox": [ + 173, + 556, + 241, + 571 + ], + "page_idx": 11 + }, + { + "type": "image", + "img_path": "images/a45880cd7de37c697125e92c3c541033a38ec7e20081fe1d6ce9b55998761b1c.jpg", + "image_caption": [], + "image_footnote": [], + "bbox": [ + 169, + 578, + 828, + 715 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": "#1 gate: $9 9 \\%$ , _observers: $1 \\%$ #2 _isDisposed: $90 \\%$ , _isStopped: $8 \\%$ , HasObservers: $2 \\%$ ", + "bbox": [ + 173, + 726, + 671, + 758 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": ". The ReturnsTo edge can help predict variables that otherwise would have been impossible. ", + "bbox": [ + 166, + 770, + 785, + 786 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": "Sample 5 ", + "text_level": 1, + "bbox": [ + 173, + 99, + 240, + 114 + ], + "page_idx": 12 + }, + { + "type": "image", + "img_path": "images/382d0a5b3bd9d9fd228858bebcaa0e6dfce97d4d8f24faa63d9010e590be558b.jpg", + "image_caption": [], + "image_footnote": [], + "bbox": [ + 166, + 121, + 828, + 534 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "#1 error: $93 \\%$ , _exception: $7 \\%$ \n#2 error: $98 \\%$ , _exception: $2 \\%$ \n#3 _gate: $100 \\%$ , _observers: $0 \\%$ \n#4 isStopped: $86 \\%$ , _isDisposed: $13 \\%$ , HasObservers: $1 \\%$ \n#5 isStopped: $91 \\%$ , _isDisposed: $9 \\%$ , HasObservers: $0 \\%$ \n#6 _exception: $100 \\%$ , error: $0 \\%$ \n#7 error: $98 \\%$ , _exception: $2 \\%$ \n#8 _exception: $9 9 \\%$ , error: $1 \\%$ ", + "bbox": [ + 174, + 545, + 679, + 684 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "$\\triangleright$ The model predicts the correct variables from all slots apart from the last. Reasoning about the last one, requires interprocedural understanding of the code across the class file. ", + "bbox": [ + 171, + 695, + 825, + 724 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "Sample 6 ", + "text_level": 1, + "bbox": [ + 173, + 99, + 241, + 114 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "private bool BecomingCommand(object message) if (ReceiveCommand #1 return true; if #2 .ToString() $= =$ #3 #4 .Tell #5 else return false; return true; \n} ", + "bbox": [ + 169, + 122, + 633, + 219 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "#1 message: $100 \\%$ , Response: $0 \\%$ , Message: $0 \\%$ #2 message: $100 \\%$ , Response: $0 \\%$ , Message: $0 \\%$ #3 Response: $91 \\%$ , Message: $9 \\%$ #4 Probe: $98 \\%$ , AskedForDelete: $2 \\%$ #5 Response: $98 \\%$ , Message: $2 \\%$ ", + "bbox": [ + 174, + 229, + 571, + 318 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": ". The model predicts correctly all usages except from the one in slot #3. Reasoning about this snippet requires additional semantic information about the intent of the code. ", + "bbox": [ + 168, + 329, + 825, + 357 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "Sample 7 ", + "text_level": 1, + "bbox": [ + 173, + 362, + 240, + 377 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "var response $=$ ResultsFilter(typeof(TResponse), #1 #2 , request); ", + "bbox": [ + 166, + 385, + 815, + 404 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "#1 httpMethod: $9 9 \\%$ , absoluteUrl: $1 \\%$ , UserName: $0 \\%$ , UserAgent: $0 \\%$ #2 absoluteUrl: $9 9 \\%$ , httpMethod: $1 \\%$ , UserName: $0 \\%$ , UserAgent: $0 \\%$ ", + "bbox": [ + 194, + 415, + 758, + 449 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "$\\triangleright$ The model knows about selecting the correct string parameters because it matches them to the formal parameter names. ", + "bbox": [ + 173, + 460, + 825, + 489 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "Sample 8 ", + "text_level": 1, + "bbox": [ + 173, + 496, + 241, + 510 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "if #1 $> =$ #2 ) throw new InvalidOperationException(Strings_Core.FAILED_CLOCK_MONITORING) ", + "bbox": [ + 187, + 517, + 843, + 547 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "#1 n: $100 \\%$ , MAXERROR: $0 \\%$ , SYNC_MAXRETRIES: $0 \\%$ #2 MAXERROR: $62 \\%$ , SYNC_MAXRETRIES: $22 \\%$ , n: $16 \\%$ ", + "bbox": [ + 174, + 559, + 596, + 593 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "$\\triangleright$ It is hard for the model to reason about conditionals, especially with rare constants as in slot #2. ", + "bbox": [ + 163, + 604, + 812, + 621 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "C NEAREST NEIGHBOR OF GGNN USAGE REPRESENTATIONS ", + "text_level": 1, + "bbox": [ + 171, + 101, + 705, + 118 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "Here we show pairs of nearest neighbors based on the cosine similarity of the learned representations $\\mathbf { u } ( t , v )$ . Each slot $t$ is marked in dark blue and all usages of $v$ are marked in yellow (i.e. variableName ). This is a set of hand-picked examples showing good and bad examples. A brief description follows after each pair. ", + "bbox": [ + 173, + 217, + 828, + 276 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "Sample 1 ", + "text_level": 1, + "bbox": [ + 173, + 281, + 240, + 296 + ], + "page_idx": 14 + }, + { + "type": "image", + "img_path": "images/3af9a60fadeb14347e4f874e0c1dc803122c0a76e9fdc856fd08d2c98658db62.jpg", + "image_caption": [], + "image_footnote": [], + "bbox": [ + 165, + 299, + 831, + 590 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "$\\triangleright$ Slots that are checked for null-ness have similar representations. ", + "bbox": [ + 171, + 595, + 606, + 611 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "Sample 2 ", + "text_level": 1, + "bbox": [ + 173, + 616, + 240, + 631 + ], + "page_idx": 14 + }, + { + "type": "image", + "img_path": "images/d1ada4fdbb0e1a6e130ce0b48d0b02926d9a7f74e9bd9bde7dc55ae52c59ef8f.jpg", + "image_caption": [], + "image_footnote": [], + "bbox": [ + 166, + 635, + 833, + 895 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "$\\triangleright$ Slots that follow similar API protocols have similar representations. Note that the function HasAddress is a local function, seen only in the testset. ", + "bbox": [ + 171, + 898, + 826, + 928 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "Sample 3 ", + "text_level": 1, + "bbox": [ + 173, + 99, + 240, + 114 + ], + "page_idx": 15 + }, + { + "type": "image", + "img_path": "images/2138340e86ff73ed6c4633db684eb824cdcf8b0c1f5f5434c9e65b177116c2f6.jpg", + "image_caption": [], + "image_footnote": [], + "bbox": [ + 169, + 117, + 831, + 296 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "$\\triangleright$ Adding elements to a collection-like object yields similar representations. ", + "bbox": [ + 173, + 301, + 668, + 316 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "D DATASET ", + "text_level": 1, + "bbox": [ + 174, + 351, + 287, + 367 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "The collected dataset and its characteristics are listed in Table 4. The full dataset as a set of projects and its parsed JSON will become available online. ", + "bbox": [ + 173, + 388, + 825, + 417 + ], + "page_idx": 15 + }, + { + "type": "table", + "img_path": "images/cbc14fa71ce2f75da12cf724042a0e7a75e4183c82fc6295ea8172f5e72f36df.jpg", + "table_caption": [ + "Table 4: Projects in our dataset. Ordered alphabetically. kLOC measures the number of non-empty lines of C# code. Projects marked with Devwere used as a development set. Projects marked with †were in the test-only dataset. The rest of the projects were split into train-validation-test. The dataset contains in total about 2.9MLOC. " + ], + "table_footnote": [], + "table_body": "
NameGit SHAkLOCsSlotsVarsDescription
Akka.NET719335a124051.3k51.2kActor-based Concurrent &Distributed Framework
AutoMapper2ca7c2b5463.7k10.7kObject-to-Object Mapping Library
BenchmarkDotNet1670ca34285.1k6.1kBenchmarking Library
BotBuilder190117c3446.4k8.7kSDK for Building Bots
choco93985688363.8k5.2kWindows Package Manager
commandline†09677b16111.1k2.3kCommand Line Parser
CommonMark.NETDevf3d54530142.6k1.4kMarkdown Parser
Dapper931c700d183.3k4.7kObject Mapper Library
EntityFrameworkfa0b7ec826333.4k39.3kObject-Relational Mapper
Hangfireffc4912f333.6k6.1kBackground Job Processing Library
Humanizertcclla77e272.4k4.4kString Manipulation and Formatting
Lean†f574bfd719026.4k28.3kAlgorithmic Trading Engine
Nancy72elf614707.5k15.7HTTP Service Framework
Newtonsoft.Json6057d9b812314.9k16.1k JSON Library
Ninject7006297f130.7k2.1kCode Injection Library
NLog643e326a758.3k11.0kLogging Library
Opserver51b032e7243.7k4.5kMonitoring System
OptiKey7d35c718346.1k3.9kAssistive On-Screen Keyboard
orleans Pollye0d6a150 0afdbc32300 3230.7k 3.8k35.6k 9.1kDistributed Virtual Actor Model
Resilience & Transient Fault Handling Library
quartznetb33e6f86499.6k9.8kScheduler
ravendbDev5523092264778.0k82.7kDocument Database
RestSharp70de357b204.0k4.5kREST and HTTP API Client Library
Rx.NET2d146fe518014.0k21.9kReactive Language Extensions
scriptcsf3cc8bcb182.7k4.3kC# Text Editor
ServiceStack6d59da7523138.0k46.2kWeb Framework
ShareX718dd71112522.3k18.1kSharing Application
SignalRfa88089e536.5k10.5kPush Notification Framework
Woxcdaf6272132.0k2.1kApplication Launcher
", + "bbox": [ + 178, + 507, + 820, + 921 + ], + "page_idx": 15 + }, + { + "type": "table", + "img_path": "images/72b636af8dc21676160a77f576e621c898d29fb40316b34c6acb8fc9ae0dc020.jpg", + "table_caption": [ + "For this work, we released a large portion of the data, with the exception of projects with a GPL license. The data can be found at https://aka.ms/iclr18-prog-graphs-dataset. Since we are excluding some projects from the data, below we report the results, averaged over three runs, on the published dataset: " + ], + "table_footnote": [], + "table_body": "
Accuracy (%)PR AUC
SEENPROJTEST84.00.976
UNSEENPROJTEST74.10.934
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Existing static analysis methods cannot detect such issues, even though a", + "type": "text", + "cross_page": true + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 296, + 397, + 309 + ], + "spans": [ + { + "bbox": [ + 105, + 296, + 397, + 309 + ], + "score": 1.0, + "content": "software engineer would easily identify this as an error from experience.", + "type": "text", + "cross_page": true + } + ], + "index": 12 + } + ], + "index": 47.5, + "bbox_fs": [ + 105, + 669, + 506, + 715 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "image", + "bbox": [ + 104, + 81, + 505, + 170 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 104, + 81, + 505, + 170 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 104, + 81, + 505, + 170 + ], + "spans": [ + { + "bbox": [ + 104, + 81, + 505, + 170 + ], + "score": 0.62, + "type": "image", + "image_path": "1bc113e2a4c0fb9e0e51adeae6b01b6e7ddf6b162f6335e7e776d43b92e616d2.jpg" + } + ] + } + ], + "index": 1, + "virtual_lines": [ + { + "bbox": [ + 104, + 81, + 505, + 110.66666666666667 + ], + "spans": [], + "index": 0 + }, + { + "bbox": [ + 104, + 110.66666666666667, + 505, + 140.33333333333334 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 104, + 140.33333333333334, + 505, + 170.0 + ], + "spans": [], + "index": 2 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 106, + 175, + 506, + 219 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 106, + 174, + 505, + 187 + ], + "spans": [ + { + "bbox": [ + 106, + 174, + 505, + 187 + ], + "score": 1.0, + "content": "Figure 1: A snippet of a detected bug in RavenDB an open-source C# project. The code has been", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 106, + 186, + 506, + 198 + ], + "spans": [ + { + "bbox": [ + 106, + 186, + 506, + 198 + ], + "score": 1.0, + "content": "slightly simplified. Our model detects correctly that the variable used in the highlighted (yellow) slot", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 198, + 505, + 209 + ], + "spans": [ + { + "bbox": [ + 105, + 198, + 505, + 209 + ], + "score": 1.0, + "content": "is incorrect. Instead, first should have been placed at the slot. We reported this problem which", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 208, + 196, + 219 + ], + "spans": [ + { + "bbox": [ + 105, + 208, + 196, + 219 + ], + "score": 1.0, + "content": "was fixed in PR 4138.", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 4.5 + } + ], + "index": 2.75 + }, + { + "type": "text", + "bbox": [ + 107, + 241, + 505, + 308 + ], + "lines": [ + { + "bbox": [ + 105, + 241, + 506, + 254 + ], + "spans": [ + { + "bbox": [ + 105, + 241, + 506, + 254 + ], + "score": 1.0, + "content": "apart in the source file. Secondly, we introduce the variable misuse prediction task (VARMISUSE), in", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 253, + 505, + 266 + ], + "spans": [ + { + "bbox": [ + 105, + 253, + 505, + 266 + ], + "score": 1.0, + "content": "which the network aims to infer which variable should be used in a program location. To illustrate the", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 263, + 505, + 276 + ], + "spans": [ + { + "bbox": [ + 105, + 263, + 505, + 276 + ], + "score": 1.0, + "content": "task, Figure 1 shows a slightly simplified snippet of a bug our model detected in a popular open-source", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 275, + 505, + 286 + ], + "spans": [ + { + "bbox": [ + 105, + 275, + 505, + 286 + ], + "score": 1.0, + "content": "project. Specifically, instead of the variable clazz, variable first should have been used in the", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 286, + 506, + 298 + ], + "spans": [ + { + "bbox": [ + 105, + 286, + 506, + 298 + ], + "score": 1.0, + "content": "yellow highlighted slot. Existing static analysis methods cannot detect such issues, even though a", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 296, + 397, + 309 + ], + "spans": [ + { + "bbox": [ + 105, + 296, + 397, + 309 + ], + "score": 1.0, + "content": "software engineer would easily identify this as an error from experience.", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 9.5 + }, + { + "type": "text", + "bbox": [ + 107, + 313, + 505, + 413 + ], + "lines": [ + { + "bbox": [ + 106, + 313, + 506, + 325 + ], + "spans": [ + { + "bbox": [ + 106, + 313, + 506, + 325 + ], + "score": 1.0, + "content": "To achieve high accuracy on these tasks, we need to learn representations of program semantics.", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 324, + 506, + 337 + ], + "spans": [ + { + "bbox": [ + 105, + 324, + 506, + 337 + ], + "score": 1.0, + "content": "For both tasks, we need to learn the semantic role of a variable (e.g., “is it a counter?”, “is it a", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 334, + 506, + 348 + ], + "spans": [ + { + "bbox": [ + 105, + 334, + 506, + 348 + ], + "score": 1.0, + "content": "filename?”). Additionally, for VARMISUSE, learning variable usage semantics (e.g., “a filename", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 344, + 505, + 360 + ], + "spans": [ + { + "bbox": [ + 105, + 344, + 505, + 360 + ], + "score": 1.0, + "content": "is needed here”) is required. This “fill the blank element” task is related to methods for learning", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 357, + 505, + 370 + ], + "spans": [ + { + "bbox": [ + 105, + 357, + 505, + 370 + ], + "score": 1.0, + "content": "distributed representations of natural language words, such as Word2Vec (Mikolov et al., 2013) and", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 367, + 505, + 381 + ], + "spans": [ + { + "bbox": [ + 105, + 367, + 505, + 381 + ], + "score": 1.0, + "content": "GLoVe (Pennington et al., 2014). However, we can learn from a much richer structure such as data", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 378, + 505, + 392 + ], + "spans": [ + { + "bbox": [ + 105, + 378, + 505, + 392 + ], + "score": 1.0, + "content": "flow information. This work is a step towards learning program representations, and we expect them", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 388, + 505, + 404 + ], + "spans": [ + { + "bbox": [ + 105, + 388, + 505, + 404 + ], + "score": 1.0, + "content": "to be valuable in a wide range of other tasks, such as code completion (“this is the variable you are", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 402, + 468, + 414 + ], + "spans": [ + { + "bbox": [ + 106, + 402, + 468, + 414 + ], + "score": 1.0, + "content": "looking for”) and more advanced bug finding (“you should lock before using this object”).", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 17 + }, + { + "type": "text", + "bbox": [ + 106, + 417, + 506, + 539 + ], + "lines": [ + { + "bbox": [ + 105, + 418, + 505, + 430 + ], + "spans": [ + { + "bbox": [ + 105, + 418, + 505, + 430 + ], + "score": 1.0, + "content": "To summarize, our contributions are: (i) We define the VARMISUSE task as a challenge for machine", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 429, + 506, + 442 + ], + "spans": [ + { + "bbox": [ + 105, + 429, + 506, + 442 + ], + "score": 1.0, + "content": "learning modeling of source code, that requires to learn (some) semantics of programs (cf. section 3).", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 439, + 506, + 453 + ], + "spans": [ + { + "bbox": [ + 105, + 439, + 506, + 453 + ], + "score": 1.0, + "content": "(ii) We present deep learning models for solving the VARNAMING and VARMISUSE tasks by modeling", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 451, + 506, + 464 + ], + "spans": [ + { + "bbox": [ + 105, + 451, + 506, + 464 + ], + "score": 1.0, + "content": "the code’s graph structure and learning program representations over those graphs (cf. section 4).", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 460, + 506, + 475 + ], + "spans": [ + { + "bbox": [ + 105, + 460, + 506, + 475 + ], + "score": 1.0, + "content": "(iii) We evaluate our models on a large dataset of 2.9 million lines of real-world source code, showing", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 473, + 506, + 485 + ], + "spans": [ + { + "bbox": [ + 105, + 473, + 223, + 485 + ], + "score": 1.0, + "content": "that our best model achieves", + "type": "text" + }, + { + "bbox": [ + 223, + 473, + 250, + 483 + ], + "score": 0.87, + "content": "3 2 . 9 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 251, + 473, + 411, + 485 + ], + "score": 1.0, + "content": "accuracy on the VARNAMING task and", + "type": "text" + }, + { + "bbox": [ + 411, + 473, + 439, + 483 + ], + "score": 0.87, + "content": "8 5 . 5 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 439, + 473, + 506, + 485 + ], + "score": 1.0, + "content": "accuracy on the", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 483, + 506, + 496 + ], + "spans": [ + { + "bbox": [ + 105, + 483, + 506, + 496 + ], + "score": 1.0, + "content": "VARMISUSE task, beating simpler baselines (cf. section 5). (iv) We document practical relevance of", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 494, + 505, + 507 + ], + "spans": [ + { + "bbox": [ + 105, + 494, + 493, + 507 + ], + "score": 1.0, + "content": "VARMISUSE by summarizing some bugs that we found in mature open-source software projects", + "type": "text" + }, + { + "bbox": [ + 493, + 495, + 505, + 506 + ], + "score": 0.29, + "content": "( c f .", + "type": "inline_equation" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 505, + 506, + 519 + ], + "spans": [ + { + "bbox": [ + 105, + 505, + 506, + 519 + ], + "score": 1.0, + "content": "subsection 5.3). Our implementation of graph neural networks (on a simpler task) can be found at", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 517, + 506, + 529 + ], + "spans": [ + { + "bbox": [ + 105, + 517, + 506, + 529 + ], + "score": 1.0, + "content": "https://github.com/Microsoft/gated-graph-neural-network-samples and", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 527, + 467, + 540 + ], + "spans": [ + { + "bbox": [ + 105, + 527, + 467, + 540 + ], + "score": 1.0, + "content": "the dataset can be found at https://aka.ms/iclr18-prog-graphs-dataset.", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 27 + }, + { + "type": "title", + "bbox": [ + 108, + 563, + 211, + 577 + ], + "lines": [ + { + "bbox": [ + 105, + 563, + 213, + 578 + ], + "spans": [ + { + "bbox": [ + 105, + 563, + 213, + 578 + ], + "score": 1.0, + "content": "2 RELATED WORK", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 33 + }, + { + "type": "text", + "bbox": [ + 107, + 594, + 505, + 649 + ], + "lines": [ + { + "bbox": [ + 106, + 594, + 505, + 606 + ], + "spans": [ + { + "bbox": [ + 106, + 594, + 505, + 606 + ], + "score": 1.0, + "content": "Our work builds upon the recent field of using machine learning for source code artifacts (Allamanis", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 604, + 506, + 618 + ], + "spans": [ + { + "bbox": [ + 105, + 604, + 506, + 618 + ], + "score": 1.0, + "content": "et al., 2017). For example, Hindle et al. (2012); Bhoopchand et al. (2016) model the code as a", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 615, + 506, + 629 + ], + "spans": [ + { + "bbox": [ + 105, + 615, + 506, + 629 + ], + "score": 1.0, + "content": "sequence of tokens, while Maddison & Tarlow (2014); Raychev et al. (2016) model the syntax tree", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 106, + 627, + 505, + 639 + ], + "spans": [ + { + "bbox": [ + 106, + 627, + 505, + 639 + ], + "score": 1.0, + "content": "structure of code. All works on language models of code find that predicting variable and method", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 637, + 308, + 651 + ], + "spans": [ + { + "bbox": [ + 105, + 637, + 308, + 651 + ], + "score": 1.0, + "content": "identifiers is one of biggest challenges in the task.", + "type": "text" + } + ], + "index": 38 + } + ], + "index": 36 + }, + { + "type": "text", + "bbox": [ + 107, + 654, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 655, + 505, + 667 + ], + "spans": [ + { + "bbox": [ + 106, + 655, + 505, + 667 + ], + "score": 1.0, + "content": "Closest to our work is the work of Allamanis et al. (2015) who learn distributed representations of", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 666, + 506, + 677 + ], + "spans": [ + { + "bbox": [ + 105, + 666, + 506, + 677 + ], + "score": 1.0, + "content": "variables using all their usages to predict their names. However, they do not use data flow information", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 677, + 506, + 689 + ], + "spans": [ + { + "bbox": [ + 105, + 677, + 506, + 689 + ], + "score": 1.0, + "content": "and we are not aware of any model that does so. Raychev et al. (2015) and Bichsel et al. 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The code has been", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 106, + 186, + 506, + 198 + ], + "spans": [ + { + "bbox": [ + 106, + 186, + 506, + 198 + ], + "score": 1.0, + "content": "slightly simplified. Our model detects correctly that the variable used in the highlighted (yellow) slot", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 198, + 505, + 209 + ], + "spans": [ + { + "bbox": [ + 105, + 198, + 505, + 209 + ], + "score": 1.0, + "content": "is incorrect. Instead, first should have been placed at the slot. We reported this problem which", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 208, + 196, + 219 + ], + "spans": [ + { + "bbox": [ + 105, + 208, + 196, + 219 + ], + "score": 1.0, + "content": "was fixed in PR 4138.", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 4.5 + } + ], + "index": 2.75 + }, + { + "type": "text", + "bbox": [ + 107, + 241, + 505, + 308 + ], + "lines": [], + "index": 9.5, + "bbox_fs": [ + 105, + 241, + 506, + 309 + ], + "lines_deleted": true + }, + { + "type": "text", + "bbox": [ + 107, + 313, + 505, + 413 + ], + "lines": [ + { + "bbox": [ + 106, + 313, + 506, + 325 + ], + "spans": [ + { + "bbox": [ + 106, + 313, + 506, + 325 + ], + "score": 1.0, + "content": "To achieve high accuracy on these tasks, we need to learn representations of program semantics.", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 324, + 506, + 337 + ], + "spans": [ + { + "bbox": [ + 105, + 324, + 506, + 337 + ], + "score": 1.0, + "content": "For both tasks, we need to learn the semantic role of a variable (e.g., “is it a counter?”, “is it a", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 334, + 506, + 348 + ], + "spans": [ + { + "bbox": [ + 105, + 334, + 506, + 348 + ], + "score": 1.0, + "content": "filename?”). Additionally, for VARMISUSE, learning variable usage semantics (e.g., “a filename", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 344, + 505, + 360 + ], + "spans": [ + { + "bbox": [ + 105, + 344, + 505, + 360 + ], + "score": 1.0, + "content": "is needed here”) is required. This “fill the blank element” task is related to methods for learning", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 357, + 505, + 370 + ], + "spans": [ + { + "bbox": [ + 105, + 357, + 505, + 370 + ], + "score": 1.0, + "content": "distributed representations of natural language words, such as Word2Vec (Mikolov et al., 2013) and", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 367, + 505, + 381 + ], + "spans": [ + { + "bbox": [ + 105, + 367, + 505, + 381 + ], + "score": 1.0, + "content": "GLoVe (Pennington et al., 2014). However, we can learn from a much richer structure such as data", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 378, + 505, + 392 + ], + "spans": [ + { + "bbox": [ + 105, + 378, + 505, + 392 + ], + "score": 1.0, + "content": "flow information. This work is a step towards learning program representations, and we expect them", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 388, + 505, + 404 + ], + "spans": [ + { + "bbox": [ + 105, + 388, + 505, + 404 + ], + "score": 1.0, + "content": "to be valuable in a wide range of other tasks, such as code completion (“this is the variable you are", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 402, + 468, + 414 + ], + "spans": [ + { + "bbox": [ + 106, + 402, + 468, + 414 + ], + "score": 1.0, + "content": "looking for”) and more advanced bug finding (“you should lock before using this object”).", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 17, + "bbox_fs": [ + 105, + 313, + 506, + 414 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 417, + 506, + 539 + ], + "lines": [ + { + "bbox": [ + 105, + 418, + 505, + 430 + ], + "spans": [ + { + "bbox": [ + 105, + 418, + 505, + 430 + ], + "score": 1.0, + "content": "To summarize, our contributions are: (i) We define the VARMISUSE task as a challenge for machine", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 429, + 506, + 442 + ], + "spans": [ + { + "bbox": [ + 105, + 429, + 506, + 442 + ], + "score": 1.0, + "content": "learning modeling of source code, that requires to learn (some) semantics of programs (cf. section 3).", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 439, + 506, + 453 + ], + "spans": [ + { + "bbox": [ + 105, + 439, + 506, + 453 + ], + "score": 1.0, + "content": "(ii) We present deep learning models for solving the VARNAMING and VARMISUSE tasks by modeling", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 451, + 506, + 464 + ], + "spans": [ + { + "bbox": [ + 105, + 451, + 506, + 464 + ], + "score": 1.0, + "content": "the code’s graph structure and learning program representations over those graphs (cf. section 4).", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 460, + 506, + 475 + ], + "spans": [ + { + "bbox": [ + 105, + 460, + 506, + 475 + ], + "score": 1.0, + "content": "(iii) We evaluate our models on a large dataset of 2.9 million lines of real-world source code, showing", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 473, + 506, + 485 + ], + "spans": [ + { + "bbox": [ + 105, + 473, + 223, + 485 + ], + "score": 1.0, + "content": "that our best model achieves", + "type": "text" + }, + { + "bbox": [ + 223, + 473, + 250, + 483 + ], + "score": 0.87, + "content": "3 2 . 9 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 251, + 473, + 411, + 485 + ], + "score": 1.0, + "content": "accuracy on the VARNAMING task and", + "type": "text" + }, + { + "bbox": [ + 411, + 473, + 439, + 483 + ], + "score": 0.87, + "content": "8 5 . 5 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 439, + 473, + 506, + 485 + ], + "score": 1.0, + "content": "accuracy on the", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 483, + 506, + 496 + ], + "spans": [ + { + "bbox": [ + 105, + 483, + 506, + 496 + ], + "score": 1.0, + "content": "VARMISUSE task, beating simpler baselines (cf. section 5). (iv) We document practical relevance of", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 494, + 505, + 507 + ], + "spans": [ + { + "bbox": [ + 105, + 494, + 493, + 507 + ], + "score": 1.0, + "content": "VARMISUSE by summarizing some bugs that we found in mature open-source software projects", + "type": "text" + }, + { + "bbox": [ + 493, + 495, + 505, + 506 + ], + "score": 0.29, + "content": "( c f .", + "type": "inline_equation" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 505, + 506, + 519 + ], + "spans": [ + { + "bbox": [ + 105, + 505, + 506, + 519 + ], + "score": 1.0, + "content": "subsection 5.3). Our implementation of graph neural networks (on a simpler task) can be found at", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 517, + 506, + 529 + ], + "spans": [ + { + "bbox": [ + 105, + 517, + 506, + 529 + ], + "score": 1.0, + "content": "https://github.com/Microsoft/gated-graph-neural-network-samples and", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 527, + 467, + 540 + ], + "spans": [ + { + "bbox": [ + 105, + 527, + 467, + 540 + ], + "score": 1.0, + "content": "the dataset can be found at https://aka.ms/iclr18-prog-graphs-dataset.", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 27, + "bbox_fs": [ + 105, + 418, + 506, + 540 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 563, + 211, + 577 + ], + "lines": [ + { + "bbox": [ + 105, + 563, + 213, + 578 + ], + "spans": [ + { + "bbox": [ + 105, + 563, + 213, + 578 + ], + "score": 1.0, + "content": "2 RELATED WORK", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 33 + }, + { + "type": "text", + "bbox": [ + 107, + 594, + 505, + 649 + ], + "lines": [ + { + "bbox": [ + 106, + 594, + 505, + 606 + ], + "spans": [ + { + "bbox": [ + 106, + 594, + 505, + 606 + ], + "score": 1.0, + "content": "Our work builds upon the recent field of using machine learning for source code artifacts (Allamanis", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 604, + 506, + 618 + ], + "spans": [ + { + "bbox": [ + 105, + 604, + 506, + 618 + ], + "score": 1.0, + "content": "et al., 2017). For example, Hindle et al. (2012); Bhoopchand et al. (2016) model the code as a", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 615, + 506, + 629 + ], + "spans": [ + { + "bbox": [ + 105, + 615, + 506, + 629 + ], + "score": 1.0, + "content": "sequence of tokens, while Maddison & Tarlow (2014); Raychev et al. (2016) model the syntax tree", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 106, + 627, + 505, + 639 + ], + "spans": [ + { + "bbox": [ + 106, + 627, + 505, + 639 + ], + "score": 1.0, + "content": "structure of code. All works on language models of code find that predicting variable and method", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 637, + 308, + 651 + ], + "spans": [ + { + "bbox": [ + 105, + 637, + 308, + 651 + ], + "score": 1.0, + "content": "identifiers is one of biggest challenges in the task.", + "type": "text" + } + ], + "index": 38 + } + ], + "index": 36, + "bbox_fs": [ + 105, + 594, + 506, + 651 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 654, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 655, + 505, + 667 + ], + "spans": [ + { + "bbox": [ + 106, + 655, + 505, + 667 + ], + "score": 1.0, + "content": "Closest to our work is the work of Allamanis et al. (2015) who learn distributed representations of", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 666, + 506, + 677 + ], + "spans": [ + { + "bbox": [ + 105, + 666, + 506, + 677 + ], + "score": 1.0, + "content": "variables using all their usages to predict their names. However, they do not use data flow information", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 677, + 506, + 689 + ], + "spans": [ + { + "bbox": [ + 105, + 677, + 506, + 689 + ], + "score": 1.0, + "content": "and we are not aware of any model that does so. Raychev et al. (2015) and Bichsel et al. (2016) use", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 688, + 505, + 699 + ], + "spans": [ + { + "bbox": [ + 105, + 688, + 505, + 699 + ], + "score": 1.0, + "content": "conditional random fields to model a variety of relationships between variables, AST elements and", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 698, + 505, + 712 + ], + "spans": [ + { + "bbox": [ + 105, + 698, + 505, + 712 + ], + "score": 1.0, + "content": "types to predict variable names and types (resp. to deobfuscate Android apps), but without considering", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 709, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 105, + 709, + 505, + 722 + ], + "score": 1.0, + "content": "the flow of data explicitly. In these works, all variable usages are deterministically known beforehand", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 721, + 452, + 733 + ], + "spans": [ + { + "bbox": [ + 105, + 721, + 452, + 733 + ], + "score": 1.0, + "content": "(as the code is complete and remains unmodified), as in Allamanis et al. (2014; 2015).", + "type": "text" + } + ], + "index": 45 + } + ], + "index": 42, + "bbox_fs": [ + 105, + 655, + 506, + 733 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 82, + 505, + 148 + ], + "lines": [ + { + "bbox": [ + 106, + 82, + 506, + 95 + ], + "spans": [ + { + "bbox": [ + 106, + 82, + 506, + 95 + ], + "score": 1.0, + "content": "Our work is remotely related to work on program synthesis using sketches (Solar-Lezama, 2008)", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 93, + 505, + 105 + ], + "spans": [ + { + "bbox": [ + 106, + 93, + 505, + 105 + ], + "score": 1.0, + "content": "and automated code transplantation (Barr et al., 2015). However, these approaches require a set of", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 105, + 506, + 117 + ], + "spans": [ + { + "bbox": [ + 105, + 105, + 506, + 117 + ], + "score": 1.0, + "content": "specifications (e.g. input-output examples, test suites) to complete the gaps, rather than statistics", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 115, + 505, + 128 + ], + "spans": [ + { + "bbox": [ + 105, + 115, + 505, + 128 + ], + "score": 1.0, + "content": "learned from big code. These approaches can be thought as complementary to ours, since we learn", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 127, + 506, + 139 + ], + "spans": [ + { + "bbox": [ + 105, + 127, + 506, + 139 + ], + "score": 1.0, + "content": "to statistically complete the gaps without any need for specifications, by learning common variable", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 138, + 211, + 149 + ], + "spans": [ + { + "bbox": [ + 105, + 138, + 211, + 149 + ], + "score": 1.0, + "content": "usage patterns from code.", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 2.5 + }, + { + "type": "text", + "bbox": [ + 107, + 154, + 505, + 221 + ], + "lines": [ + { + "bbox": [ + 105, + 152, + 507, + 168 + ], + "spans": [ + { + "bbox": [ + 105, + 152, + 507, + 168 + ], + "score": 1.0, + "content": "Neural networks on graphs (Gori et al., 2005; Li et al., 2015; Defferrard et al., 2016; Kipf & Welling,", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 164, + 506, + 178 + ], + "spans": [ + { + "bbox": [ + 105, + 164, + 506, + 178 + ], + "score": 1.0, + "content": "2016; Gilmer et al., 2017) adapt a variety of deep learning methods to graph-structured input. They", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 177, + 506, + 188 + ], + "spans": [ + { + "bbox": [ + 106, + 177, + 506, + 188 + ], + "score": 1.0, + "content": "have been used in a series of applications, such as link prediction and classification (Grover &", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 187, + 506, + 200 + ], + "spans": [ + { + "bbox": [ + 105, + 187, + 506, + 200 + ], + "score": 1.0, + "content": "Leskovec, 2016) and semantic role labeling in NLP (Marcheggiani & Titov, 2017). Somewhat", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 198, + 506, + 211 + ], + "spans": [ + { + "bbox": [ + 105, + 198, + 506, + 211 + ], + "score": 1.0, + "content": "related to source code is the work of Wang et al. (2017) who learn graph-based representations of", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 209, + 368, + 222 + ], + "spans": [ + { + "bbox": [ + 105, + 209, + 368, + 222 + ], + "score": 1.0, + "content": "mathematical formulas for premise selection in theorem proving.", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 8.5 + }, + { + "type": "title", + "bbox": [ + 108, + 241, + 245, + 254 + ], + "lines": [ + { + "bbox": [ + 104, + 239, + 246, + 256 + ], + "spans": [ + { + "bbox": [ + 104, + 239, + 246, + 256 + ], + "score": 1.0, + "content": "3 THE VARMISUSE TASK", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 12 + }, + { + "type": "text", + "bbox": [ + 107, + 269, + 505, + 335 + ], + "lines": [ + { + "bbox": [ + 105, + 269, + 505, + 282 + ], + "spans": [ + { + "bbox": [ + 105, + 269, + 505, + 282 + ], + "score": 1.0, + "content": "Detecting variable misuses in code is a task that requires understanding and reasoning about program", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 280, + 505, + 293 + ], + "spans": [ + { + "bbox": [ + 105, + 280, + 505, + 293 + ], + "score": 1.0, + "content": "semantics. To successfully tackle the task one needs to infer the role and function of the program", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 291, + 505, + 304 + ], + "spans": [ + { + "bbox": [ + 105, + 291, + 505, + 304 + ], + "score": 1.0, + "content": "elements and understand how they relate. For example, given a program such as Fig. 1, the task is", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 302, + 504, + 314 + ], + "spans": [ + { + "bbox": [ + 105, + 302, + 293, + 314 + ], + "score": 1.0, + "content": "to automatically detect that the marked use of", + "type": "text" + }, + { + "bbox": [ + 293, + 303, + 324, + 313 + ], + "score": 0.6, + "content": "\\mathtt { C 1 a z z }", + "type": "inline_equation" + }, + { + "bbox": [ + 324, + 302, + 504, + 314 + ], + "score": 1.0, + "content": "is a mistake and that first should be used", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 313, + 505, + 325 + ], + "spans": [ + { + "bbox": [ + 105, + 313, + 505, + 325 + ], + "score": 1.0, + "content": "instead. While this task resembles standard code completion, it differs significantly in its scope and", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 324, + 433, + 337 + ], + "spans": [ + { + "bbox": [ + 105, + 324, + 433, + 337 + ], + "score": 1.0, + "content": "purpose, by considering only variable identifiers and a mostly complete program.", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 15.5 + }, + { + "type": "text", + "bbox": [ + 107, + 352, + 505, + 441 + ], + "lines": [ + { + "bbox": [ + 105, + 352, + 505, + 365 + ], + "spans": [ + { + "bbox": [ + 105, + 352, + 403, + 365 + ], + "score": 1.0, + "content": "Task Description We view a source code file as a sequence of tokens", + "type": "text" + }, + { + "bbox": [ + 403, + 353, + 462, + 364 + ], + "score": 0.92, + "content": "t _ { 0 } \\ldots t _ { N } = \\mathcal { T }", + "type": "inline_equation" + }, + { + "bbox": [ + 462, + 352, + 505, + 365 + ], + "score": 1.0, + "content": ", in which", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 363, + 506, + 376 + ], + "spans": [ + { + "bbox": [ + 105, + 363, + 159, + 376 + ], + "score": 1.0, + "content": "some tokens", + "type": "text" + }, + { + "bbox": [ + 160, + 364, + 204, + 375 + ], + "score": 0.9, + "content": "t _ { \\lambda _ { 0 } } , t _ { \\lambda _ { 1 } } \\ldots", + "type": "inline_equation" + }, + { + "bbox": [ + 205, + 363, + 333, + 376 + ], + "score": 1.0, + "content": "are variables. Furthermore, let", + "type": "text" + }, + { + "bbox": [ + 334, + 363, + 367, + 374 + ], + "score": 0.92, + "content": "\\mathbb { V } _ { t } \\subset \\mathbb { V }", + "type": "inline_equation" + }, + { + "bbox": [ + 367, + 363, + 506, + 376 + ], + "score": 1.0, + "content": "refer to the set of all type-correct", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 374, + 506, + 387 + ], + "spans": [ + { + "bbox": [ + 105, + 374, + 255, + 387 + ], + "score": 1.0, + "content": "variables in scope at the location of", + "type": "text" + }, + { + "bbox": [ + 256, + 375, + 261, + 385 + ], + "score": 0.61, + "content": "t", + "type": "inline_equation" + }, + { + "bbox": [ + 261, + 374, + 426, + 387 + ], + "score": 1.0, + "content": ", i.e., those variables that can be used at", + "type": "text" + }, + { + "bbox": [ + 426, + 375, + 432, + 384 + ], + "score": 0.73, + "content": "t", + "type": "inline_equation" + }, + { + "bbox": [ + 432, + 374, + 506, + 387 + ], + "score": 1.0, + "content": "without raising a", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 385, + 505, + 398 + ], + "spans": [ + { + "bbox": [ + 105, + 385, + 233, + 398 + ], + "score": 1.0, + "content": "compiler error. We call a token", + "type": "text" + }, + { + "bbox": [ + 233, + 385, + 253, + 396 + ], + "score": 0.89, + "content": "t o k _ { \\lambda }", + "type": "inline_equation" + }, + { + "bbox": [ + 253, + 385, + 505, + 398 + ], + "score": 1.0, + "content": "where we want to predict the correct variable usage a slot. 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For training and evaluation purposes, a correct solution is one that simply matches the ground", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 418, + 505, + 430 + ], + "spans": [ + { + "bbox": [ + 105, + 418, + 505, + 430 + ], + "score": 1.0, + "content": "truth, but note that in practice, several possible assignments could be considered correct (i.e., when", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 429, + 320, + 442 + ], + "spans": [ + { + "bbox": [ + 105, + 429, + 320, + 442 + ], + "score": 1.0, + "content": "several variables refer to the same value in memory).", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 22.5 + }, + { + "type": "title", + "bbox": [ + 107, + 461, + 293, + 474 + ], + "lines": [ + { + "bbox": [ + 105, + 461, + 294, + 475 + ], + "spans": [ + { + "bbox": [ + 105, + 461, + 294, + 475 + ], + "score": 1.0, + "content": "4 MODEL: PROGRAMS AS GRAPHS", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 27 + }, + { + "type": "text", + "bbox": [ + 107, + 489, + 504, + 522 + ], + "lines": [ + { + "bbox": [ + 105, + 489, + 505, + 502 + ], + "spans": [ + { + "bbox": [ + 105, + 489, + 505, + 502 + ], + "score": 1.0, + "content": "In this section, we discuss how to transform program source code into program graphs and learn", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 501, + 505, + 513 + ], + "spans": [ + { + "bbox": [ + 105, + 501, + 505, + 513 + ], + "score": 1.0, + "content": "representations over them. These program graphs not only encode the program text but also the", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 511, + 399, + 523 + ], + "spans": [ + { + "bbox": [ + 106, + 511, + 399, + 523 + ], + "score": 1.0, + "content": "semantic information that can be obtained using standard compiler tools.", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 29 + }, + { + "type": "text", + "bbox": [ + 107, + 539, + 505, + 596 + ], + "lines": [ + { + "bbox": [ + 106, + 540, + 506, + 552 + ], + "spans": [ + { + "bbox": [ + 106, + 540, + 506, + 552 + ], + "score": 1.0, + "content": "Gated Graph Neural Networks Our work builds on Gated Graph Neural Networks (Li et al.,", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 550, + 505, + 563 + ], + "spans": [ + { + "bbox": [ + 106, + 550, + 324, + 563 + ], + "score": 1.0, + "content": "2015) (GGNN) and we summarize them here. A graph", + "type": "text" + }, + { + "bbox": [ + 324, + 551, + 385, + 562 + ], + "score": 0.91, + "content": "\\mathcal { G } = ( \\nu , \\pmb { \\varepsilon } , \\pmb { X } )", + "type": "inline_equation" + }, + { + "bbox": [ + 386, + 550, + 505, + 563 + ], + "score": 1.0, + "content": "is composed of a set of nodes", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 107, + 560, + 505, + 574 + ], + "spans": [ + { + "bbox": [ + 107, + 562, + 115, + 572 + ], + "score": 0.66, + "content": "\\nu", + "type": "inline_equation" + }, + { + "bbox": [ + 115, + 560, + 176, + 574 + ], + "score": 1.0, + "content": ", node features", + "type": "text" + }, + { + "bbox": [ + 177, + 562, + 188, + 572 + ], + "score": 0.79, + "content": "\\boldsymbol { X }", + "type": "inline_equation" + }, + { + "bbox": [ + 188, + 560, + 319, + 574 + ], + "score": 1.0, + "content": ", and a list of directed edge sets", + "type": "text" + }, + { + "bbox": [ + 320, + 562, + 394, + 574 + ], + "score": 0.89, + "content": "\\pmb { \\mathcal { E } } = ( \\mathcal { E } _ { 1 } , \\ldots , \\mathcal { E } _ { K } )", + "type": "inline_equation" + }, + { + "bbox": [ + 394, + 560, + 423, + 574 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 424, + 562, + 434, + 572 + ], + "score": 0.81, + "content": "K", + "type": "inline_equation" + }, + { + "bbox": [ + 434, + 560, + 505, + 574 + ], + "score": 1.0, + "content": "is the number of", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 104, + 569, + 506, + 589 + ], + "spans": [ + { + "bbox": [ + 104, + 569, + 228, + 589 + ], + "score": 1.0, + "content": "edge types. We annotate each", + "type": "text" + }, + { + "bbox": [ + 228, + 574, + 254, + 584 + ], + "score": 0.9, + "content": "v \\in \\mathcal V", + "type": "inline_equation" + }, + { + "bbox": [ + 254, + 569, + 357, + 589 + ], + "score": 1.0, + "content": "with a real-valued vector", + "type": "text" + }, + { + "bbox": [ + 357, + 573, + 402, + 584 + ], + "score": 0.9, + "content": "\\pmb { x } ^ { ( v ) } \\in \\mathbb { R } ^ { D }", + "type": "inline_equation" + }, + { + "bbox": [ + 402, + 569, + 506, + 589 + ], + "score": 1.0, + "content": "representing the features", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 584, + 358, + 597 + ], + "spans": [ + { + "bbox": [ + 105, + 584, + 358, + 597 + ], + "score": 1.0, + "content": "of the node (e.g., the embedding of a string label of that node).", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 33 + }, + { + "type": "text", + "bbox": [ + 107, + 601, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 601, + 506, + 615 + ], + "spans": [ + { + "bbox": [ + 105, + 601, + 205, + 615 + ], + "score": 1.0, + "content": "We associate every node", + "type": "text" + }, + { + "bbox": [ + 205, + 605, + 212, + 613 + ], + "score": 0.74, + "content": "v", + "type": "inline_equation" + }, + { + "bbox": [ + 212, + 601, + 287, + 615 + ], + "score": 1.0, + "content": "with a state vector", + "type": "text" + }, + { + "bbox": [ + 288, + 601, + 306, + 613 + ], + "score": 0.9, + "content": "\\mathbf { \\Omega } _ { h } ( v )", + "type": "inline_equation" + }, + { + "bbox": [ + 306, + 601, + 432, + 615 + ], + "score": 1.0, + "content": ", initialized from the node label", + "type": "text" + }, + { + "bbox": [ + 432, + 601, + 450, + 613 + ], + "score": 0.91, + "content": "\\pmb { x } ^ { ( v ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 450, + 601, + 506, + 615 + ], + "score": 1.0, + "content": ". The sizes of", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 106, + 614, + 505, + 626 + ], + "spans": [ + { + "bbox": [ + 106, + 614, + 505, + 626 + ], + "score": 1.0, + "content": "the state vector and feature vector are typically the same, but we can use larger state vectors through", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 624, + 504, + 637 + ], + "spans": [ + { + "bbox": [ + 105, + 624, + 497, + 637 + ], + "score": 1.0, + "content": "padding of node features. To propagate information throughout the graph, “messages” of type", + "type": "text" + }, + { + "bbox": [ + 497, + 625, + 504, + 635 + ], + "score": 0.78, + "content": "k", + "type": "inline_equation" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 635, + 506, + 649 + ], + "spans": [ + { + "bbox": [ + 105, + 635, + 182, + 649 + ], + "score": 1.0, + "content": "are sent from each", + "type": "text" + }, + { + "bbox": [ + 182, + 638, + 189, + 646 + ], + "score": 0.72, + "content": "v", + "type": "inline_equation" + }, + { + "bbox": [ + 190, + 635, + 506, + 649 + ], + "score": 1.0, + "content": "to its neighbors, where each message is computed from its current state vector", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 646, + 507, + 665 + ], + "spans": [ + { + "bbox": [ + 105, + 648, + 136, + 664 + ], + "score": 1.0, + "content": "as m(vk", + "type": "text" + }, + { + "bbox": [ + 118, + 646, + 191, + 662 + ], + "score": 0.93, + "content": "m _ { k } ^ { ( v ) } = f _ { k } ( { \\pmb h } ^ { ( v ) } )", + "type": "inline_equation" + }, + { + "bbox": [ + 191, + 646, + 221, + 665 + ], + "score": 1.0, + "content": ". Here,", + "type": "text" + }, + { + "bbox": [ + 222, + 649, + 232, + 661 + ], + "score": 0.88, + "content": "f _ { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 233, + 646, + 507, + 665 + ], + "score": 1.0, + "content": "can be an arbitrary function; we choose a linear layer in our case.", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 660, + 506, + 673 + ], + "spans": [ + { + "bbox": [ + 105, + 660, + 506, + 673 + ], + "score": 1.0, + "content": "By computing messages for all graph edges at the same time, all states can be updated at the same", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 670, + 506, + 685 + ], + "spans": [ + { + "bbox": [ + 105, + 670, + 276, + 685 + ], + "score": 1.0, + "content": "time. In particular, a new state for a node", + "type": "text" + }, + { + "bbox": [ + 276, + 673, + 283, + 681 + ], + "score": 0.78, + "content": "v", + "type": "inline_equation" + }, + { + "bbox": [ + 283, + 670, + 506, + 685 + ], + "score": 1.0, + "content": "is computed by aggregating all incoming messages as", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 106, + 679, + 506, + 702 + ], + "spans": [ + { + "bbox": [ + 106, + 682, + 183, + 697 + ], + "score": 0.92, + "content": "\\tilde { m } ^ { ( v ) } = g ( \\{ m _ { k } ^ { ( u ) } \\ |", + "type": "inline_equation" + }, + { + "bbox": [ + 183, + 679, + 280, + 702 + ], + "score": 1.0, + "content": "there is an edge of type", + "type": "text" + }, + { + "bbox": [ + 280, + 685, + 287, + 695 + ], + "score": 0.81, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 288, + 679, + 310, + 702 + ], + "score": 1.0, + "content": "from", + "type": "text" + }, + { + "bbox": [ + 310, + 687, + 317, + 695 + ], + "score": 0.64, + "content": "u", + "type": "inline_equation" + }, + { + "bbox": [ + 317, + 679, + 328, + 702 + ], + "score": 1.0, + "content": "to", + "type": "text" + }, + { + "bbox": [ + 329, + 685, + 339, + 697 + ], + "score": 0.81, + "content": "v \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 340, + 679, + 348, + 702 + ], + "score": 1.0, + "content": ").", + "type": "text" + }, + { + "bbox": [ + 348, + 687, + 355, + 696 + ], + "score": 0.76, + "content": "g", + "type": "inline_equation" + }, + { + "bbox": [ + 355, + 679, + 506, + 702 + ], + "score": 1.0, + "content": "is an aggregation function, which we", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 696, + 506, + 711 + ], + "spans": [ + { + "bbox": [ + 105, + 696, + 396, + 711 + ], + "score": 1.0, + "content": "implement as elementwise summation. Given the aggregated message", + "type": "text" + }, + { + "bbox": [ + 396, + 696, + 418, + 708 + ], + "score": 0.9, + "content": "\\tilde { m } ^ { ( v ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 418, + 696, + 506, + 711 + ], + "score": 1.0, + "content": "and the current state", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 706, + 506, + 722 + ], + "spans": [ + { + "bbox": [ + 105, + 706, + 133, + 722 + ], + "score": 1.0, + "content": "vector", + "type": "text" + }, + { + "bbox": [ + 133, + 708, + 151, + 720 + ], + "score": 0.9, + "content": "\\mathbf { \\Omega } _ { h } ( v )", + "type": "inline_equation" + }, + { + "bbox": [ + 152, + 706, + 185, + 722 + ], + "score": 1.0, + "content": "of node", + "type": "text" + }, + { + "bbox": [ + 185, + 711, + 191, + 720 + ], + "score": 0.73, + "content": "v", + "type": "inline_equation" + }, + { + "bbox": [ + 192, + 706, + 312, + 722 + ], + "score": 1.0, + "content": ", the state of the next time step", + "type": "text" + }, + { + "bbox": [ + 312, + 709, + 333, + 720 + ], + "score": 0.89, + "content": "\\boldsymbol { h ^ { \\prime } } ^ { ( v ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 333, + 706, + 394, + 722 + ], + "score": 1.0, + "content": "is computed as", + "type": "text" + }, + { + "bbox": [ + 395, + 708, + 502, + 721 + ], + "score": 0.91, + "content": "\\pmb { h } ^ { \\prime ( v ) } = \\mathbf { G } \\mathbf { R } \\mathbf { U } ( \\tilde { \\pmb { m } } ^ { ( v ) } , \\pmb { h } ^ { ( v ) } )", + "type": "inline_equation" + }, + { + "bbox": [ + 503, + 706, + 506, + 722 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 106, + 720, + 504, + 732 + ], + "spans": [ + { + "bbox": [ + 106, + 720, + 504, + 732 + ], + "score": 1.0, + "content": "where GRU is the recurrent cell function of gated recurrent unit (GRU) (Cho et al., 2014). The", + "type": "text" + } + ], + "index": 46 + } + ], + "index": 41 + } + ], + "page_idx": 2, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 108, + 27, + 293, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 26, + 294, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 26, + 294, + 38 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2018", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 302, + 751, + 309, + 760 + ], + "lines": [ + { + "bbox": [ + 301, + 750, + 310, + 762 + ], + "spans": [ + { + "bbox": [ + 301, + 750, + 310, + 762 + ], + "score": 1.0, + "content": "3", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 82, + 505, + 148 + ], + "lines": [ + { + "bbox": [ + 106, + 82, + 506, + 95 + ], + "spans": [ + { + "bbox": [ + 106, + 82, + 506, + 95 + ], + "score": 1.0, + "content": "Our work is remotely related to work on program synthesis using sketches (Solar-Lezama, 2008)", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 93, + 505, + 105 + ], + "spans": [ + { + "bbox": [ + 106, + 93, + 505, + 105 + ], + "score": 1.0, + "content": "and automated code transplantation (Barr et al., 2015). However, these approaches require a set of", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 105, + 506, + 117 + ], + "spans": [ + { + "bbox": [ + 105, + 105, + 506, + 117 + ], + "score": 1.0, + "content": "specifications (e.g. input-output examples, test suites) to complete the gaps, rather than statistics", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 115, + 505, + 128 + ], + "spans": [ + { + "bbox": [ + 105, + 115, + 505, + 128 + ], + "score": 1.0, + "content": "learned from big code. These approaches can be thought as complementary to ours, since we learn", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 127, + 506, + 139 + ], + "spans": [ + { + "bbox": [ + 105, + 127, + 506, + 139 + ], + "score": 1.0, + "content": "to statistically complete the gaps without any need for specifications, by learning common variable", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 138, + 211, + 149 + ], + "spans": [ + { + "bbox": [ + 105, + 138, + 211, + 149 + ], + "score": 1.0, + "content": "usage patterns from code.", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 2.5, + "bbox_fs": [ + 105, + 82, + 506, + 149 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 154, + 505, + 221 + ], + "lines": [ + { + "bbox": [ + 105, + 152, + 507, + 168 + ], + "spans": [ + { + "bbox": [ + 105, + 152, + 507, + 168 + ], + "score": 1.0, + "content": "Neural networks on graphs (Gori et al., 2005; Li et al., 2015; Defferrard et al., 2016; Kipf & Welling,", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 164, + 506, + 178 + ], + "spans": [ + { + "bbox": [ + 105, + 164, + 506, + 178 + ], + "score": 1.0, + "content": "2016; Gilmer et al., 2017) adapt a variety of deep learning methods to graph-structured input. They", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 177, + 506, + 188 + ], + "spans": [ + { + "bbox": [ + 106, + 177, + 506, + 188 + ], + "score": 1.0, + "content": "have been used in a series of applications, such as link prediction and classification (Grover &", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 187, + 506, + 200 + ], + "spans": [ + { + "bbox": [ + 105, + 187, + 506, + 200 + ], + "score": 1.0, + "content": "Leskovec, 2016) and semantic role labeling in NLP (Marcheggiani & Titov, 2017). Somewhat", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 198, + 506, + 211 + ], + "spans": [ + { + "bbox": [ + 105, + 198, + 506, + 211 + ], + "score": 1.0, + "content": "related to source code is the work of Wang et al. (2017) who learn graph-based representations of", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 209, + 368, + 222 + ], + "spans": [ + { + "bbox": [ + 105, + 209, + 368, + 222 + ], + "score": 1.0, + "content": "mathematical formulas for premise selection in theorem proving.", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 8.5, + "bbox_fs": [ + 105, + 152, + 507, + 222 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 241, + 245, + 254 + ], + "lines": [ + { + "bbox": [ + 104, + 239, + 246, + 256 + ], + "spans": [ + { + "bbox": [ + 104, + 239, + 246, + 256 + ], + "score": 1.0, + "content": "3 THE VARMISUSE TASK", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 12 + }, + { + "type": "text", + "bbox": [ + 107, + 269, + 505, + 335 + ], + "lines": [ + { + "bbox": [ + 105, + 269, + 505, + 282 + ], + "spans": [ + { + "bbox": [ + 105, + 269, + 505, + 282 + ], + "score": 1.0, + "content": "Detecting variable misuses in code is a task that requires understanding and reasoning about program", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 280, + 505, + 293 + ], + "spans": [ + { + "bbox": [ + 105, + 280, + 505, + 293 + ], + "score": 1.0, + "content": "semantics. To successfully tackle the task one needs to infer the role and function of the program", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 291, + 505, + 304 + ], + "spans": [ + { + "bbox": [ + 105, + 291, + 505, + 304 + ], + "score": 1.0, + "content": "elements and understand how they relate. For example, given a program such as Fig. 1, the task is", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 302, + 504, + 314 + ], + "spans": [ + { + "bbox": [ + 105, + 302, + 293, + 314 + ], + "score": 1.0, + "content": "to automatically detect that the marked use of", + "type": "text" + }, + { + "bbox": [ + 293, + 303, + 324, + 313 + ], + "score": 0.6, + "content": "\\mathtt { C 1 a z z }", + "type": "inline_equation" + }, + { + "bbox": [ + 324, + 302, + 504, + 314 + ], + "score": 1.0, + "content": "is a mistake and that first should be used", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 313, + 505, + 325 + ], + "spans": [ + { + "bbox": [ + 105, + 313, + 505, + 325 + ], + "score": 1.0, + "content": "instead. While this task resembles standard code completion, it differs significantly in its scope and", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 324, + 433, + 337 + ], + "spans": [ + { + "bbox": [ + 105, + 324, + 433, + 337 + ], + "score": 1.0, + "content": "purpose, by considering only variable identifiers and a mostly complete program.", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 15.5, + "bbox_fs": [ + 105, + 269, + 505, + 337 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 352, + 505, + 441 + ], + "lines": [ + { + "bbox": [ + 105, + 352, + 505, + 365 + ], + "spans": [ + { + "bbox": [ + 105, + 352, + 403, + 365 + ], + "score": 1.0, + "content": "Task Description We view a source code file as a sequence of tokens", + "type": "text" + }, + { + "bbox": [ + 403, + 353, + 462, + 364 + ], + "score": 0.92, + "content": "t _ { 0 } \\ldots t _ { N } = \\mathcal { T }", + "type": "inline_equation" + }, + { + "bbox": [ + 462, + 352, + 505, + 365 + ], + "score": 1.0, + "content": ", in which", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 363, + 506, + 376 + ], + "spans": [ + { + "bbox": [ + 105, + 363, + 159, + 376 + ], + "score": 1.0, + "content": "some tokens", + "type": "text" + }, + { + "bbox": [ + 160, + 364, + 204, + 375 + ], + "score": 0.9, + "content": "t _ { \\lambda _ { 0 } } , t _ { \\lambda _ { 1 } } \\ldots", + "type": "inline_equation" + }, + { + "bbox": [ + 205, + 363, + 333, + 376 + ], + "score": 1.0, + "content": "are variables. Furthermore, let", + "type": "text" + }, + { + "bbox": [ + 334, + 363, + 367, + 374 + ], + "score": 0.92, + "content": "\\mathbb { V } _ { t } \\subset \\mathbb { V }", + "type": "inline_equation" + }, + { + "bbox": [ + 367, + 363, + 506, + 376 + ], + "score": 1.0, + "content": "refer to the set of all type-correct", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 374, + 506, + 387 + ], + "spans": [ + { + "bbox": [ + 105, + 374, + 255, + 387 + ], + "score": 1.0, + "content": "variables in scope at the location of", + "type": "text" + }, + { + "bbox": [ + 256, + 375, + 261, + 385 + ], + "score": 0.61, + "content": "t", + "type": "inline_equation" + }, + { + "bbox": [ + 261, + 374, + 426, + 387 + ], + "score": 1.0, + "content": ", i.e., those variables that can be used at", + "type": "text" + }, + { + "bbox": [ + 426, + 375, + 432, + 384 + ], + "score": 0.73, + "content": "t", + "type": "inline_equation" + }, + { + "bbox": [ + 432, + 374, + 506, + 387 + ], + "score": 1.0, + "content": "without raising a", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 385, + 505, + 398 + ], + "spans": [ + { + "bbox": [ + 105, + 385, + 233, + 398 + ], + "score": 1.0, + "content": "compiler error. We call a token", + "type": "text" + }, + { + "bbox": [ + 233, + 385, + 253, + 396 + ], + "score": 0.89, + "content": "t o k _ { \\lambda }", + "type": "inline_equation" + }, + { + "bbox": [ + 253, + 385, + 505, + 398 + ], + "score": 1.0, + "content": "where we want to predict the correct variable usage a slot. We", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 396, + 505, + 410 + ], + "spans": [ + { + "bbox": [ + 105, + 396, + 247, + 410 + ], + "score": 1.0, + "content": "define a separate task for each slot", + "type": "text" + }, + { + "bbox": [ + 247, + 397, + 257, + 407 + ], + "score": 0.86, + "content": "t _ { \\lambda }", + "type": "inline_equation" + }, + { + "bbox": [ + 258, + 396, + 288, + 410 + ], + "score": 1.0, + "content": ": Given", + "type": "text" + }, + { + "bbox": [ + 289, + 397, + 332, + 408 + ], + "score": 0.91, + "content": "t _ { 0 } \\ldots t _ { \\lambda - 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 332, + 396, + 350, + 410 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 351, + 397, + 403, + 408 + ], + "score": 0.92, + "content": "t _ { \\lambda + 1 } , \\dots , t _ { N }", + "type": "inline_equation" + }, + { + "bbox": [ + 404, + 396, + 471, + 410 + ], + "score": 1.0, + "content": ", correctly select", + "type": "text" + }, + { + "bbox": [ + 471, + 397, + 482, + 407 + ], + "score": 0.87, + "content": "t _ { \\lambda }", + "type": "inline_equation" + }, + { + "bbox": [ + 482, + 396, + 505, + 410 + ], + "score": 1.0, + "content": "from", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 406, + 506, + 420 + ], + "spans": [ + { + "bbox": [ + 106, + 407, + 123, + 419 + ], + "score": 0.88, + "content": "\\mathbb { V } _ { t _ { \\lambda } }", + "type": "inline_equation" + }, + { + "bbox": [ + 123, + 406, + 506, + 420 + ], + "score": 1.0, + "content": ". For training and evaluation purposes, a correct solution is one that simply matches the ground", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 418, + 505, + 430 + ], + "spans": [ + { + "bbox": [ + 105, + 418, + 505, + 430 + ], + "score": 1.0, + "content": "truth, but note that in practice, several possible assignments could be considered correct (i.e., when", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 429, + 320, + 442 + ], + "spans": [ + { + "bbox": [ + 105, + 429, + 320, + 442 + ], + "score": 1.0, + "content": "several variables refer to the same value in memory).", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 22.5, + "bbox_fs": [ + 105, + 352, + 506, + 442 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 461, + 293, + 474 + ], + "lines": [ + { + "bbox": [ + 105, + 461, + 294, + 475 + ], + "spans": [ + { + "bbox": [ + 105, + 461, + 294, + 475 + ], + "score": 1.0, + "content": "4 MODEL: PROGRAMS AS GRAPHS", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 27 + }, + { + "type": "text", + "bbox": [ + 107, + 489, + 504, + 522 + ], + "lines": [ + { + "bbox": [ + 105, + 489, + 505, + 502 + ], + "spans": [ + { + "bbox": [ + 105, + 489, + 505, + 502 + ], + "score": 1.0, + "content": "In this section, we discuss how to transform program source code into program graphs and learn", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 501, + 505, + 513 + ], + "spans": [ + { + "bbox": [ + 105, + 501, + 505, + 513 + ], + "score": 1.0, + "content": "representations over them. These program graphs not only encode the program text but also the", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 511, + 399, + 523 + ], + "spans": [ + { + "bbox": [ + 106, + 511, + 399, + 523 + ], + "score": 1.0, + "content": "semantic information that can be obtained using standard compiler tools.", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 29, + "bbox_fs": [ + 105, + 489, + 505, + 523 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 539, + 505, + 596 + ], + "lines": [ + { + "bbox": [ + 106, + 540, + 506, + 552 + ], + "spans": [ + { + "bbox": [ + 106, + 540, + 506, + 552 + ], + "score": 1.0, + "content": "Gated Graph Neural Networks Our work builds on Gated Graph Neural Networks (Li et al.,", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 550, + 505, + 563 + ], + "spans": [ + { + "bbox": [ + 106, + 550, + 324, + 563 + ], + "score": 1.0, + "content": "2015) (GGNN) and we summarize them here. A graph", + "type": "text" + }, + { + "bbox": [ + 324, + 551, + 385, + 562 + ], + "score": 0.91, + "content": "\\mathcal { G } = ( \\nu , \\pmb { \\varepsilon } , \\pmb { X } )", + "type": "inline_equation" + }, + { + "bbox": [ + 386, + 550, + 505, + 563 + ], + "score": 1.0, + "content": "is composed of a set of nodes", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 107, + 560, + 505, + 574 + ], + "spans": [ + { + "bbox": [ + 107, + 562, + 115, + 572 + ], + "score": 0.66, + "content": "\\nu", + "type": "inline_equation" + }, + { + "bbox": [ + 115, + 560, + 176, + 574 + ], + "score": 1.0, + "content": ", node features", + "type": "text" + }, + { + "bbox": [ + 177, + 562, + 188, + 572 + ], + "score": 0.79, + "content": "\\boldsymbol { X }", + "type": "inline_equation" + }, + { + "bbox": [ + 188, + 560, + 319, + 574 + ], + "score": 1.0, + "content": ", and a list of directed edge sets", + "type": "text" + }, + { + "bbox": [ + 320, + 562, + 394, + 574 + ], + "score": 0.89, + "content": "\\pmb { \\mathcal { E } } = ( \\mathcal { E } _ { 1 } , \\ldots , \\mathcal { E } _ { K } )", + "type": "inline_equation" + }, + { + "bbox": [ + 394, + 560, + 423, + 574 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 424, + 562, + 434, + 572 + ], + "score": 0.81, + "content": "K", + "type": "inline_equation" + }, + { + "bbox": [ + 434, + 560, + 505, + 574 + ], + "score": 1.0, + "content": "is the number of", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 104, + 569, + 506, + 589 + ], + "spans": [ + { + "bbox": [ + 104, + 569, + 228, + 589 + ], + "score": 1.0, + "content": "edge types. We annotate each", + "type": "text" + }, + { + "bbox": [ + 228, + 574, + 254, + 584 + ], + "score": 0.9, + "content": "v \\in \\mathcal V", + "type": "inline_equation" + }, + { + "bbox": [ + 254, + 569, + 357, + 589 + ], + "score": 1.0, + "content": "with a real-valued vector", + "type": "text" + }, + { + "bbox": [ + 357, + 573, + 402, + 584 + ], + "score": 0.9, + "content": "\\pmb { x } ^ { ( v ) } \\in \\mathbb { R } ^ { D }", + "type": "inline_equation" + }, + { + "bbox": [ + 402, + 569, + 506, + 589 + ], + "score": 1.0, + "content": "representing the features", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 584, + 358, + 597 + ], + "spans": [ + { + "bbox": [ + 105, + 584, + 358, + 597 + ], + "score": 1.0, + "content": "of the node (e.g., the embedding of a string label of that node).", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 33, + "bbox_fs": [ + 104, + 540, + 506, + 597 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 601, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 601, + 506, + 615 + ], + "spans": [ + { + "bbox": [ + 105, + 601, + 205, + 615 + ], + "score": 1.0, + "content": "We associate every node", + "type": "text" + }, + { + "bbox": [ + 205, + 605, + 212, + 613 + ], + "score": 0.74, + "content": "v", + "type": "inline_equation" + }, + { + "bbox": [ + 212, + 601, + 287, + 615 + ], + "score": 1.0, + "content": "with a state vector", + "type": "text" + }, + { + "bbox": [ + 288, + 601, + 306, + 613 + ], + "score": 0.9, + "content": "\\mathbf { \\Omega } _ { h } ( v )", + "type": "inline_equation" + }, + { + "bbox": [ + 306, + 601, + 432, + 615 + ], + "score": 1.0, + "content": ", initialized from the node label", + "type": "text" + }, + { + "bbox": [ + 432, + 601, + 450, + 613 + ], + "score": 0.91, + "content": "\\pmb { x } ^ { ( v ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 450, + 601, + 506, + 615 + ], + "score": 1.0, + "content": ". The sizes of", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 106, + 614, + 505, + 626 + ], + "spans": [ + { + "bbox": [ + 106, + 614, + 505, + 626 + ], + "score": 1.0, + "content": "the state vector and feature vector are typically the same, but we can use larger state vectors through", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 624, + 504, + 637 + ], + "spans": [ + { + "bbox": [ + 105, + 624, + 497, + 637 + ], + "score": 1.0, + "content": "padding of node features. To propagate information throughout the graph, “messages” of type", + "type": "text" + }, + { + "bbox": [ + 497, + 625, + 504, + 635 + ], + "score": 0.78, + "content": "k", + "type": "inline_equation" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 635, + 506, + 649 + ], + "spans": [ + { + "bbox": [ + 105, + 635, + 182, + 649 + ], + "score": 1.0, + "content": "are sent from each", + "type": "text" + }, + { + "bbox": [ + 182, + 638, + 189, + 646 + ], + "score": 0.72, + "content": "v", + "type": "inline_equation" + }, + { + "bbox": [ + 190, + 635, + 506, + 649 + ], + "score": 1.0, + "content": "to its neighbors, where each message is computed from its current state vector", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 646, + 507, + 665 + ], + "spans": [ + { + "bbox": [ + 105, + 648, + 136, + 664 + ], + "score": 1.0, + "content": "as m(vk", + "type": "text" + }, + { + "bbox": [ + 118, + 646, + 191, + 662 + ], + "score": 0.93, + "content": "m _ { k } ^ { ( v ) } = f _ { k } ( { \\pmb h } ^ { ( v ) } )", + "type": "inline_equation" + }, + { + "bbox": [ + 191, + 646, + 221, + 665 + ], + "score": 1.0, + "content": ". Here,", + "type": "text" + }, + { + "bbox": [ + 222, + 649, + 232, + 661 + ], + "score": 0.88, + "content": "f _ { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 233, + 646, + 507, + 665 + ], + "score": 1.0, + "content": "can be an arbitrary function; we choose a linear layer in our case.", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 660, + 506, + 673 + ], + "spans": [ + { + "bbox": [ + 105, + 660, + 506, + 673 + ], + "score": 1.0, + "content": "By computing messages for all graph edges at the same time, all states can be updated at the same", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 670, + 506, + 685 + ], + "spans": [ + { + "bbox": [ + 105, + 670, + 276, + 685 + ], + "score": 1.0, + "content": "time. 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Then, we use", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 259, + 380, + 272 + ], + "spans": [ + { + "bbox": [ + 105, + 259, + 380, + 272 + ], + "score": 1.0, + "content": "the state vectors from the last time step as the node representations.1", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 17.5 + }, + { + "type": "text", + "bbox": [ + 106, + 290, + 505, + 390 + ], + "lines": [ + { + "bbox": [ + 106, + 291, + 504, + 303 + ], + "spans": [ + { + "bbox": [ + 106, + 291, + 504, + 303 + ], + "score": 1.0, + "content": "Program Graphs We represent program source code as graphs and use different edge types to", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 300, + 506, + 316 + ], + "spans": [ + { + "bbox": [ + 105, + 300, + 506, + 316 + ], + "score": 1.0, + "content": "model syntactic and semantic relationships between different tokens. The backbone of a program", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 312, + 506, + 326 + ], + "spans": [ + { + "bbox": [ + 105, + 312, + 506, + 326 + ], + "score": 1.0, + "content": "graph is the program’s abstract syntax tree (AST), consisting of syntax nodes (corresponding to non-", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 324, + 506, + 337 + ], + "spans": [ + { + "bbox": [ + 105, + 324, + 506, + 337 + ], + "score": 1.0, + "content": "terminals in the programming language’s grammar) and syntax tokens (corresponding to terminals).", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 334, + 506, + 348 + ], + "spans": [ + { + "bbox": [ + 105, + 334, + 506, + 348 + ], + "score": 1.0, + "content": "We label syntax nodes with the name of the nonterminal from the program’s grammar, whereas", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 104, + 345, + 506, + 358 + ], + "spans": [ + { + "bbox": [ + 104, + 345, + 506, + 358 + ], + "score": 1.0, + "content": "syntax tokens are labeled with the string that they represent. We use Child edges to connect nodes", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 356, + 505, + 370 + ], + "spans": [ + { + "bbox": [ + 105, + 356, + 505, + 370 + ], + "score": 1.0, + "content": "according to the AST. As this does not induce an order on children of a syntax node, we additionally", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 367, + 505, + 380 + ], + "spans": [ + { + "bbox": [ + 105, + 367, + 505, + 380 + ], + "score": 1.0, + "content": "add NextToken edges connecting each syntax token to its successor. An example of this is shown in", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 104, + 378, + 139, + 392 + ], + "spans": [ + { + "bbox": [ + 104, + 378, + 139, + 392 + ], + "score": 1.0, + "content": "Fig. 2a.", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 23 + }, + { + "type": "text", + "bbox": [ + 106, + 395, + 505, + 505 + ], + "lines": [ + { + "bbox": [ + 105, + 394, + 506, + 409 + ], + "spans": [ + { + "bbox": [ + 105, + 394, + 506, + 409 + ], + "score": 1.0, + "content": "To capture the flow of control and data through a program, we add additional edges connecting", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 404, + 505, + 420 + ], + "spans": [ + { + "bbox": [ + 105, + 404, + 453, + 420 + ], + "score": 1.0, + "content": "different uses and updates of syntax tokens corresponding to variables. For such a token", + "type": "text" + }, + { + "bbox": [ + 454, + 408, + 460, + 416 + ], + "score": 0.68, + "content": "v", + "type": "inline_equation" + }, + { + "bbox": [ + 460, + 404, + 476, + 420 + ], + "score": 1.0, + "content": ", let", + "type": "text" + }, + { + "bbox": [ + 477, + 406, + 505, + 418 + ], + "score": 0.92, + "content": "\\mathcal { D } ^ { R } ( v )", + "type": "inline_equation" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 417, + 506, + 430 + ], + "spans": [ + { + "bbox": [ + 105, + 417, + 506, + 430 + ], + "score": 1.0, + "content": "be the set of syntax tokens at which the variable could have been used last. This set may contain", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 428, + 506, + 441 + ], + "spans": [ + { + "bbox": [ + 105, + 428, + 506, + 441 + ], + "score": 1.0, + "content": "several nodes (for example, when using a variable after a conditional in which it was used in both", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 438, + 506, + 452 + ], + "spans": [ + { + "bbox": [ + 105, + 438, + 506, + 452 + ], + "score": 1.0, + "content": "branches), and even syntax tokens that follow in the program code (in the case of loops). Similarly,", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 104, + 448, + 506, + 464 + ], + "spans": [ + { + "bbox": [ + 104, + 448, + 119, + 464 + ], + "score": 1.0, + "content": "let", + "type": "text" + }, + { + "bbox": [ + 120, + 449, + 151, + 462 + ], + "score": 0.91, + "content": "\\mathcal { D } ^ { W } ( v )", + "type": "inline_equation" + }, + { + "bbox": [ + 151, + 448, + 506, + 464 + ], + "score": 1.0, + "content": "be the set of syntax tokens at which the variable was last written to. Using these, we", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 104, + 457, + 507, + 475 + ], + "spans": [ + { + "bbox": [ + 104, + 457, + 324, + 475 + ], + "score": 1.0, + "content": "add LastRead (resp. 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An example of such semantic edges is shown in", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 104, + 493, + 141, + 508 + ], + "spans": [ + { + "bbox": [ + 104, + 493, + 141, + 508 + ], + "score": 1.0, + "content": "Fig. 2b.", + "type": "text" + } + ], + "index": 37 + } + ], + "index": 32.5 + }, + { + "type": "text", + "bbox": [ + 106, + 510, + 506, + 632 + ], + "lines": [ + { + "bbox": [ + 106, + 511, + 506, + 523 + ], + "spans": [ + { + "bbox": [ + 106, + 511, + 506, + 523 + ], + "score": 1.0, + "content": "We extend the graph to chain all uses of the same variable using LastLexicalUse edges (independent", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 522, + 506, + 534 + ], + "spans": [ + { + "bbox": [ + 105, + 522, + 506, + 534 + ], + "score": 1.0, + "content": "of data flow, i.e., in if (...) { ... v ...} else { ... v ...}, we link the two oc-", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 104, + 531, + 506, + 546 + ], + "spans": [ + { + "bbox": [ + 104, + 531, + 157, + 546 + ], + "score": 1.0, + "content": "currences of", + "type": "text" + }, + { + "bbox": [ + 158, + 535, + 164, + 543 + ], + "score": 0.61, + "content": "v", + "type": "inline_equation" + }, + { + "bbox": [ + 164, + 531, + 506, + 546 + ], + "score": 1.0, + "content": "). We also connect return tokens to the method declaration using ReturnsTo edges", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 542, + 505, + 556 + ], + "spans": [ + { + "bbox": [ + 105, + 542, + 505, + 556 + ], + "score": 1.0, + "content": "(this creates a “shortcut” to its name and type). Inspired by Rice et al. (2017), we connect arguments", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 104, + 553, + 506, + 569 + ], + "spans": [ + { + "bbox": [ + 104, + 553, + 506, + 569 + ], + "score": 1.0, + "content": "in method calls to the formal parameters that they are matched to with FormalArgName edges,", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 566, + 506, + 578 + ], + "spans": [ + { + "bbox": [ + 105, + 566, + 506, + 578 + ], + "score": 1.0, + "content": "i.e., if we observe a call Foo(bar) and a method declaration Foo(InputStream stream),", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 577, + 506, + 589 + ], + "spans": [ + { + "bbox": [ + 105, + 577, + 506, + 589 + ], + "score": 1.0, + "content": "we connect the bar token to the stream token. Finally, we connect every token corresponding", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 587, + 506, + 599 + ], + "spans": [ + { + "bbox": [ + 105, + 587, + 506, + 599 + ], + "score": 1.0, + "content": "to a variable to enclosing guard expressions that use the variable with GuardedBy and Guarded-", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 106, + 597, + 506, + 611 + ], + "spans": [ + { + "bbox": [ + 106, + 597, + 279, + 611 + ], + "score": 1.0, + "content": "ByNegation edges. For example, in if (", + "type": "text" + }, + { + "bbox": [ + 279, + 599, + 312, + 610 + ], + "score": 0.69, + "content": "\\mathrm { ~ \\bf ~ \\cdot ~ x ~ } > \\mathrm { ~ \\bf ~ y ~ }", + "type": "inline_equation" + }, + { + "bbox": [ + 313, + 597, + 333, + 611 + ], + "score": 1.0, + "content": ") {", + "type": "text" + }, + { + "bbox": [ + 335, + 600, + 426, + 610 + ], + "score": 1.0, + "content": "... x ...} else", + "type": "text" + }, + { + "bbox": [ + 431, + 599, + 502, + 611 + ], + "score": 0.42, + "content": "\\{ \\ \\begin{array} { r l } { \\mathrm { ~ \\texttt ~ { ~ . ~ . ~ } ~ } \\underline { { \\mathrm { ~ y ~ } } } \\mathrm { ~ . ~ . ~ . ~ } \\} } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 502, + 600, + 506, + 610 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 609, + 506, + 623 + ], + "spans": [ + { + "bbox": [ + 105, + 609, + 242, + 623 + ], + "score": 1.0, + "content": "we add a GuardedBy edge from", + "type": "text" + }, + { + "bbox": [ + 243, + 611, + 251, + 621 + ], + "score": 0.33, + "content": "\\underline { { \\boldsymbol { \\mathrm { X } } } }", + "type": "inline_equation" + }, + { + "bbox": [ + 252, + 609, + 506, + 623 + ], + "score": 1.0, + "content": "(resp. a GuardedByNegation edge from y) to the AST node", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 105, + 620, + 212, + 634 + ], + "spans": [ + { + "bbox": [ + 105, + 620, + 176, + 634 + ], + "score": 1.0, + "content": "corresponding to", + "type": "text" + }, + { + "bbox": [ + 176, + 622, + 207, + 632 + ], + "score": 0.81, + "content": "\\mathrm { ~ ~ { ~ x ~ } ~ } > \\mathrm { ~ ~ { ~ y ~ } ~ }", + "type": "inline_equation" + }, + { + "bbox": [ + 208, + 620, + 212, + 634 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 48 + } + ], + "index": 43 + }, + { + "type": "text", + "bbox": [ + 108, + 637, + 504, + 671 + ], + "lines": [ + { + "bbox": [ + 106, + 637, + 505, + 650 + ], + "spans": [ + { + "bbox": [ + 106, + 637, + 505, + 650 + ], + "score": 1.0, + "content": "Finally, for all types of edges we introduce their respective backwards edges (transposing the", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 106, + 648, + 505, + 662 + ], + "spans": [ + { + "bbox": [ + 106, + 648, + 505, + 662 + ], + "score": 1.0, + "content": "adjacency matrix), doubling the number of edges and edge types. Backwards edges help with", + "type": "text" + } + ], + "index": 50 + }, + { + "bbox": [ + 105, + 659, + 457, + 672 + ], + "spans": [ + { + "bbox": [ + 105, + 659, + 457, + 672 + ], + "score": 1.0, + "content": "propagating information faster across the GGNN and make the model more expressive.", + "type": "text" + } + ], + "index": 51 + } + ], + "index": 50 + } + ], + "page_idx": 3, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 106, + 691, + 505, + 731 + ], + "lines": [ + { + "bbox": [ + 119, + 690, + 506, + 703 + ], + "spans": [ + { + "bbox": [ + 119, + 690, + 506, + 703 + ], + "score": 1.0, + "content": "1Graph Convolutional Networks (GCN) (Kipf & Welling, 2016; Schlichtkrull et al., 2017) would be a simpler", + "type": "text" + } + ] + }, + { + "bbox": [ + 106, + 701, + 505, + 713 + ], + "spans": [ + { + "bbox": [ + 106, + 701, + 505, + 713 + ], + "score": 1.0, + "content": "replacement for GGNNs. They correspond to the special case of GGNNs in which no gated recurrent units are", + "type": "text" + } + ] + }, + { + "bbox": [ + 106, + 712, + 505, + 723 + ], + "spans": [ + { + "bbox": [ + 106, + 712, + 505, + 723 + ], + "score": 1.0, + "content": "used for state updates and the number of propagation steps per GGNN layer is fixed to 1. Instead, several layers", + "type": "text" + } + ] + }, + { + "bbox": [ + 105, + 721, + 367, + 732 + ], + "spans": [ + { + "bbox": [ + 105, + 721, + 367, + 732 + ], + "score": 1.0, + "content": "are used. 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Then, we use", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 259, + 380, + 272 + ], + "spans": [ + { + "bbox": [ + 105, + 259, + 380, + 272 + ], + "score": 1.0, + "content": "the state vectors from the last time step as the node representations.1", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 17.5, + "bbox_fs": [ + 105, + 247, + 505, + 272 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 290, + 505, + 390 + ], + "lines": [ + { + "bbox": [ + 106, + 291, + 504, + 303 + ], + "spans": [ + { + "bbox": [ + 106, + 291, + 504, + 303 + ], + "score": 1.0, + "content": "Program Graphs We represent program source code as graphs and use different edge types to", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 300, + 506, + 316 + ], + "spans": [ + { + "bbox": [ + 105, + 300, + 506, + 316 + ], + "score": 1.0, + "content": "model syntactic and semantic relationships between different tokens. The backbone of a program", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 312, + 506, + 326 + ], + "spans": [ + { + "bbox": [ + 105, + 312, + 506, + 326 + ], + "score": 1.0, + "content": "graph is the program’s abstract syntax tree (AST), consisting of syntax nodes (corresponding to non-", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 324, + 506, + 337 + ], + "spans": [ + { + "bbox": [ + 105, + 324, + 506, + 337 + ], + "score": 1.0, + "content": "terminals in the programming language’s grammar) and syntax tokens (corresponding to terminals).", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 334, + 506, + 348 + ], + "spans": [ + { + "bbox": [ + 105, + 334, + 506, + 348 + ], + "score": 1.0, + "content": "We label syntax nodes with the name of the nonterminal from the program’s grammar, whereas", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 104, + 345, + 506, + 358 + ], + "spans": [ + { + "bbox": [ + 104, + 345, + 506, + 358 + ], + "score": 1.0, + "content": "syntax tokens are labeled with the string that they represent. We use Child edges to connect nodes", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 356, + 505, + 370 + ], + "spans": [ + { + "bbox": [ + 105, + 356, + 505, + 370 + ], + "score": 1.0, + "content": "according to the AST. As this does not induce an order on children of a syntax node, we additionally", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 367, + 505, + 380 + ], + "spans": [ + { + "bbox": [ + 105, + 367, + 505, + 380 + ], + "score": 1.0, + "content": "add NextToken edges connecting each syntax token to its successor. An example of this is shown in", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 104, + 378, + 139, + 392 + ], + "spans": [ + { + "bbox": [ + 104, + 378, + 139, + 392 + ], + "score": 1.0, + "content": "Fig. 2a.", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 23, + "bbox_fs": [ + 104, + 291, + 506, + 392 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 395, + 505, + 505 + ], + "lines": [ + { + "bbox": [ + 105, + 394, + 506, + 409 + ], + "spans": [ + { + "bbox": [ + 105, + 394, + 506, + 409 + ], + "score": 1.0, + "content": "To capture the flow of control and data through a program, we add additional edges connecting", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 404, + 505, + 420 + ], + "spans": [ + { + "bbox": [ + 105, + 404, + 453, + 420 + ], + "score": 1.0, + "content": "different uses and updates of syntax tokens corresponding to variables. For such a token", + "type": "text" + }, + { + "bbox": [ + 454, + 408, + 460, + 416 + ], + "score": 0.68, + "content": "v", + "type": "inline_equation" + }, + { + "bbox": [ + 460, + 404, + 476, + 420 + ], + "score": 1.0, + "content": ", let", + "type": "text" + }, + { + "bbox": [ + 477, + 406, + 505, + 418 + ], + "score": 0.92, + "content": "\\mathcal { D } ^ { R } ( v )", + "type": "inline_equation" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 417, + 506, + 430 + ], + "spans": [ + { + "bbox": [ + 105, + 417, + 506, + 430 + ], + "score": 1.0, + "content": "be the set of syntax tokens at which the variable could have been used last. This set may contain", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 428, + 506, + 441 + ], + "spans": [ + { + "bbox": [ + 105, + 428, + 506, + 441 + ], + "score": 1.0, + "content": "several nodes (for example, when using a variable after a conditional in which it was used in both", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 438, + 506, + 452 + ], + "spans": [ + { + "bbox": [ + 105, + 438, + 506, + 452 + ], + "score": 1.0, + "content": "branches), and even syntax tokens that follow in the program code (in the case of loops). Similarly,", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 104, + 448, + 506, + 464 + ], + "spans": [ + { + "bbox": [ + 104, + 448, + 119, + 464 + ], + "score": 1.0, + "content": "let", + "type": "text" + }, + { + "bbox": [ + 120, + 449, + 151, + 462 + ], + "score": 0.91, + "content": "\\mathcal { D } ^ { W } ( v )", + "type": "inline_equation" + }, + { + "bbox": [ + 151, + 448, + 506, + 464 + ], + "score": 1.0, + "content": "be the set of syntax tokens at which the variable was last written to. Using these, we", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 104, + 457, + 507, + 475 + ], + "spans": [ + { + "bbox": [ + 104, + 457, + 324, + 475 + ], + "score": 1.0, + "content": "add LastRead (resp. LastWrite) edges connecting", + "type": "text" + }, + { + "bbox": [ + 324, + 463, + 331, + 471 + ], + "score": 0.67, + "content": "v", + "type": "inline_equation" + }, + { + "bbox": [ + 331, + 457, + 409, + 475 + ], + "score": 1.0, + "content": "to all elements of", + "type": "text" + }, + { + "bbox": [ + 409, + 460, + 437, + 473 + ], + "score": 0.93, + "content": "\\mathcal { D } ^ { R } ( v )", + "type": "inline_equation" + }, + { + "bbox": [ + 438, + 457, + 468, + 475 + ], + "score": 1.0, + "content": "(resp.", + "type": "text" + }, + { + "bbox": [ + 468, + 460, + 502, + 473 + ], + "score": 0.91, + "content": "\\tilde { \\mathcal { D } } ^ { W } ( v ) )", + "type": "inline_equation" + }, + { + "bbox": [ + 502, + 457, + 507, + 475 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 471, + 506, + 486 + ], + "spans": [ + { + "bbox": [ + 105, + 471, + 314, + 486 + ], + "score": 1.0, + "content": "Additionally, whenever we observe an assignment", + "type": "text" + }, + { + "bbox": [ + 315, + 474, + 355, + 483 + ], + "score": 0.85, + "content": "v = e x p r", + "type": "inline_equation" + }, + { + "bbox": [ + 356, + 471, + 408, + 486 + ], + "score": 1.0, + "content": ", we connect", + "type": "text" + }, + { + "bbox": [ + 409, + 474, + 416, + 482 + ], + "score": 0.64, + "content": "v", + "type": "inline_equation" + }, + { + "bbox": [ + 416, + 471, + 506, + 486 + ], + "score": 1.0, + "content": "to all variable tokens", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 483, + 505, + 496 + ], + "spans": [ + { + "bbox": [ + 105, + 483, + 505, + 496 + ], + "score": 1.0, + "content": "occurring in expr using ComputedFrom edges. An example of such semantic edges is shown in", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 104, + 493, + 141, + 508 + ], + "spans": [ + { + "bbox": [ + 104, + 493, + 141, + 508 + ], + "score": 1.0, + "content": "Fig. 2b.", + "type": "text" + } + ], + "index": 37 + } + ], + "index": 32.5, + "bbox_fs": [ + 104, + 394, + 507, + 508 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 510, + 506, + 632 + ], + "lines": [ + { + "bbox": [ + 106, + 511, + 506, + 523 + ], + "spans": [ + { + "bbox": [ + 106, + 511, + 506, + 523 + ], + "score": 1.0, + "content": "We extend the graph to chain all uses of the same variable using LastLexicalUse edges (independent", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 522, + 506, + 534 + ], + "spans": [ + { + "bbox": [ + 105, + 522, + 506, + 534 + ], + "score": 1.0, + "content": "of data flow, i.e., in if (...) { ... v ...} else { ... v ...}, we link the two oc-", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 104, + 531, + 506, + 546 + ], + "spans": [ + { + "bbox": [ + 104, + 531, + 157, + 546 + ], + "score": 1.0, + "content": "currences of", + "type": "text" + }, + { + "bbox": [ + 158, + 535, + 164, + 543 + ], + "score": 0.61, + "content": "v", + "type": "inline_equation" + }, + { + "bbox": [ + 164, + 531, + 506, + 546 + ], + "score": 1.0, + "content": "). We also connect return tokens to the method declaration using ReturnsTo edges", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 542, + 505, + 556 + ], + "spans": [ + { + "bbox": [ + 105, + 542, + 505, + 556 + ], + "score": 1.0, + "content": "(this creates a “shortcut” to its name and type). Inspired by Rice et al. (2017), we connect arguments", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 104, + 553, + 506, + 569 + ], + "spans": [ + { + "bbox": [ + 104, + 553, + 506, + 569 + ], + "score": 1.0, + "content": "in method calls to the formal parameters that they are matched to with FormalArgName edges,", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 566, + 506, + 578 + ], + "spans": [ + { + "bbox": [ + 105, + 566, + 506, + 578 + ], + "score": 1.0, + "content": "i.e., if we observe a call Foo(bar) and a method declaration Foo(InputStream stream),", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 577, + 506, + 589 + ], + "spans": [ + { + "bbox": [ + 105, + 577, + 506, + 589 + ], + "score": 1.0, + "content": "we connect the bar token to the stream token. Finally, we connect every token corresponding", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 587, + 506, + 599 + ], + "spans": [ + { + "bbox": [ + 105, + 587, + 506, + 599 + ], + "score": 1.0, + "content": "to a variable to enclosing guard expressions that use the variable with GuardedBy and Guarded-", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 106, + 597, + 506, + 611 + ], + "spans": [ + { + "bbox": [ + 106, + 597, + 279, + 611 + ], + "score": 1.0, + "content": "ByNegation edges. For example, in if (", + "type": "text" + }, + { + "bbox": [ + 279, + 599, + 312, + 610 + ], + "score": 0.69, + "content": "\\mathrm { ~ \\bf ~ \\cdot ~ x ~ } > \\mathrm { ~ \\bf ~ y ~ }", + "type": "inline_equation" + }, + { + "bbox": [ + 313, + 597, + 333, + 611 + ], + "score": 1.0, + "content": ") {", + "type": "text" + }, + { + "bbox": [ + 335, + 600, + 426, + 610 + ], + "score": 1.0, + "content": "... x ...} else", + "type": "text" + }, + { + "bbox": [ + 431, + 599, + 502, + 611 + ], + "score": 0.42, + "content": "\\{ \\ \\begin{array} { r l } { \\mathrm { ~ \\texttt ~ { ~ . ~ . ~ } ~ } \\underline { { \\mathrm { ~ y ~ } } } \\mathrm { ~ . ~ . ~ . ~ } \\} } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 502, + 600, + 506, + 610 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 609, + 506, + 623 + ], + "spans": [ + { + "bbox": [ + 105, + 609, + 242, + 623 + ], + "score": 1.0, + "content": "we add a GuardedBy edge from", + "type": "text" + }, + { + "bbox": [ + 243, + 611, + 251, + 621 + ], + "score": 0.33, + "content": "\\underline { { \\boldsymbol { \\mathrm { X } } } }", + "type": "inline_equation" + }, + { + "bbox": [ + 252, + 609, + 506, + 623 + ], + "score": 1.0, + "content": "(resp. a GuardedByNegation edge from y) to the AST node", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 105, + 620, + 212, + 634 + ], + "spans": [ + { + "bbox": [ + 105, + 620, + 176, + 634 + ], + "score": 1.0, + "content": "corresponding to", + "type": "text" + }, + { + "bbox": [ + 176, + 622, + 207, + 632 + ], + "score": 0.81, + "content": "\\mathrm { ~ ~ { ~ x ~ } ~ } > \\mathrm { ~ ~ { ~ y ~ } ~ }", + "type": "inline_equation" + }, + { + "bbox": [ + 208, + 620, + 212, + 634 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 48 + } + ], + "index": 43, + "bbox_fs": [ + 104, + 511, + 506, + 634 + ] + }, + { + "type": "text", + "bbox": [ + 108, + 637, + 504, + 671 + ], + "lines": [ + { + "bbox": [ + 106, + 637, + 505, + 650 + ], + "spans": [ + { + "bbox": [ + 106, + 637, + 505, + 650 + ], + "score": 1.0, + "content": "Finally, for all types of edges we introduce their respective backwards edges (transposing the", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 106, + 648, + 505, + 662 + ], + "spans": [ + { + "bbox": [ + 106, + 648, + 505, + 662 + ], + "score": 1.0, + "content": "adjacency matrix), doubling the number of edges and edge types. Backwards edges help with", + "type": "text" + } + ], + "index": 50 + }, + { + "bbox": [ + 105, + 659, + 457, + 672 + ], + "spans": [ + { + "bbox": [ + 105, + 659, + 457, + 672 + ], + "score": 1.0, + "content": "propagating information faster across the GGNN and make the model more expressive.", + "type": "text" + } + ], + "index": 51 + } + ], + "index": 50, + "bbox_fs": [ + 105, + 637, + 505, + 672 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 82, + 505, + 236 + ], + "lines": [ + { + "bbox": [ + 105, + 82, + 505, + 95 + ], + "spans": [ + { + "bbox": [ + 105, + 82, + 505, + 95 + ], + "score": 1.0, + "content": "Leveraging Variable Type Information We assume a statically typed language and that the", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 93, + 505, + 106 + ], + "spans": [ + { + "bbox": [ + 105, + 93, + 399, + 106 + ], + "score": 1.0, + "content": "source code can be compiled, and thus each variable has a (known) type", + "type": "text" + }, + { + "bbox": [ + 400, + 93, + 419, + 106 + ], + "score": 0.91, + "content": "\\tau ( v )", + "type": "inline_equation" + }, + { + "bbox": [ + 420, + 93, + 505, + 106 + ], + "score": 1.0, + "content": ". To use it, we define", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 104, + 506, + 117 + ], + "spans": [ + { + "bbox": [ + 105, + 104, + 237, + 117 + ], + "score": 1.0, + "content": "a learnable embedding function", + "type": "text" + }, + { + "bbox": [ + 237, + 104, + 257, + 116 + ], + "score": 0.9, + "content": "\\mathbf { r } ( \\tau )", + "type": "inline_equation" + }, + { + "bbox": [ + 257, + 104, + 506, + 117 + ], + "score": 1.0, + "content": "for known types and additionally define an “UNKTYPE” for", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 115, + 505, + 128 + ], + "spans": [ + { + "bbox": [ + 105, + 115, + 505, + 128 + ], + "score": 1.0, + "content": "all unknown/unrepresented types. We also leverage the rich type hierarchy that is available in many", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 126, + 507, + 139 + ], + "spans": [ + { + "bbox": [ + 105, + 127, + 358, + 139 + ], + "score": 1.0, + "content": "object-oriented languages. For this, we map a variable’s type", + "type": "text" + }, + { + "bbox": [ + 359, + 126, + 379, + 138 + ], + "score": 0.92, + "content": "\\tau ( v )", + "type": "inline_equation" + }, + { + "bbox": [ + 379, + 127, + 507, + 139 + ], + "score": 1.0, + "content": "to the set of its supertypes, i.e.", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 107, + 136, + 505, + 151 + ], + "spans": [ + { + "bbox": [ + 107, + 137, + 184, + 149 + ], + "score": 0.93, + "content": "\\tau ^ { * } \\bar { ( } v ) = \\{ \\tau : \\tau ( v ) \\quad", + "type": "inline_equation" + }, + { + "bbox": [ + 184, + 136, + 253, + 151 + ], + "score": 1.0, + "content": "implements type", + "type": "text" + }, + { + "bbox": [ + 254, + 137, + 306, + 149 + ], + "score": 0.93, + "content": "\\tau \\} \\cup \\{ \\bar { \\tau } ( v ) \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 306, + 136, + 481, + 151 + ], + "score": 1.0, + "content": ". We then compute the type representation", + "type": "text" + }, + { + "bbox": [ + 482, + 137, + 505, + 149 + ], + "score": 0.91, + "content": "\\mathbf { r } ^ { * } ( v )", + "type": "inline_equation" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 147, + 506, + 161 + ], + "spans": [ + { + "bbox": [ + 105, + 147, + 162, + 161 + ], + "score": 1.0, + "content": "of a variable", + "type": "text" + }, + { + "bbox": [ + 162, + 151, + 169, + 158 + ], + "score": 0.69, + "content": "v", + "type": "inline_equation" + }, + { + "bbox": [ + 169, + 147, + 312, + 161 + ], + "score": 1.0, + "content": "as the element-wise maximum of", + "type": "text" + }, + { + "bbox": [ + 312, + 148, + 396, + 160 + ], + "score": 0.92, + "content": "\\{ \\mathbf { r } ( \\tau ) : \\tau \\in \\tau ^ { * } ( v ) \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 396, + 147, + 506, + 161 + ], + "score": 1.0, + "content": ". We chose the maximum", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 104, + 158, + 505, + 173 + ], + "spans": [ + { + "bbox": [ + 104, + 158, + 505, + 173 + ], + "score": 1.0, + "content": "here, as it is a natural pooling operation for representing partial ordering relations (such as type", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 169, + 506, + 183 + ], + "spans": [ + { + "bbox": [ + 105, + 169, + 218, + 183 + ], + "score": 1.0, + "content": "lattices). Using all types in", + "type": "text" + }, + { + "bbox": [ + 218, + 171, + 242, + 182 + ], + "score": 0.92, + "content": "\\tau ^ { * } ( v )", + "type": "inline_equation" + }, + { + "bbox": [ + 242, + 169, + 506, + 183 + ], + "score": 1.0, + "content": "allows us to generalize to unseen types that implement common", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 181, + 507, + 195 + ], + "spans": [ + { + "bbox": [ + 105, + 181, + 291, + 195 + ], + "score": 1.0, + "content": "supertypes or interfaces. For example, List", + "type": "text" + }, + { + "bbox": [ + 292, + 182, + 311, + 192 + ], + "score": 0.79, + "content": "{ \\tt < K > }", + "type": "inline_equation" + }, + { + "bbox": [ + 311, + 181, + 507, + 195 + ], + "score": 1.0, + "content": "has multiple concrete types (e.g. List,", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 192, + 506, + 205 + ], + "spans": [ + { + "bbox": [ + 105, + 192, + 506, + 205 + ], + "score": 1.0, + "content": "List). Nevertheless, these types implement a common interface (IList) and share", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 203, + 505, + 216 + ], + "spans": [ + { + "bbox": [ + 105, + 203, + 452, + 216 + ], + "score": 1.0, + "content": "common characteristics. During training, we randomly select a non-empty subset of", + "type": "text" + }, + { + "bbox": [ + 452, + 203, + 476, + 215 + ], + "score": 0.93, + "content": "\\tau ^ { * } ( v )", + "type": "inline_equation" + }, + { + "bbox": [ + 476, + 203, + 505, + 216 + ], + "score": 1.0, + "content": "which", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 214, + 505, + 227 + ], + "spans": [ + { + "bbox": [ + 105, + 214, + 505, + 227 + ], + "score": 1.0, + "content": "ensures training of all known types in the lattice. This acts both like a dropout mechanism and allows", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 225, + 362, + 238 + ], + "spans": [ + { + "bbox": [ + 105, + 225, + 362, + 238 + ], + "score": 1.0, + "content": "us to learn a good representation for all types in the type lattice.", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 6.5 + }, + { + "type": "text", + "bbox": [ + 107, + 249, + 505, + 326 + ], + "lines": [ + { + "bbox": [ + 106, + 249, + 505, + 260 + ], + "spans": [ + { + "bbox": [ + 106, + 249, + 505, + 260 + ], + "score": 1.0, + "content": "Initial Node Representation To compute the initial node state, we combine information from the", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 259, + 505, + 273 + ], + "spans": [ + { + "bbox": [ + 105, + 259, + 505, + 273 + ], + "score": 1.0, + "content": "textual representation of the token and its type. Concretely, we split the name of a node representing", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 271, + 505, + 283 + ], + "spans": [ + { + "bbox": [ + 105, + 271, + 505, + 283 + ], + "score": 1.0, + "content": "a token into subtokens (e.g. classTypes will be split into two subtokens class and types) on", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 282, + 505, + 294 + ], + "spans": [ + { + "bbox": [ + 106, + 282, + 505, + 294 + ], + "score": 1.0, + "content": "camelCase and pascal_case. We then average the embeddings of all subtokens to retrieve", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 292, + 506, + 306 + ], + "spans": [ + { + "bbox": [ + 105, + 292, + 478, + 306 + ], + "score": 1.0, + "content": "an embedding for the node name. Finally, we concatenate the learned type representation", + "type": "text" + }, + { + "bbox": [ + 479, + 293, + 502, + 304 + ], + "score": 0.92, + "content": "\\mathbf { r } ^ { * } ( v )", + "type": "inline_equation" + }, + { + "bbox": [ + 502, + 292, + 506, + 306 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 303, + 505, + 316 + ], + "spans": [ + { + "bbox": [ + 105, + 303, + 505, + 316 + ], + "score": 1.0, + "content": "computed as discussed earlier, with the node name representation, and pass it through a linear layer", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 315, + 356, + 327 + ], + "spans": [ + { + "bbox": [ + 105, + 315, + 356, + 327 + ], + "score": 1.0, + "content": "to obtain the initial representations for each node in the graph.", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 17 + }, + { + "type": "text", + "bbox": [ + 107, + 338, + 505, + 448 + ], + "lines": [ + { + "bbox": [ + 105, + 338, + 506, + 351 + ], + "spans": [ + { + "bbox": [ + 105, + 338, + 447, + 351 + ], + "score": 1.0, + "content": "Programs Graphs for VARNAMING Given a program and an existing variable", + "type": "text" + }, + { + "bbox": [ + 447, + 341, + 453, + 349 + ], + "score": 0.69, + "content": "v", + "type": "inline_equation" + }, + { + "bbox": [ + 454, + 338, + 506, + 351 + ], + "score": 1.0, + "content": ", we build a", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 349, + 505, + 362 + ], + "spans": [ + { + "bbox": [ + 105, + 349, + 505, + 362 + ], + "score": 1.0, + "content": "program graph as discussed above and then replace the variable name in all corresponding variable", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 361, + 505, + 372 + ], + "spans": [ + { + "bbox": [ + 106, + 361, + 188, + 372 + ], + "score": 1.0, + "content": "tokens by a special", + "type": "text" + }, + { + "bbox": [ + 189, + 361, + 226, + 371 + ], + "score": 0.68, + "content": "{ < } S \\mathrm { L O T } >", + "type": "inline_equation" + }, + { + "bbox": [ + 226, + 361, + 505, + 372 + ], + "score": 1.0, + "content": "token. To predict a name, we use the initial node labels computed", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 371, + 505, + 384 + ], + "spans": [ + { + "bbox": [ + 105, + 371, + 505, + 384 + ], + "score": 1.0, + "content": "as the concatenation of learnable token embeddings and type embeddings as discussed above, run", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 380, + 506, + 397 + ], + "spans": [ + { + "bbox": [ + 105, + 380, + 506, + 397 + ], + "score": 1.0, + "content": "GGNN propagation for 8 time steps2 and then compute a variable usage representation by averaging", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 393, + 506, + 406 + ], + "spans": [ + { + "bbox": [ + 105, + 393, + 217, + 406 + ], + "score": 1.0, + "content": "the representations for all", + "type": "text" + }, + { + "bbox": [ + 217, + 394, + 254, + 404 + ], + "score": 0.52, + "content": "{ < } S \\mathrm { L O T } >", + "type": "inline_equation" + }, + { + "bbox": [ + 255, + 393, + 506, + 406 + ], + "score": 1.0, + "content": "tokens. This representation is then used as the initial state", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 403, + 506, + 417 + ], + "spans": [ + { + "bbox": [ + 105, + 403, + 506, + 417 + ], + "score": 1.0, + "content": "of a one-layer GRU, which predicts the target name as a sequence of subtokens (e.g., the name", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 415, + 505, + 428 + ], + "spans": [ + { + "bbox": [ + 105, + 415, + 505, + 428 + ], + "score": 1.0, + "content": "inputStreamBuffer is treated as the sequence [input, stream, buffer]). We train this", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 425, + 506, + 441 + ], + "spans": [ + { + "bbox": [ + 105, + 425, + 506, + 441 + ], + "score": 1.0, + "content": "graph2seq architecture using a maximum likelihood objective. In section 5, we report the accuracy", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 437, + 408, + 450 + ], + "spans": [ + { + "bbox": [ + 105, + 437, + 408, + 450 + ], + "score": 1.0, + "content": "for predicting the exact name and the F1 score for predicting its subtokens.", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 25.5 + }, + { + "type": "text", + "bbox": [ + 107, + 461, + 505, + 571 + ], + "lines": [ + { + "bbox": [ + 106, + 461, + 506, + 474 + ], + "spans": [ + { + "bbox": [ + 106, + 461, + 506, + 474 + ], + "score": 1.0, + "content": "Program Graphs for VARMISUSE To model VARMISUSE with program graphs we need to", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 472, + 506, + 485 + ], + "spans": [ + { + "bbox": [ + 106, + 472, + 361, + 485 + ], + "score": 1.0, + "content": "modify the graph. First, to compute a context representation", + "type": "text" + }, + { + "bbox": [ + 362, + 472, + 379, + 484 + ], + "score": 0.91, + "content": "\\mathbf { } c ( t )", + "type": "inline_equation" + }, + { + "bbox": [ + 379, + 472, + 421, + 485 + ], + "score": 1.0, + "content": "for a slot", + "type": "text" + }, + { + "bbox": [ + 421, + 473, + 427, + 482 + ], + "score": 0.61, + "content": "t", + "type": "inline_equation" + }, + { + "bbox": [ + 428, + 472, + 506, + 485 + ], + "score": 1.0, + "content": "where we want to", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 483, + 506, + 496 + ], + "spans": [ + { + "bbox": [ + 105, + 483, + 292, + 496 + ], + "score": 1.0, + "content": "predict the used variable, we insert a new node", + "type": "text" + }, + { + "bbox": [ + 293, + 485, + 324, + 495 + ], + "score": 0.89, + "content": "v _ { < S \\tt L O T > }", + "type": "inline_equation" + }, + { + "bbox": [ + 325, + 483, + 394, + 496 + ], + "score": 1.0, + "content": "at the position of", + "type": "text" + }, + { + "bbox": [ + 395, + 484, + 399, + 493 + ], + "score": 0.72, + "content": "t", + "type": "inline_equation" + }, + { + "bbox": [ + 400, + 483, + 506, + 496 + ], + "score": 1.0, + "content": ", corresponding to a “hole”", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 494, + 506, + 506 + ], + "spans": [ + { + "bbox": [ + 105, + 494, + 506, + 506 + ], + "score": 1.0, + "content": "at this point, and connect it to the remaining graph using all applicable edges that do not depend", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 505, + 506, + 517 + ], + "spans": [ + { + "bbox": [ + 105, + 505, + 506, + 517 + ], + "score": 1.0, + "content": "on the chosen variable at the slot (i.e., everything but LastUse, LastWrite, LastLexicalUse, and", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 515, + 506, + 529 + ], + "spans": [ + { + "bbox": [ + 105, + 515, + 367, + 529 + ], + "score": 1.0, + "content": "GuardedBy edges). Then, to compute the usage representation", + "type": "text" + }, + { + "bbox": [ + 367, + 516, + 396, + 528 + ], + "score": 0.92, + "content": "\\mathbf { u } ( t , v )", + "type": "inline_equation" + }, + { + "bbox": [ + 396, + 515, + 506, + 529 + ], + "score": 1.0, + "content": "of each candidate variable", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 106, + 527, + 505, + 540 + ], + "spans": [ + { + "bbox": [ + 106, + 529, + 113, + 537 + ], + "score": 0.69, + "content": "v", + "type": "inline_equation" + }, + { + "bbox": [ + 113, + 527, + 303, + 540 + ], + "score": 1.0, + "content": "at the target slot, we insert a “candidate” node", + "type": "text" + }, + { + "bbox": [ + 303, + 528, + 320, + 539 + ], + "score": 0.88, + "content": "v _ { t , v }", + "type": "inline_equation" + }, + { + "bbox": [ + 320, + 527, + 348, + 540 + ], + "score": 1.0, + "content": "for all", + "type": "text" + }, + { + "bbox": [ + 348, + 529, + 355, + 537 + ], + "score": 0.75, + "content": "v", + "type": "inline_equation" + }, + { + "bbox": [ + 355, + 527, + 366, + 540 + ], + "score": 1.0, + "content": "in", + "type": "text" + }, + { + "bbox": [ + 366, + 528, + 378, + 538 + ], + "score": 0.88, + "content": "\\mathbb { V } _ { t }", + "type": "inline_equation" + }, + { + "bbox": [ + 378, + 527, + 505, + 540 + ], + "score": 1.0, + "content": ", and connect it to the graph by", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 537, + 505, + 550 + ], + "spans": [ + { + "bbox": [ + 105, + 537, + 505, + 550 + ], + "score": 1.0, + "content": "inserting the LastUse, LastWrite and LastLexicalUse edges that would be used if the variable were", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 549, + 505, + 561 + ], + "spans": [ + { + "bbox": [ + 105, + 549, + 505, + 561 + ], + "score": 1.0, + "content": "to be used at this slot. Each of these candidate nodes represents the speculative placement of the", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 559, + 210, + 573 + ], + "spans": [ + { + "bbox": [ + 105, + 559, + 210, + 573 + ], + "score": 1.0, + "content": "variable within the scope.", + "type": "text" + } + ], + "index": 40 + } + ], + "index": 35.5 + }, + { + "type": "text", + "bbox": [ + 107, + 577, + 505, + 634 + ], + "lines": [ + { + "bbox": [ + 107, + 578, + 505, + 588 + ], + "spans": [ + { + "bbox": [ + 107, + 578, + 505, + 588 + ], + "score": 1.0, + "content": "Using the initial node representations, concatenated with an extra bit that is set to one for the candidate", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 586, + 506, + 601 + ], + "spans": [ + { + "bbox": [ + 105, + 586, + 132, + 601 + ], + "score": 1.0, + "content": "nodes", + "type": "text" + }, + { + "bbox": [ + 132, + 589, + 149, + 600 + ], + "score": 0.88, + "content": "v _ { t , v }", + "type": "inline_equation" + }, + { + "bbox": [ + 149, + 586, + 506, + 601 + ], + "score": 1.0, + "content": ", we run GGNN propagation for 8 time steps.2 The context and usage representation are", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 599, + 506, + 613 + ], + "spans": [ + { + "bbox": [ + 105, + 599, + 273, + 613 + ], + "score": 1.0, + "content": "then the final node states of the nodes, i.e.,", + "type": "text" + }, + { + "bbox": [ + 273, + 599, + 339, + 612 + ], + "score": 0.9, + "content": "\\pmb { c } ( t ) = \\pmb { h } ^ { ( v _ { < \\mathrm { S L O T } > } ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 339, + 599, + 357, + 613 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 357, + 599, + 425, + 612 + ], + "score": 0.91, + "content": "\\mathbf { u } ( t , v ) = h ^ { ( v _ { t , v } ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 425, + 599, + 506, + 613 + ], + "score": 1.0, + "content": ". Finally, the correct", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 610, + 506, + 625 + ], + "spans": [ + { + "bbox": [ + 105, + 610, + 307, + 625 + ], + "score": 1.0, + "content": "variable usage at the location is computed as arg", + "type": "text" + }, + { + "bbox": [ + 307, + 612, + 397, + 623 + ], + "score": 0.83, + "content": "\\textstyle \\operatorname* { m a x } _ { v } W [ c ( t ) , \\mathbf { u } ( t , v ) ]", + "type": "inline_equation" + }, + { + "bbox": [ + 398, + 610, + 426, + 625 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 426, + 612, + 439, + 622 + ], + "score": 0.59, + "content": "W", + "type": "inline_equation" + }, + { + "bbox": [ + 439, + 610, + 506, + 625 + ], + "score": 1.0, + "content": "is a linear layer", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 621, + 455, + 635 + ], + "spans": [ + { + "bbox": [ + 105, + 621, + 227, + 635 + ], + "score": 1.0, + "content": "that uses the concatenation of", + "type": "text" + }, + { + "bbox": [ + 227, + 622, + 244, + 634 + ], + "score": 0.92, + "content": "\\mathbf { } c ( t )", + "type": "inline_equation" + }, + { + "bbox": [ + 245, + 621, + 262, + 635 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 263, + 622, + 291, + 634 + ], + "score": 0.93, + "content": "\\mathbf { u } ( t , v )", + "type": "inline_equation" + }, + { + "bbox": [ + 291, + 621, + 455, + 635 + ], + "score": 1.0, + "content": ". We train using a max-margin objective.", + "type": "text" + } + ], + "index": 45 + } + ], + "index": 43 + }, + { + "type": "title", + "bbox": [ + 107, + 648, + 209, + 659 + ], + "lines": [ + { + "bbox": [ + 105, + 647, + 210, + 660 + ], + "spans": [ + { + "bbox": [ + 105, + 647, + 210, + 660 + ], + "score": 1.0, + "content": "4.1 IMPLEMENTATION", + "type": "text" + } + ], + "index": 46 + } + ], + "index": 46 + }, + { + "type": "text", + "bbox": [ + 108, + 668, + 505, + 713 + ], + "lines": [ + { + "bbox": [ + 105, + 667, + 505, + 681 + ], + "spans": [ + { + "bbox": [ + 105, + 667, + 505, + 681 + ], + "score": 1.0, + "content": "Using GGNNs for sets of large, diverse graphs requires some engineering effort, as efficient batching", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 105, + 678, + 505, + 691 + ], + "spans": [ + { + "bbox": [ + 105, + 678, + 505, + 691 + ], + "score": 1.0, + "content": "is hard in the presence of diverse shapes. An important observation is that large graphs are normally", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 105, + 690, + 505, + 703 + ], + "spans": [ + { + "bbox": [ + 105, + 690, + 505, + 703 + ], + "score": 1.0, + "content": "very sparse, and thus a representation of edges as an adjacency list would usually be advantageous", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 106, + 702, + 505, + 713 + ], + "spans": [ + { + "bbox": [ + 106, + 702, + 505, + 713 + ], + "score": 1.0, + "content": "to reduce memory consumption. In our case, this can be easily implemented using a sparse tensor", + "type": "text" + } + ], + "index": 50 + } + ], + "index": 48.5 + } + ], + "page_idx": 4, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 107, + 27, + 293, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 26, + 294, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 26, + 294, + 38 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2018", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 114, + 721, + 503, + 732 + ], + "lines": [ + { + "bbox": [ + 118, + 719, + 505, + 734 + ], + "spans": [ + { + "bbox": [ + 118, + 719, + 505, + 734 + ], + "score": 1.0, + "content": "2We found fewer steps to be insufficient for good results and more propagation steps to not help substantially.", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 302, + 751, + 308, + 760 + ], + "lines": [ + { + "bbox": [ + 302, + 750, + 309, + 763 + ], + "spans": [ + { + "bbox": [ + 302, + 750, + 309, + 763 + ], + "score": 1.0, + "content": "5", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 82, + 505, + 236 + ], + "lines": [ + { + "bbox": [ + 105, + 82, + 505, + 95 + ], + "spans": [ + { + "bbox": [ + 105, + 82, + 505, + 95 + ], + "score": 1.0, + "content": "Leveraging Variable Type Information We assume a statically typed language and that the", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 93, + 505, + 106 + ], + "spans": [ + { + "bbox": [ + 105, + 93, + 399, + 106 + ], + "score": 1.0, + "content": "source code can be compiled, and thus each variable has a (known) type", + "type": "text" + }, + { + "bbox": [ + 400, + 93, + 419, + 106 + ], + "score": 0.91, + "content": "\\tau ( v )", + "type": "inline_equation" + }, + { + "bbox": [ + 420, + 93, + 505, + 106 + ], + "score": 1.0, + "content": ". To use it, we define", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 104, + 506, + 117 + ], + "spans": [ + { + "bbox": [ + 105, + 104, + 237, + 117 + ], + "score": 1.0, + "content": "a learnable embedding function", + "type": "text" + }, + { + "bbox": [ + 237, + 104, + 257, + 116 + ], + "score": 0.9, + "content": "\\mathbf { r } ( \\tau )", + "type": "inline_equation" + }, + { + "bbox": [ + 257, + 104, + 506, + 117 + ], + "score": 1.0, + "content": "for known types and additionally define an “UNKTYPE” for", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 115, + 505, + 128 + ], + "spans": [ + { + "bbox": [ + 105, + 115, + 505, + 128 + ], + "score": 1.0, + "content": "all unknown/unrepresented types. We also leverage the rich type hierarchy that is available in many", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 126, + 507, + 139 + ], + "spans": [ + { + "bbox": [ + 105, + 127, + 358, + 139 + ], + "score": 1.0, + "content": "object-oriented languages. For this, we map a variable’s type", + "type": "text" + }, + { + "bbox": [ + 359, + 126, + 379, + 138 + ], + "score": 0.92, + "content": "\\tau ( v )", + "type": "inline_equation" + }, + { + "bbox": [ + 379, + 127, + 507, + 139 + ], + "score": 1.0, + "content": "to the set of its supertypes, i.e.", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 107, + 136, + 505, + 151 + ], + "spans": [ + { + "bbox": [ + 107, + 137, + 184, + 149 + ], + "score": 0.93, + "content": "\\tau ^ { * } \\bar { ( } v ) = \\{ \\tau : \\tau ( v ) \\quad", + "type": "inline_equation" + }, + { + "bbox": [ + 184, + 136, + 253, + 151 + ], + "score": 1.0, + "content": "implements type", + "type": "text" + }, + { + "bbox": [ + 254, + 137, + 306, + 149 + ], + "score": 0.93, + "content": "\\tau \\} \\cup \\{ \\bar { \\tau } ( v ) \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 306, + 136, + 481, + 151 + ], + "score": 1.0, + "content": ". We then compute the type representation", + "type": "text" + }, + { + "bbox": [ + 482, + 137, + 505, + 149 + ], + "score": 0.91, + "content": "\\mathbf { r } ^ { * } ( v )", + "type": "inline_equation" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 147, + 506, + 161 + ], + "spans": [ + { + "bbox": [ + 105, + 147, + 162, + 161 + ], + "score": 1.0, + "content": "of a variable", + "type": "text" + }, + { + "bbox": [ + 162, + 151, + 169, + 158 + ], + "score": 0.69, + "content": "v", + "type": "inline_equation" + }, + { + "bbox": [ + 169, + 147, + 312, + 161 + ], + "score": 1.0, + "content": "as the element-wise maximum of", + "type": "text" + }, + { + "bbox": [ + 312, + 148, + 396, + 160 + ], + "score": 0.92, + "content": "\\{ \\mathbf { r } ( \\tau ) : \\tau \\in \\tau ^ { * } ( v ) \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 396, + 147, + 506, + 161 + ], + "score": 1.0, + "content": ". We chose the maximum", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 104, + 158, + 505, + 173 + ], + "spans": [ + { + "bbox": [ + 104, + 158, + 505, + 173 + ], + "score": 1.0, + "content": "here, as it is a natural pooling operation for representing partial ordering relations (such as type", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 169, + 506, + 183 + ], + "spans": [ + { + "bbox": [ + 105, + 169, + 218, + 183 + ], + "score": 1.0, + "content": "lattices). Using all types in", + "type": "text" + }, + { + "bbox": [ + 218, + 171, + 242, + 182 + ], + "score": 0.92, + "content": "\\tau ^ { * } ( v )", + "type": "inline_equation" + }, + { + "bbox": [ + 242, + 169, + 506, + 183 + ], + "score": 1.0, + "content": "allows us to generalize to unseen types that implement common", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 181, + 507, + 195 + ], + "spans": [ + { + "bbox": [ + 105, + 181, + 291, + 195 + ], + "score": 1.0, + "content": "supertypes or interfaces. For example, List", + "type": "text" + }, + { + "bbox": [ + 292, + 182, + 311, + 192 + ], + "score": 0.79, + "content": "{ \\tt < K > }", + "type": "inline_equation" + }, + { + "bbox": [ + 311, + 181, + 507, + 195 + ], + "score": 1.0, + "content": "has multiple concrete types (e.g. List,", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 192, + 506, + 205 + ], + "spans": [ + { + "bbox": [ + 105, + 192, + 506, + 205 + ], + "score": 1.0, + "content": "List). Nevertheless, these types implement a common interface (IList) and share", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 203, + 505, + 216 + ], + "spans": [ + { + "bbox": [ + 105, + 203, + 452, + 216 + ], + "score": 1.0, + "content": "common characteristics. During training, we randomly select a non-empty subset of", + "type": "text" + }, + { + "bbox": [ + 452, + 203, + 476, + 215 + ], + "score": 0.93, + "content": "\\tau ^ { * } ( v )", + "type": "inline_equation" + }, + { + "bbox": [ + 476, + 203, + 505, + 216 + ], + "score": 1.0, + "content": "which", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 214, + 505, + 227 + ], + "spans": [ + { + "bbox": [ + 105, + 214, + 505, + 227 + ], + "score": 1.0, + "content": "ensures training of all known types in the lattice. This acts both like a dropout mechanism and allows", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 225, + 362, + 238 + ], + "spans": [ + { + "bbox": [ + 105, + 225, + 362, + 238 + ], + "score": 1.0, + "content": "us to learn a good representation for all types in the type lattice.", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 6.5, + "bbox_fs": [ + 104, + 82, + 507, + 238 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 249, + 505, + 326 + ], + "lines": [ + { + "bbox": [ + 106, + 249, + 505, + 260 + ], + "spans": [ + { + "bbox": [ + 106, + 249, + 505, + 260 + ], + "score": 1.0, + "content": "Initial Node Representation To compute the initial node state, we combine information from the", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 259, + 505, + 273 + ], + "spans": [ + { + "bbox": [ + 105, + 259, + 505, + 273 + ], + "score": 1.0, + "content": "textual representation of the token and its type. Concretely, we split the name of a node representing", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 271, + 505, + 283 + ], + "spans": [ + { + "bbox": [ + 105, + 271, + 505, + 283 + ], + "score": 1.0, + "content": "a token into subtokens (e.g. classTypes will be split into two subtokens class and types) on", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 282, + 505, + 294 + ], + "spans": [ + { + "bbox": [ + 106, + 282, + 505, + 294 + ], + "score": 1.0, + "content": "camelCase and pascal_case. We then average the embeddings of all subtokens to retrieve", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 292, + 506, + 306 + ], + "spans": [ + { + "bbox": [ + 105, + 292, + 478, + 306 + ], + "score": 1.0, + "content": "an embedding for the node name. Finally, we concatenate the learned type representation", + "type": "text" + }, + { + "bbox": [ + 479, + 293, + 502, + 304 + ], + "score": 0.92, + "content": "\\mathbf { r } ^ { * } ( v )", + "type": "inline_equation" + }, + { + "bbox": [ + 502, + 292, + 506, + 306 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 303, + 505, + 316 + ], + "spans": [ + { + "bbox": [ + 105, + 303, + 505, + 316 + ], + "score": 1.0, + "content": "computed as discussed earlier, with the node name representation, and pass it through a linear layer", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 315, + 356, + 327 + ], + "spans": [ + { + "bbox": [ + 105, + 315, + 356, + 327 + ], + "score": 1.0, + "content": "to obtain the initial representations for each node in the graph.", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 17, + "bbox_fs": [ + 105, + 249, + 506, + 327 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 338, + 505, + 448 + ], + "lines": [ + { + "bbox": [ + 105, + 338, + 506, + 351 + ], + "spans": [ + { + "bbox": [ + 105, + 338, + 447, + 351 + ], + "score": 1.0, + "content": "Programs Graphs for VARNAMING Given a program and an existing variable", + "type": "text" + }, + { + "bbox": [ + 447, + 341, + 453, + 349 + ], + "score": 0.69, + "content": "v", + "type": "inline_equation" + }, + { + "bbox": [ + 454, + 338, + 506, + 351 + ], + "score": 1.0, + "content": ", we build a", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 349, + 505, + 362 + ], + "spans": [ + { + "bbox": [ + 105, + 349, + 505, + 362 + ], + "score": 1.0, + "content": "program graph as discussed above and then replace the variable name in all corresponding variable", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 361, + 505, + 372 + ], + "spans": [ + { + "bbox": [ + 106, + 361, + 188, + 372 + ], + "score": 1.0, + "content": "tokens by a special", + "type": "text" + }, + { + "bbox": [ + 189, + 361, + 226, + 371 + ], + "score": 0.68, + "content": "{ < } S \\mathrm { L O T } >", + "type": "inline_equation" + }, + { + "bbox": [ + 226, + 361, + 505, + 372 + ], + "score": 1.0, + "content": "token. To predict a name, we use the initial node labels computed", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 371, + 505, + 384 + ], + "spans": [ + { + "bbox": [ + 105, + 371, + 505, + 384 + ], + "score": 1.0, + "content": "as the concatenation of learnable token embeddings and type embeddings as discussed above, run", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 380, + 506, + 397 + ], + "spans": [ + { + "bbox": [ + 105, + 380, + 506, + 397 + ], + "score": 1.0, + "content": "GGNN propagation for 8 time steps2 and then compute a variable usage representation by averaging", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 393, + 506, + 406 + ], + "spans": [ + { + "bbox": [ + 105, + 393, + 217, + 406 + ], + "score": 1.0, + "content": "the representations for all", + "type": "text" + }, + { + "bbox": [ + 217, + 394, + 254, + 404 + ], + "score": 0.52, + "content": "{ < } S \\mathrm { L O T } >", + "type": "inline_equation" + }, + { + "bbox": [ + 255, + 393, + 506, + 406 + ], + "score": 1.0, + "content": "tokens. This representation is then used as the initial state", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 403, + 506, + 417 + ], + "spans": [ + { + "bbox": [ + 105, + 403, + 506, + 417 + ], + "score": 1.0, + "content": "of a one-layer GRU, which predicts the target name as a sequence of subtokens (e.g., the name", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 415, + 505, + 428 + ], + "spans": [ + { + "bbox": [ + 105, + 415, + 505, + 428 + ], + "score": 1.0, + "content": "inputStreamBuffer is treated as the sequence [input, stream, buffer]). We train this", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 425, + 506, + 441 + ], + "spans": [ + { + "bbox": [ + 105, + 425, + 506, + 441 + ], + "score": 1.0, + "content": "graph2seq architecture using a maximum likelihood objective. In section 5, we report the accuracy", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 437, + 408, + 450 + ], + "spans": [ + { + "bbox": [ + 105, + 437, + 408, + 450 + ], + "score": 1.0, + "content": "for predicting the exact name and the F1 score for predicting its subtokens.", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 25.5, + "bbox_fs": [ + 105, + 338, + 506, + 450 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 461, + 505, + 571 + ], + "lines": [ + { + "bbox": [ + 106, + 461, + 506, + 474 + ], + "spans": [ + { + "bbox": [ + 106, + 461, + 506, + 474 + ], + "score": 1.0, + "content": "Program Graphs for VARMISUSE To model VARMISUSE with program graphs we need to", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 472, + 506, + 485 + ], + "spans": [ + { + "bbox": [ + 106, + 472, + 361, + 485 + ], + "score": 1.0, + "content": "modify the graph. First, to compute a context representation", + "type": "text" + }, + { + "bbox": [ + 362, + 472, + 379, + 484 + ], + "score": 0.91, + "content": "\\mathbf { } c ( t )", + "type": "inline_equation" + }, + { + "bbox": [ + 379, + 472, + 421, + 485 + ], + "score": 1.0, + "content": "for a slot", + "type": "text" + }, + { + "bbox": [ + 421, + 473, + 427, + 482 + ], + "score": 0.61, + "content": "t", + "type": "inline_equation" + }, + { + "bbox": [ + 428, + 472, + 506, + 485 + ], + "score": 1.0, + "content": "where we want to", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 483, + 506, + 496 + ], + "spans": [ + { + "bbox": [ + 105, + 483, + 292, + 496 + ], + "score": 1.0, + "content": "predict the used variable, we insert a new node", + "type": "text" + }, + { + "bbox": [ + 293, + 485, + 324, + 495 + ], + "score": 0.89, + "content": "v _ { < S \\tt L O T > }", + "type": "inline_equation" + }, + { + "bbox": [ + 325, + 483, + 394, + 496 + ], + "score": 1.0, + "content": "at the position of", + "type": "text" + }, + { + "bbox": [ + 395, + 484, + 399, + 493 + ], + "score": 0.72, + "content": "t", + "type": "inline_equation" + }, + { + "bbox": [ + 400, + 483, + 506, + 496 + ], + "score": 1.0, + "content": ", corresponding to a “hole”", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 494, + 506, + 506 + ], + "spans": [ + { + "bbox": [ + 105, + 494, + 506, + 506 + ], + "score": 1.0, + "content": "at this point, and connect it to the remaining graph using all applicable edges that do not depend", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 505, + 506, + 517 + ], + "spans": [ + { + "bbox": [ + 105, + 505, + 506, + 517 + ], + "score": 1.0, + "content": "on the chosen variable at the slot (i.e., everything but LastUse, LastWrite, LastLexicalUse, and", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 515, + 506, + 529 + ], + "spans": [ + { + "bbox": [ + 105, + 515, + 367, + 529 + ], + "score": 1.0, + "content": "GuardedBy edges). Then, to compute the usage representation", + "type": "text" + }, + { + "bbox": [ + 367, + 516, + 396, + 528 + ], + "score": 0.92, + "content": "\\mathbf { u } ( t , v )", + "type": "inline_equation" + }, + { + "bbox": [ + 396, + 515, + 506, + 529 + ], + "score": 1.0, + "content": "of each candidate variable", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 106, + 527, + 505, + 540 + ], + "spans": [ + { + "bbox": [ + 106, + 529, + 113, + 537 + ], + "score": 0.69, + "content": "v", + "type": "inline_equation" + }, + { + "bbox": [ + 113, + 527, + 303, + 540 + ], + "score": 1.0, + "content": "at the target slot, we insert a “candidate” node", + "type": "text" + }, + { + "bbox": [ + 303, + 528, + 320, + 539 + ], + "score": 0.88, + "content": "v _ { t , v }", + "type": "inline_equation" + }, + { + "bbox": [ + 320, + 527, + 348, + 540 + ], + "score": 1.0, + "content": "for all", + "type": "text" + }, + { + "bbox": [ + 348, + 529, + 355, + 537 + ], + "score": 0.75, + "content": "v", + "type": "inline_equation" + }, + { + "bbox": [ + 355, + 527, + 366, + 540 + ], + "score": 1.0, + "content": "in", + "type": "text" + }, + { + "bbox": [ + 366, + 528, + 378, + 538 + ], + "score": 0.88, + "content": "\\mathbb { V } _ { t }", + "type": "inline_equation" + }, + { + "bbox": [ + 378, + 527, + 505, + 540 + ], + "score": 1.0, + "content": ", and connect it to the graph by", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 537, + 505, + 550 + ], + "spans": [ + { + "bbox": [ + 105, + 537, + 505, + 550 + ], + "score": 1.0, + "content": "inserting the LastUse, LastWrite and LastLexicalUse edges that would be used if the variable were", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 549, + 505, + 561 + ], + "spans": [ + { + "bbox": [ + 105, + 549, + 505, + 561 + ], + "score": 1.0, + "content": "to be used at this slot. Each of these candidate nodes represents the speculative placement of the", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 559, + 210, + 573 + ], + "spans": [ + { + "bbox": [ + 105, + 559, + 210, + 573 + ], + "score": 1.0, + "content": "variable within the scope.", + "type": "text" + } + ], + "index": 40 + } + ], + "index": 35.5, + "bbox_fs": [ + 105, + 461, + 506, + 573 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 577, + 505, + 634 + ], + "lines": [ + { + "bbox": [ + 107, + 578, + 505, + 588 + ], + "spans": [ + { + "bbox": [ + 107, + 578, + 505, + 588 + ], + "score": 1.0, + "content": "Using the initial node representations, concatenated with an extra bit that is set to one for the candidate", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 586, + 506, + 601 + ], + "spans": [ + { + "bbox": [ + 105, + 586, + 132, + 601 + ], + "score": 1.0, + "content": "nodes", + "type": "text" + }, + { + "bbox": [ + 132, + 589, + 149, + 600 + ], + "score": 0.88, + "content": "v _ { t , v }", + "type": "inline_equation" + }, + { + "bbox": [ + 149, + 586, + 506, + 601 + ], + "score": 1.0, + "content": ", we run GGNN propagation for 8 time steps.2 The context and usage representation are", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 599, + 506, + 613 + ], + "spans": [ + { + "bbox": [ + 105, + 599, + 273, + 613 + ], + "score": 1.0, + "content": "then the final node states of the nodes, i.e.,", + "type": "text" + }, + { + "bbox": [ + 273, + 599, + 339, + 612 + ], + "score": 0.9, + "content": "\\pmb { c } ( t ) = \\pmb { h } ^ { ( v _ { < \\mathrm { S L O T } > } ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 339, + 599, + 357, + 613 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 357, + 599, + 425, + 612 + ], + "score": 0.91, + "content": "\\mathbf { u } ( t , v ) = h ^ { ( v _ { t , v } ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 425, + 599, + 506, + 613 + ], + "score": 1.0, + "content": ". Finally, the correct", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 610, + 506, + 625 + ], + "spans": [ + { + "bbox": [ + 105, + 610, + 307, + 625 + ], + "score": 1.0, + "content": "variable usage at the location is computed as arg", + "type": "text" + }, + { + "bbox": [ + 307, + 612, + 397, + 623 + ], + "score": 0.83, + "content": "\\textstyle \\operatorname* { m a x } _ { v } W [ c ( t ) , \\mathbf { u } ( t , v ) ]", + "type": "inline_equation" + }, + { + "bbox": [ + 398, + 610, + 426, + 625 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 426, + 612, + 439, + 622 + ], + "score": 0.59, + "content": "W", + "type": "inline_equation" + }, + { + "bbox": [ + 439, + 610, + 506, + 625 + ], + "score": 1.0, + "content": "is a linear layer", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 621, + 455, + 635 + ], + "spans": [ + { + "bbox": [ + 105, + 621, + 227, + 635 + ], + "score": 1.0, + "content": "that uses the concatenation of", + "type": "text" + }, + { + "bbox": [ + 227, + 622, + 244, + 634 + ], + "score": 0.92, + "content": "\\mathbf { } c ( t )", + "type": "inline_equation" + }, + { + "bbox": [ + 245, + 621, + 262, + 635 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 263, + 622, + 291, + 634 + ], + "score": 0.93, + "content": "\\mathbf { u } ( t , v )", + "type": "inline_equation" + }, + { + "bbox": [ + 291, + 621, + 455, + 635 + ], + "score": 1.0, + "content": ". We train using a max-margin objective.", + "type": "text" + } + ], + "index": 45 + } + ], + "index": 43, + "bbox_fs": [ + 105, + 578, + 506, + 635 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 648, + 209, + 659 + ], + "lines": [ + { + "bbox": [ + 105, + 647, + 210, + 660 + ], + "spans": [ + { + "bbox": [ + 105, + 647, + 210, + 660 + ], + "score": 1.0, + "content": "4.1 IMPLEMENTATION", + "type": "text" + } + ], + "index": 46 + } + ], + "index": 46 + }, + { + "type": "text", + "bbox": [ + 108, + 668, + 505, + 713 + ], + "lines": [ + { + "bbox": [ + 105, + 667, + 505, + 681 + ], + "spans": [ + { + "bbox": [ + 105, + 667, + 505, + 681 + ], + "score": 1.0, + "content": "Using GGNNs for sets of large, diverse graphs requires some engineering effort, as efficient batching", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 105, + 678, + 505, + 691 + ], + "spans": [ + { + "bbox": [ + 105, + 678, + 505, + 691 + ], + "score": 1.0, + "content": "is hard in the presence of diverse shapes. An important observation is that large graphs are normally", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 105, + 690, + 505, + 703 + ], + "spans": [ + { + "bbox": [ + 105, + 690, + 505, + 703 + ], + "score": 1.0, + "content": "very sparse, and thus a representation of edges as an adjacency list would usually be advantageous", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 106, + 702, + 505, + 713 + ], + "spans": [ + { + "bbox": [ + 106, + 702, + 505, + 713 + ], + "score": 1.0, + "content": "to reduce memory consumption. In our case, this can be easily implemented using a sparse tensor", + "type": "text" + } + ], + "index": 50 + }, + { + "bbox": [ + 105, + 82, + 507, + 95 + ], + "spans": [ + { + "bbox": [ + 105, + 82, + 507, + 95 + ], + "score": 1.0, + "content": "representation, allowing large batch sizes that exploit the parallelism of modern GPUs efficiently.", + "type": "text", + "cross_page": true + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 93, + 506, + 106 + ], + "spans": [ + { + "bbox": [ + 105, + 93, + 506, + 106 + ], + "score": 1.0, + "content": "A second key insight is to represent a batch of graphs as one large graph with many disconnected", + "type": "text", + "cross_page": true + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 105, + 506, + 118 + ], + "spans": [ + { + "bbox": [ + 105, + 105, + 506, + 118 + ], + "score": 1.0, + "content": "components. This just requires appropriate pre-processing to make node identities unique. As this", + "type": "text", + "cross_page": true + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 115, + 505, + 128 + ], + "spans": [ + { + "bbox": [ + 105, + 115, + 505, + 128 + ], + "score": 1.0, + "content": "makes batch construction somewhat CPU-intensive, we found it useful to prepare minibatches on", + "type": "text", + "cross_page": true + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 127, + 506, + 140 + ], + "spans": [ + { + "bbox": [ + 105, + 127, + 506, + 140 + ], + "score": 1.0, + "content": "a separate thread. Our TensorFlow (Abadi et al., 2016) implementation scales to 55 graphs per", + "type": "text", + "cross_page": true + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 137, + 506, + 150 + ], + "spans": [ + { + "bbox": [ + 105, + 137, + 506, + 150 + ], + "score": 1.0, + "content": "second during training and 219 graphs per second during test-time using a single NVidia GeForce", + "type": "text", + "cross_page": true + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 147, + 506, + 162 + ], + "spans": [ + { + "bbox": [ + 105, + 147, + 506, + 162 + ], + "score": 1.0, + "content": "GTX Titan X with graphs having on average 2,228 (median 936) nodes and 8,350 (median 3,274)", + "type": "text", + "cross_page": true + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 159, + 506, + 172 + ], + "spans": [ + { + "bbox": [ + 105, + 159, + 506, + 172 + ], + "score": 1.0, + "content": "edges and 8 GGNN unrolling iterations, all 20 edge types (forward and backward edges for 10", + "type": "text", + "cross_page": true + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 171, + 506, + 183 + ], + "spans": [ + { + "bbox": [ + 105, + 171, + 506, + 183 + ], + "score": 1.0, + "content": "original edge types) and the size of the hidden layer set to 64. The number of types of edges", + "type": "text", + "cross_page": true + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 181, + 506, + 194 + ], + "spans": [ + { + "bbox": [ + 105, + 181, + 506, + 194 + ], + "score": 1.0, + "content": "in the GGNN contributes proportionally to the running time. For example, a GGNN run for our", + "type": "text", + "cross_page": true + } + ], + "index": 9 + }, + { + "bbox": [ + 104, + 191, + 506, + 205 + ], + "spans": [ + { + "bbox": [ + 104, + 191, + 506, + 205 + ], + "score": 1.0, + "content": "ablation study using only the two most common edge types (NextToken, Child) achieves 105", + "type": "text", + "cross_page": true + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 203, + 507, + 216 + ], + "spans": [ + { + "bbox": [ + 105, + 203, + 507, + 216 + ], + "score": 1.0, + "content": "graphs/second during training and 419 graphs/second at test time with the same hyperparameters.", + "type": "text", + "cross_page": true + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 214, + 506, + 227 + ], + "spans": [ + { + "bbox": [ + 106, + 214, + 506, + 227 + ], + "score": 1.0, + "content": "Our (generic) implementation of GGNNs is available at https://github.com/Microsoft/", + "type": "text", + "cross_page": true + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 226, + 460, + 237 + ], + "spans": [ + { + "bbox": [ + 106, + 226, + 460, + 237 + ], + "score": 1.0, + "content": "gated-graph-neural-network-samples, using a simpler demonstration task.", + "type": "text", + "cross_page": true + } + ], + "index": 13 + } + ], + "index": 48.5, + "bbox_fs": [ + 105, + 667, + 505, + 713 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 82, + 505, + 237 + ], + "lines": [ + { + "bbox": [ + 105, + 82, + 507, + 95 + ], + "spans": [ + { + "bbox": [ + 105, + 82, + 507, + 95 + ], + "score": 1.0, + "content": "representation, allowing large batch sizes that exploit the parallelism of modern GPUs efficiently.", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 93, + 506, + 106 + ], + "spans": [ + { + "bbox": [ + 105, + 93, + 506, + 106 + ], + "score": 1.0, + "content": "A second key insight is to represent a batch of graphs as one large graph with many disconnected", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 105, + 506, + 118 + ], + "spans": [ + { + "bbox": [ + 105, + 105, + 506, + 118 + ], + "score": 1.0, + "content": "components. This just requires appropriate pre-processing to make node identities unique. As this", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 115, + 505, + 128 + ], + "spans": [ + { + "bbox": [ + 105, + 115, + 505, + 128 + ], + "score": 1.0, + "content": "makes batch construction somewhat CPU-intensive, we found it useful to prepare minibatches on", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 127, + 506, + 140 + ], + "spans": [ + { + "bbox": [ + 105, + 127, + 506, + 140 + ], + "score": 1.0, + "content": "a separate thread. 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The number of types of edges", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 181, + 506, + 194 + ], + "spans": [ + { + "bbox": [ + 105, + 181, + 506, + 194 + ], + "score": 1.0, + "content": "in the GGNN contributes proportionally to the running time. 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We then filtered out", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 337, + 506, + 351 + ], + "spans": [ + { + "bbox": [ + 105, + 337, + 506, + 351 + ], + "score": 1.0, + "content": "projects that we could not (easily) compile in full using Roslyn3, as we require a compilation to", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 349, + 506, + 362 + ], + "spans": [ + { + "bbox": [ + 105, + 349, + 506, + 362 + ], + "score": 1.0, + "content": "extract precise type information for the code (including those types present in external libraries). Our", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 359, + 506, + 372 + ], + "spans": [ + { + "bbox": [ + 105, + 359, + 506, + 372 + ], + "score": 1.0, + "content": "final dataset contains 29 projects from a diverse set of domains (compilers, databases, . . . ) with about", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 370, + 404, + 383 + ], + "spans": [ + { + "bbox": [ + 105, + 370, + 404, + 383 + ], + "score": 1.0, + "content": "2.9 million non-empty lines of code. A full table is shown in Appendix D.", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 17.5 + }, + { + "type": "text", + "bbox": [ + 107, + 387, + 505, + 453 + ], + "lines": [ + { + "bbox": [ + 105, + 387, + 506, + 400 + ], + "spans": [ + { + "bbox": [ + 105, + 387, + 506, + 400 + ], + "score": 1.0, + "content": "For the task of detecting variable misuses, we collect data from all projects by selecting all variable", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 398, + 506, + 411 + ], + "spans": [ + { + "bbox": [ + 106, + 398, + 506, + 411 + ], + "score": 1.0, + "content": "usage locations, filtering out variable declarations, where at least one other type-compatible replace-", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 408, + 506, + 421 + ], + "spans": [ + { + "bbox": [ + 105, + 408, + 506, + 421 + ], + "score": 1.0, + "content": "ment variable is in scope. The task is then to infer the correct variable that originally existed in that", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 420, + 506, + 433 + ], + "spans": [ + { + "bbox": [ + 105, + 420, + 506, + 433 + ], + "score": 1.0, + "content": "location. Thus, by construction there is at least one type-correct replacement variable, i.e. picking it", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 430, + 506, + 444 + ], + "spans": [ + { + "bbox": [ + 105, + 430, + 506, + 444 + ], + "score": 1.0, + "content": "would not raise an error during type checking. In our test datasets, at each slot there are on average", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 442, + 340, + 455 + ], + "spans": [ + { + "bbox": [ + 105, + 442, + 299, + 455 + ], + "score": 1.0, + "content": "3.8 type-correct alternative variables (median 3,", + "type": "text" + }, + { + "bbox": [ + 299, + 442, + 333, + 453 + ], + "score": 0.83, + "content": "\\sigma = 2 . 6", + "type": "inline_equation" + }, + { + "bbox": [ + 334, + 442, + 340, + 455 + ], + "score": 1.0, + "content": ").", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 23.5 + }, + { + "type": "text", + "bbox": [ + 107, + 459, + 505, + 514 + ], + "lines": [ + { + "bbox": [ + 105, + 458, + 505, + 472 + ], + "spans": [ + { + "bbox": [ + 105, + 458, + 505, + 472 + ], + "score": 1.0, + "content": "From our dataset, we selected two projects as our development set. From the rest of the projects, we", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 469, + 505, + 482 + ], + "spans": [ + { + "bbox": [ + 106, + 469, + 505, + 482 + ], + "score": 1.0, + "content": "selected three projects for UNSEENPROJTEST to allow testing on projects with completely unknown", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 481, + 505, + 493 + ], + "spans": [ + { + "bbox": [ + 105, + 481, + 505, + 493 + ], + "score": 1.0, + "content": "structure and types. We split the remaining 23 projects into train/validation/test sets in the proportion", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 492, + 505, + 504 + ], + "spans": [ + { + "bbox": [ + 106, + 492, + 505, + 504 + ], + "score": 1.0, + "content": "60-10-30, splitting along files (i.e., all examples from one source file are in the same set). We call the", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 503, + 279, + 515 + ], + "spans": [ + { + "bbox": [ + 105, + 503, + 279, + 515 + ], + "score": 1.0, + "content": "test set obtained like this SEENPROJTEST.", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 29 + }, + { + "type": "text", + "bbox": [ + 106, + 550, + 505, + 682 + ], + "lines": [ + { + "bbox": [ + 106, + 550, + 504, + 561 + ], + "spans": [ + { + "bbox": [ + 106, + 550, + 504, + 561 + ], + "score": 1.0, + "content": "Baselines For VARMISUSE, we consider two bidirectional RNN-based baselines. The local model", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 562, + 506, + 573 + ], + "spans": [ + { + "bbox": [ + 106, + 562, + 506, + 573 + ], + "score": 1.0, + "content": "(LOC) is a simple two-layer bidirectional GRU run over the tokens before and after the target location.", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 572, + 506, + 585 + ], + "spans": [ + { + "bbox": [ + 105, + 572, + 176, + 585 + ], + "score": 1.0, + "content": "For this baseline,", + "type": "text" + }, + { + "bbox": [ + 176, + 572, + 193, + 583 + ], + "score": 0.89, + "content": "\\mathbf { } c ( t )", + "type": "inline_equation" + }, + { + "bbox": [ + 194, + 572, + 506, + 585 + ], + "score": 1.0, + "content": "is set to the slot representation computed by the RNN, and the usage context of", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 106, + 583, + 505, + 595 + ], + "spans": [ + { + "bbox": [ + 106, + 583, + 162, + 595 + ], + "score": 1.0, + "content": "each variable", + "type": "text" + }, + { + "bbox": [ + 162, + 583, + 191, + 595 + ], + "score": 0.92, + "content": "\\mathbf { u } ( t , v )", + "type": "inline_equation" + }, + { + "bbox": [ + 191, + 583, + 505, + 595 + ], + "score": 1.0, + "content": "is the embedding of the name and type of the variable, computed in the same", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 594, + 506, + 607 + ], + "spans": [ + { + "bbox": [ + 105, + 594, + 506, + 607 + ], + "score": 1.0, + "content": "way as the initial node labels in the GGNN. This baseline allows us to evaluate how important the", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 604, + 505, + 617 + ], + "spans": [ + { + "bbox": [ + 105, + 604, + 505, + 617 + ], + "score": 1.0, + "content": "usage context information is for this task. The flat dataflow model (AVGBIRNN) is an extension to", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 616, + 505, + 628 + ], + "spans": [ + { + "bbox": [ + 105, + 616, + 253, + 628 + ], + "score": 1.0, + "content": "LOC, where the usage representation", + "type": "text" + }, + { + "bbox": [ + 254, + 616, + 282, + 628 + ], + "score": 0.93, + "content": "\\mathbf { u } ( t , v )", + "type": "inline_equation" + }, + { + "bbox": [ + 282, + 616, + 505, + 628 + ], + "score": 1.0, + "content": "is computed using another two-layer bidirectional RNN", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 626, + 505, + 640 + ], + "spans": [ + { + "bbox": [ + 105, + 626, + 505, + 640 + ], + "score": 1.0, + "content": "run over the tokens before/after each usage, and then averaging over the computed representations", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 106, + 638, + 505, + 650 + ], + "spans": [ + { + "bbox": [ + 106, + 638, + 191, + 650 + ], + "score": 1.0, + "content": "at the variable token", + "type": "text" + }, + { + "bbox": [ + 191, + 640, + 198, + 648 + ], + "score": 0.7, + "content": "v", + "type": "inline_equation" + }, + { + "bbox": [ + 198, + 638, + 278, + 650 + ], + "score": 1.0, + "content": ". The local context,", + "type": "text" + }, + { + "bbox": [ + 278, + 638, + 295, + 650 + ], + "score": 0.9, + "content": "\\mathbf { } c ( t )", + "type": "inline_equation" + }, + { + "bbox": [ + 296, + 638, + 505, + 650 + ], + "score": 1.0, + "content": ", is identical to LOC. AVGBIRNN is a significantly", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 648, + 506, + 662 + ], + "spans": [ + { + "bbox": [ + 105, + 648, + 506, + 662 + ], + "score": 1.0, + "content": "stronger baseline that already takes some structural information into account, as the averaging over all", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 106, + 660, + 505, + 672 + ], + "spans": [ + { + "bbox": [ + 106, + 660, + 505, + 672 + ], + "score": 1.0, + "content": "variables usages helps with long-range dependencies. Both models pick the variable that maximizes", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 107, + 668, + 164, + 685 + ], + "spans": [ + { + "bbox": [ + 107, + 670, + 158, + 683 + ], + "score": 0.93, + "content": "\\boldsymbol { c } ( t ) ^ { T } \\mathbf { u } ( t , v )", + "type": "inline_equation" + }, + { + "bbox": [ + 158, + 668, + 164, + 685 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 43 + } + ], + "index": 37.5 + }, + { + "type": "text", + "bbox": [ + 107, + 687, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 688, + 505, + 700 + ], + "spans": [ + { + "bbox": [ + 105, + 688, + 505, + 700 + ], + "score": 1.0, + "content": "For VARNAMING, we replace LOC by AVGLBL, which uses a log-bilinear model for 4 left and 4", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 699, + 506, + 712 + ], + "spans": [ + { + "bbox": [ + 105, + 699, + 506, + 712 + ], + "score": 1.0, + "content": "right context tokens of each variable usage, and then averages over these context representations (this", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 710, + 507, + 722 + ], + "spans": [ + { + "bbox": [ + 105, + 710, + 507, + 722 + ], + "score": 1.0, + "content": "corresponds to the model in Allamanis et al. (2015)). We also test AVGBIRNN on VARNAMING,", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 720, + 434, + 732 + ], + "spans": [ + { + "bbox": [ + 105, + 720, + 434, + 732 + ], + "score": 1.0, + "content": "which essentially replaces the log-bilinear context model by a bidirectional RNN.", + "type": "text" + } + ], + "index": 47 + } + ], + "index": 45.5 + } + ], + "page_idx": 5, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 108, + 27, + 293, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 26, + 294, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 26, + 294, + 38 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2018", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 302, + 752, + 308, + 760 + ], + "lines": [ + { + "bbox": [ + 302, + 751, + 309, + 762 + ], + "spans": [ + { + "bbox": [ + 302, + 751, + 309, + 762 + ], + "score": 1.0, + "content": "6", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 82, + 505, + 237 + ], + "lines": [], + "index": 6.5, + "bbox_fs": [ + 104, + 82, + 507, + 237 + ], + "lines_deleted": true + }, + { + "type": "title", + "bbox": [ + 108, + 276, + 192, + 289 + ], + "lines": [ + { + "bbox": [ + 104, + 274, + 195, + 291 + ], + "spans": [ + { + "bbox": [ + 104, + 274, + 195, + 291 + ], + "score": 1.0, + "content": "5 EVALUATION", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 14 + }, + { + "type": "text", + "bbox": [ + 107, + 315, + 505, + 381 + ], + "lines": [ + { + "bbox": [ + 105, + 314, + 507, + 327 + ], + "spans": [ + { + "bbox": [ + 105, + 315, + 412, + 327 + ], + "score": 1.0, + "content": "Dataset We collected a dataset for the VARMISUSE task from open source", + "type": "text" + }, + { + "bbox": [ + 413, + 314, + 426, + 326 + ], + "score": 0.82, + "content": "C ^ { \\# }", + "type": "inline_equation" + }, + { + "bbox": [ + 426, + 315, + 507, + 327 + ], + "score": 1.0, + "content": "projects on GitHub.", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 326, + 506, + 339 + ], + "spans": [ + { + "bbox": [ + 105, + 326, + 506, + 339 + ], + "score": 1.0, + "content": "To select projects, we picked the top-starred (non-fork) projects in GitHub. We then filtered out", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 337, + 506, + 351 + ], + "spans": [ + { + "bbox": [ + 105, + 337, + 506, + 351 + ], + "score": 1.0, + "content": "projects that we could not (easily) compile in full using Roslyn3, as we require a compilation to", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 349, + 506, + 362 + ], + "spans": [ + { + "bbox": [ + 105, + 349, + 506, + 362 + ], + "score": 1.0, + "content": "extract precise type information for the code (including those types present in external libraries). Our", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 359, + 506, + 372 + ], + "spans": [ + { + "bbox": [ + 105, + 359, + 506, + 372 + ], + "score": 1.0, + "content": "final dataset contains 29 projects from a diverse set of domains (compilers, databases, . . . ) with about", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 370, + 404, + 383 + ], + "spans": [ + { + "bbox": [ + 105, + 370, + 404, + 383 + ], + "score": 1.0, + "content": "2.9 million non-empty lines of code. A full table is shown in Appendix D.", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 17.5, + "bbox_fs": [ + 105, + 314, + 507, + 383 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 387, + 505, + 453 + ], + "lines": [ + { + "bbox": [ + 105, + 387, + 506, + 400 + ], + "spans": [ + { + "bbox": [ + 105, + 387, + 506, + 400 + ], + "score": 1.0, + "content": "For the task of detecting variable misuses, we collect data from all projects by selecting all variable", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 398, + 506, + 411 + ], + "spans": [ + { + "bbox": [ + 106, + 398, + 506, + 411 + ], + "score": 1.0, + "content": "usage locations, filtering out variable declarations, where at least one other type-compatible replace-", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 408, + 506, + 421 + ], + "spans": [ + { + "bbox": [ + 105, + 408, + 506, + 421 + ], + "score": 1.0, + "content": "ment variable is in scope. The task is then to infer the correct variable that originally existed in that", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 420, + 506, + 433 + ], + "spans": [ + { + "bbox": [ + 105, + 420, + 506, + 433 + ], + "score": 1.0, + "content": "location. Thus, by construction there is at least one type-correct replacement variable, i.e. picking it", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 430, + 506, + 444 + ], + "spans": [ + { + "bbox": [ + 105, + 430, + 506, + 444 + ], + "score": 1.0, + "content": "would not raise an error during type checking. In our test datasets, at each slot there are on average", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 442, + 340, + 455 + ], + "spans": [ + { + "bbox": [ + 105, + 442, + 299, + 455 + ], + "score": 1.0, + "content": "3.8 type-correct alternative variables (median 3,", + "type": "text" + }, + { + "bbox": [ + 299, + 442, + 333, + 453 + ], + "score": 0.83, + "content": "\\sigma = 2 . 6", + "type": "inline_equation" + }, + { + "bbox": [ + 334, + 442, + 340, + 455 + ], + "score": 1.0, + "content": ").", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 23.5, + "bbox_fs": [ + 105, + 387, + 506, + 455 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 459, + 505, + 514 + ], + "lines": [ + { + "bbox": [ + 105, + 458, + 505, + 472 + ], + "spans": [ + { + "bbox": [ + 105, + 458, + 505, + 472 + ], + "score": 1.0, + "content": "From our dataset, we selected two projects as our development set. From the rest of the projects, we", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 469, + 505, + 482 + ], + "spans": [ + { + "bbox": [ + 106, + 469, + 505, + 482 + ], + "score": 1.0, + "content": "selected three projects for UNSEENPROJTEST to allow testing on projects with completely unknown", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 481, + 505, + 493 + ], + "spans": [ + { + "bbox": [ + 105, + 481, + 505, + 493 + ], + "score": 1.0, + "content": "structure and types. We split the remaining 23 projects into train/validation/test sets in the proportion", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 492, + 505, + 504 + ], + "spans": [ + { + "bbox": [ + 106, + 492, + 505, + 504 + ], + "score": 1.0, + "content": "60-10-30, splitting along files (i.e., all examples from one source file are in the same set). We call the", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 503, + 279, + 515 + ], + "spans": [ + { + "bbox": [ + 105, + 503, + 279, + 515 + ], + "score": 1.0, + "content": "test set obtained like this SEENPROJTEST.", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 29, + "bbox_fs": [ + 105, + 458, + 505, + 515 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 550, + 505, + 682 + ], + "lines": [ + { + "bbox": [ + 106, + 550, + 504, + 561 + ], + "spans": [ + { + "bbox": [ + 106, + 550, + 504, + 561 + ], + "score": 1.0, + "content": "Baselines For VARMISUSE, we consider two bidirectional RNN-based baselines. The local model", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 562, + 506, + 573 + ], + "spans": [ + { + "bbox": [ + 106, + 562, + 506, + 573 + ], + "score": 1.0, + "content": "(LOC) is a simple two-layer bidirectional GRU run over the tokens before and after the target location.", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 572, + 506, + 585 + ], + "spans": [ + { + "bbox": [ + 105, + 572, + 176, + 585 + ], + "score": 1.0, + "content": "For this baseline,", + "type": "text" + }, + { + "bbox": [ + 176, + 572, + 193, + 583 + ], + "score": 0.89, + "content": "\\mathbf { } c ( t )", + "type": "inline_equation" + }, + { + "bbox": [ + 194, + 572, + 506, + 585 + ], + "score": 1.0, + "content": "is set to the slot representation computed by the RNN, and the usage context of", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 106, + 583, + 505, + 595 + ], + "spans": [ + { + "bbox": [ + 106, + 583, + 162, + 595 + ], + "score": 1.0, + "content": "each variable", + "type": "text" + }, + { + "bbox": [ + 162, + 583, + 191, + 595 + ], + "score": 0.92, + "content": "\\mathbf { u } ( t , v )", + "type": "inline_equation" + }, + { + "bbox": [ + 191, + 583, + 505, + 595 + ], + "score": 1.0, + "content": "is the embedding of the name and type of the variable, computed in the same", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 594, + 506, + 607 + ], + "spans": [ + { + "bbox": [ + 105, + 594, + 506, + 607 + ], + "score": 1.0, + "content": "way as the initial node labels in the GGNN. This baseline allows us to evaluate how important the", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 604, + 505, + 617 + ], + "spans": [ + { + "bbox": [ + 105, + 604, + 505, + 617 + ], + "score": 1.0, + "content": "usage context information is for this task. The flat dataflow model (AVGBIRNN) is an extension to", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 616, + 505, + 628 + ], + "spans": [ + { + "bbox": [ + 105, + 616, + 253, + 628 + ], + "score": 1.0, + "content": "LOC, where the usage representation", + "type": "text" + }, + { + "bbox": [ + 254, + 616, + 282, + 628 + ], + "score": 0.93, + "content": "\\mathbf { u } ( t , v )", + "type": "inline_equation" + }, + { + "bbox": [ + 282, + 616, + 505, + 628 + ], + "score": 1.0, + "content": "is computed using another two-layer bidirectional RNN", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 626, + 505, + 640 + ], + "spans": [ + { + "bbox": [ + 105, + 626, + 505, + 640 + ], + "score": 1.0, + "content": "run over the tokens before/after each usage, and then averaging over the computed representations", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 106, + 638, + 505, + 650 + ], + "spans": [ + { + "bbox": [ + 106, + 638, + 191, + 650 + ], + "score": 1.0, + "content": "at the variable token", + "type": "text" + }, + { + "bbox": [ + 191, + 640, + 198, + 648 + ], + "score": 0.7, + "content": "v", + "type": "inline_equation" + }, + { + "bbox": [ + 198, + 638, + 278, + 650 + ], + "score": 1.0, + "content": ". The local context,", + "type": "text" + }, + { + "bbox": [ + 278, + 638, + 295, + 650 + ], + "score": 0.9, + "content": "\\mathbf { } c ( t )", + "type": "inline_equation" + }, + { + "bbox": [ + 296, + 638, + 505, + 650 + ], + "score": 1.0, + "content": ", is identical to LOC. AVGBIRNN is a significantly", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 648, + 506, + 662 + ], + "spans": [ + { + "bbox": [ + 105, + 648, + 506, + 662 + ], + "score": 1.0, + "content": "stronger baseline that already takes some structural information into account, as the averaging over all", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 106, + 660, + 505, + 672 + ], + "spans": [ + { + "bbox": [ + 106, + 660, + 505, + 672 + ], + "score": 1.0, + "content": "variables usages helps with long-range dependencies. Both models pick the variable that maximizes", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 107, + 668, + 164, + 685 + ], + "spans": [ + { + "bbox": [ + 107, + 670, + 158, + 683 + ], + "score": 0.93, + "content": "\\boldsymbol { c } ( t ) ^ { T } \\mathbf { u } ( t , v )", + "type": "inline_equation" + }, + { + "bbox": [ + 158, + 668, + 164, + 685 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 43 + } + ], + "index": 37.5, + "bbox_fs": [ + 105, + 550, + 506, + 685 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 687, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 688, + 505, + 700 + ], + "spans": [ + { + "bbox": [ + 105, + 688, + 505, + 700 + ], + "score": 1.0, + "content": "For VARNAMING, we replace LOC by AVGLBL, which uses a log-bilinear model for 4 left and 4", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 699, + 506, + 712 + ], + "spans": [ + { + "bbox": [ + 105, + 699, + 506, + 712 + ], + "score": 1.0, + "content": "right context tokens of each variable usage, and then averages over these context representations (this", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 710, + 507, + 722 + ], + "spans": [ + { + "bbox": [ + 105, + 710, + 507, + 722 + ], + "score": 1.0, + "content": "corresponds to the model in Allamanis et al. (2015)). We also test AVGBIRNN on VARNAMING,", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 720, + 434, + 732 + ], + "spans": [ + { + "bbox": [ + 105, + 720, + 434, + 732 + ], + "score": 1.0, + "content": "which essentially replaces the log-bilinear context model by a bidirectional RNN.", + "type": "text" + } + ], + "index": 47 + } + ], + "index": 45.5, + "bbox_fs": [ + 105, + 688, + 507, + 732 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "table", + "bbox": [ + 106, + 114, + 504, + 200 + ], + "blocks": [ + { + "type": "table_caption", + "bbox": [ + 106, + 80, + 506, + 113 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 105, + 79, + 505, + 93 + ], + "spans": [ + { + "bbox": [ + 105, + 79, + 505, + 93 + ], + "score": 1.0, + "content": "Table 1: Evaluation of models. SEENPROJTEST refers to the test set containing projects that have", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 91, + 507, + 103 + ], + "spans": [ + { + "bbox": [ + 105, + 91, + 507, + 103 + ], + "score": 1.0, + "content": "files in the training set, UNSEENPROJTEST refers to projects that have no files in the training data.", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 101, + 236, + 115 + ], + "spans": [ + { + "bbox": [ + 105, + 101, + 236, + 115 + ], + "score": 1.0, + "content": "Results averaged over two runs.", + "type": "text" + } + ], + "index": 2 + } + ], + "index": 1 + }, + { + "type": "table_body", + "bbox": [ + 106, + 114, + 504, + 200 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 106, + 114, + 504, + 200 + ], + "spans": [ + { + "bbox": [ + 106, + 114, + 504, + 200 + ], + "score": 0.976, + "html": "
SEENPROJTESTUNSEENPROJTEST
LocAVGLBLAVGBIRNNGGNNLocAVGLBLAVGBIRNNGGNN
VARMISUSE
Accuracy (%)50.0二73.785.528.960.278.2
PR AUC0.7880.9410.9800.6110.8950.958
VARNAMING Accuracy (%)36.142.9
53.622.723.444.0
F1 (%)44.050.165.830.632.062.0
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Ablation DescriptionAccuracy (%) VARMISUSE
VARNAMING
Standard Model (reported in Table 1)85.553.6
Only NextToken, Child, LastUse_LastWrite edges80.631.2
Only semantic edges (all but NextToken, Child) Only syntax edges (NextToken, Child)78.4 55.352.9 34.3
Node Labels: Tokens instead of subtokens
Node Labels:Disabled85.6 84.334.5
31.8
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AVGLBL and AVGBIRNN, which capture information", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 389, + 506, + 405 + ], + "spans": [ + { + "bbox": [ + 105, + 389, + 506, + 405 + ], + "score": 1.0, + "content": "from many variable usage sites, but do not explicitly encode the rich structure of the problem, still lag", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 401, + 505, + 414 + ], + "spans": [ + { + "bbox": [ + 105, + 401, + 505, + 414 + ], + "score": 1.0, + "content": "behind the GGNN by a wide margin. The performance difference is larger for VARMISUSE, since", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 104, + 410, + 433, + 427 + ], + "spans": [ + { + "bbox": [ + 104, + 410, + 433, + 427 + ], + "score": 1.0, + "content": "the structure and the semantics of code are far more important within this setting.", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 13 + }, + { + "type": "text", + "bbox": [ + 107, + 437, + 505, + 492 + ], + "lines": [ + { + "bbox": [ + 106, + 436, + 505, + 450 + ], + "spans": [ + { + "bbox": [ + 106, + 436, + 505, + 450 + ], + "score": 1.0, + "content": "Generalization to new projects Generalizing across a diverse set of source code projects with", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 447, + 505, + 460 + ], + "spans": [ + { + "bbox": [ + 105, + 447, + 505, + 460 + ], + "score": 1.0, + "content": "different domains is an important challenge in machine learning. We repeat the evaluation using the", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 458, + 506, + 471 + ], + "spans": [ + { + "bbox": [ + 105, + 458, + 506, + 471 + ], + "score": 1.0, + "content": "UNSEENPROJTEST set stemming from projects that have no files in the training set. The right side of", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 469, + 505, + 482 + ], + "spans": [ + { + "bbox": [ + 105, + 469, + 505, + 482 + ], + "score": 1.0, + "content": "Table 1 shows that our models still achieve good performance, although it is slightly lower compared", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 481, + 506, + 493 + ], + "spans": [ + { + "bbox": [ + 105, + 481, + 506, + 493 + ], + "score": 1.0, + "content": "to SEENPROJTEST. This is expected since the type lattice is mostly unknown in UNSEENPROJTEST.", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 18 + }, + { + "type": "text", + "bbox": [ + 108, + 497, + 504, + 531 + ], + "lines": [ + { + "bbox": [ + 105, + 496, + 507, + 511 + ], + "spans": [ + { + "bbox": [ + 105, + 496, + 507, + 511 + ], + "score": 1.0, + "content": "We believe that the dominant problem in applying a trained model to an unknown project (i.e.,", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 509, + 506, + 521 + ], + "spans": [ + { + "bbox": [ + 106, + 509, + 506, + 521 + ], + "score": 1.0, + "content": "domain) is the fact that its type hierarchy is unknown and the used vocabulary (e.g. in variables,", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 520, + 323, + 532 + ], + "spans": [ + { + "bbox": [ + 106, + 520, + 323, + 532 + ], + "score": 1.0, + "content": "method and class names, etc.) can differ substantially.", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 22 + }, + { + "type": "text", + "bbox": [ + 107, + 543, + 505, + 632 + ], + "lines": [ + { + "bbox": [ + 106, + 543, + 505, + 556 + ], + "spans": [ + { + "bbox": [ + 106, + 543, + 505, + 556 + ], + "score": 1.0, + "content": "Ablation Study To study the effect of some of the design choices for our models, we have run", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 555, + 505, + 567 + ], + "spans": [ + { + "bbox": [ + 105, + 555, + 505, + 567 + ], + "score": 1.0, + "content": "some additional experiments and show their results in Table 2. First, we varied the edges used in", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 566, + 506, + 579 + ], + "spans": [ + { + "bbox": [ + 106, + 566, + 506, + 579 + ], + "score": 1.0, + "content": "the program graph. We find that restricting the model to syntactic information has a large impact", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 577, + 506, + 589 + ], + "spans": [ + { + "bbox": [ + 105, + 577, + 506, + 589 + ], + "score": 1.0, + "content": "on performance on both tasks, whereas restricting it to semantic edges seems to mostly impact", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 587, + 505, + 600 + ], + "spans": [ + { + "bbox": [ + 105, + 587, + 505, + 600 + ], + "score": 1.0, + "content": "performance on VARMISUSE. Similarly, the ComputedFrom, FormalArgName and ReturnsTo", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 599, + 505, + 611 + ], + "spans": [ + { + "bbox": [ + 105, + 599, + 505, + 611 + ], + "score": 1.0, + "content": "edges give a small boost on VARMISUSE, but greatly improve performance on VARNAMING. As", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 609, + 506, + 623 + ], + "spans": [ + { + "bbox": [ + 105, + 609, + 506, + 623 + ], + "score": 1.0, + "content": "evidenced by the experiments with the node label representation, syntax node and token names seem", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 620, + 442, + 633 + ], + "spans": [ + { + "bbox": [ + 105, + 620, + 442, + 633 + ], + "score": 1.0, + "content": "to matter little for VARMISUSE, but naturally have a great impact on VARNAMING.", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 27.5 + }, + { + "type": "title", + "bbox": [ + 108, + 646, + 248, + 658 + ], + "lines": [ + { + "bbox": [ + 105, + 645, + 250, + 660 + ], + "spans": [ + { + "bbox": [ + 105, + 645, + 250, + 660 + ], + "score": 1.0, + "content": "5.2 QUALITATIVE EVALUATION", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 32 + }, + { + "type": "text", + "bbox": [ + 108, + 667, + 504, + 701 + ], + "lines": [ + { + "bbox": [ + 106, + 667, + 506, + 681 + ], + "spans": [ + { + "bbox": [ + 106, + 667, + 506, + 681 + ], + "score": 1.0, + "content": "Figure 3 illustrates the predictions that GGNN makes on a sample test snippet. The snippet recursively", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 677, + 506, + 691 + ], + "spans": [ + { + "bbox": [ + 105, + 677, + 506, + 691 + ], + "score": 1.0, + "content": "searches for the global directives file by gradually descending into the root folder. 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SEENPROJTESTUNSEENPROJTEST
LocAVGLBLAVGBIRNNGGNNLocAVGLBLAVGBIRNNGGNN
VARMISUSE
Accuracy (%)50.0二73.785.528.960.278.2
PR AUC0.7880.9410.9800.6110.8950.958
VARNAMING Accuracy (%)36.142.9
53.622.723.444.0
F1 (%)44.050.165.830.632.062.0
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Ablation DescriptionAccuracy (%) VARMISUSE
VARNAMING
Standard Model (reported in Table 1)85.553.6
Only NextToken, Child, LastUse_LastWrite edges80.631.2
Only semantic edges (all but NextToken, Child) Only syntax edges (NextToken, Child)78.4 55.352.9 34.3
Node Labels: Tokens instead of subtokens
Node Labels:Disabled85.6 84.334.5
31.8
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AVGLBL and AVGBIRNN, which capture information", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 389, + 506, + 405 + ], + "spans": [ + { + "bbox": [ + 105, + 389, + 506, + 405 + ], + "score": 1.0, + "content": "from many variable usage sites, but do not explicitly encode the rich structure of the problem, still lag", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 401, + 505, + 414 + ], + "spans": [ + { + "bbox": [ + 105, + 401, + 505, + 414 + ], + "score": 1.0, + "content": "behind the GGNN by a wide margin. The performance difference is larger for VARMISUSE, since", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 104, + 410, + 433, + 427 + ], + "spans": [ + { + "bbox": [ + 104, + 410, + 433, + 427 + ], + "score": 1.0, + "content": "the structure and the semantics of code are far more important within this setting.", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 13, + "bbox_fs": [ + 104, + 367, + 506, + 427 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 437, + 505, + 492 + ], + "lines": [ + { + "bbox": [ + 106, + 436, + 505, + 450 + ], + "spans": [ + { + "bbox": [ + 106, + 436, + 505, + 450 + ], + "score": 1.0, + "content": "Generalization to new projects Generalizing across a diverse set of source code projects with", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 447, + 505, + 460 + ], + "spans": [ + { + "bbox": [ + 105, + 447, + 505, + 460 + ], + "score": 1.0, + "content": "different domains is an important challenge in machine learning. We repeat the evaluation using the", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 458, + 506, + 471 + ], + "spans": [ + { + "bbox": [ + 105, + 458, + 506, + 471 + ], + "score": 1.0, + "content": "UNSEENPROJTEST set stemming from projects that have no files in the training set. The right side of", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 469, + 505, + 482 + ], + "spans": [ + { + "bbox": [ + 105, + 469, + 505, + 482 + ], + "score": 1.0, + "content": "Table 1 shows that our models still achieve good performance, although it is slightly lower compared", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 481, + 506, + 493 + ], + "spans": [ + { + "bbox": [ + 105, + 481, + 506, + 493 + ], + "score": 1.0, + "content": "to SEENPROJTEST. This is expected since the type lattice is mostly unknown in UNSEENPROJTEST.", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 18, + "bbox_fs": [ + 105, + 436, + 506, + 493 + ] + }, + { + "type": "text", + "bbox": [ + 108, + 497, + 504, + 531 + ], + "lines": [ + { + "bbox": [ + 105, + 496, + 507, + 511 + ], + "spans": [ + { + "bbox": [ + 105, + 496, + 507, + 511 + ], + "score": 1.0, + "content": "We believe that the dominant problem in applying a trained model to an unknown project (i.e.,", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 509, + 506, + 521 + ], + "spans": [ + { + "bbox": [ + 106, + 509, + 506, + 521 + ], + "score": 1.0, + "content": "domain) is the fact that its type hierarchy is unknown and the used vocabulary (e.g. in variables,", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 520, + 323, + 532 + ], + "spans": [ + { + "bbox": [ + 106, + 520, + 323, + 532 + ], + "score": 1.0, + "content": "method and class names, etc.) can differ substantially.", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 22, + "bbox_fs": [ + 105, + 496, + 507, + 532 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 543, + 505, + 632 + ], + "lines": [ + { + "bbox": [ + 106, + 543, + 505, + 556 + ], + "spans": [ + { + "bbox": [ + 106, + 543, + 505, + 556 + ], + "score": 1.0, + "content": "Ablation Study To study the effect of some of the design choices for our models, we have run", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 555, + 505, + 567 + ], + "spans": [ + { + "bbox": [ + 105, + 555, + 505, + 567 + ], + "score": 1.0, + "content": "some additional experiments and show their results in Table 2. First, we varied the edges used in", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 566, + 506, + 579 + ], + "spans": [ + { + "bbox": [ + 106, + 566, + 506, + 579 + ], + "score": 1.0, + "content": "the program graph. We find that restricting the model to syntactic information has a large impact", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 577, + 506, + 589 + ], + "spans": [ + { + "bbox": [ + 105, + 577, + 506, + 589 + ], + "score": 1.0, + "content": "on performance on both tasks, whereas restricting it to semantic edges seems to mostly impact", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 587, + 505, + 600 + ], + "spans": [ + { + "bbox": [ + 105, + 587, + 505, + 600 + ], + "score": 1.0, + "content": "performance on VARMISUSE. Similarly, the ComputedFrom, FormalArgName and ReturnsTo", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 599, + 505, + 611 + ], + "spans": [ + { + "bbox": [ + 105, + 599, + 505, + 611 + ], + "score": 1.0, + "content": "edges give a small boost on VARMISUSE, but greatly improve performance on VARNAMING. As", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 609, + 506, + 623 + ], + "spans": [ + { + "bbox": [ + 105, + 609, + 506, + 623 + ], + "score": 1.0, + "content": "evidenced by the experiments with the node label representation, syntax node and token names seem", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 620, + 442, + 633 + ], + "spans": [ + { + "bbox": [ + 105, + 620, + 442, + 633 + ], + "score": 1.0, + "content": "to matter little for VARMISUSE, but naturally have a great impact on VARNAMING.", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 27.5, + "bbox_fs": [ + 105, + 543, + 506, + 633 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 646, + 248, + 658 + ], + "lines": [ + { + "bbox": [ + 105, + 645, + 250, + 660 + ], + "spans": [ + { + "bbox": [ + 105, + 645, + 250, + 660 + ], + "score": 1.0, + "content": "5.2 QUALITATIVE EVALUATION", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 32 + }, + { + "type": "text", + "bbox": [ + 108, + 667, + 504, + 701 + ], + "lines": [ + { + "bbox": [ + 106, + 667, + 506, + 681 + ], + "spans": [ + { + "bbox": [ + 106, + 667, + 506, + 681 + ], + "score": 1.0, + "content": "Figure 3 illustrates the predictions that GGNN makes on a sample test snippet. The snippet recursively", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 677, + 506, + 691 + ], + "spans": [ + { + "bbox": [ + 105, + 677, + 506, + 691 + ], + "score": 1.0, + "content": "searches for the global directives file by gradually descending into the root folder. Reasoning about", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 106, + 689, + 505, + 702 + ], + "spans": [ + { + "bbox": [ + 106, + 689, + 505, + 702 + ], + "score": 1.0, + "content": "the correct variable usages is hard, even for humans, but the GGNN correctly predicts the variable", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 505, + 506, + 518 + ], + "spans": [ + { + "bbox": [ + 105, + 505, + 506, + 518 + ], + "score": 1.0, + "content": "usages at all locations except two (slot 1 and 8). As a software engineer is writing the code, it is", + "type": "text", + "cross_page": true + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 516, + 505, + 530 + ], + "spans": [ + { + "bbox": [ + 105, + 516, + 505, + 530 + ], + "score": 1.0, + "content": "imaginable that she may make a mistake misusing one variable in the place of another. Since all", + "type": "text", + "cross_page": true + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 526, + 506, + 542 + ], + "spans": [ + { + "bbox": [ + 105, + 526, + 506, + 542 + ], + "score": 1.0, + "content": "variables are string variables, no type errors will be raised. As the probabilities in Fig. 3 suggest", + "type": "text", + "cross_page": true + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 540, + 505, + 551 + ], + "spans": [ + { + "bbox": [ + 106, + 540, + 505, + 551 + ], + "score": 1.0, + "content": "most potential variable misuses can be flagged by the model yielding valuable warnings to software", + "type": "text", + "cross_page": true + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 550, + 411, + 563 + ], + "spans": [ + { + "bbox": [ + 106, + 550, + 411, + 563 + ], + "score": 1.0, + "content": "engineers. Additional samples with comments can be found in Appendix B.", + "type": "text", + "cross_page": true + } + ], + "index": 18 + } + ], + "index": 34, + "bbox_fs": [ + 105, + 667, + 506, + 702 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "image", + "bbox": [ + 104, + 84, + 507, + 311 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 104, + 84, + 507, + 311 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 104, + 84, + 507, + 311 + ], + "spans": [ + { + "bbox": [ + 104, + 84, + 507, + 311 + ], + "score": 0.838, + "type": "image", + "image_path": "cd8c9c73a863608276a7923a4347ae44724d33325aed2f76dd26669048968bd2.jpg" + } + ] + } + ], + "index": 1, + "virtual_lines": [ + { + "bbox": [ + 104, + 84, + 507, + 159.66666666666669 + ], + "spans": [], + "index": 0 + }, + { + "bbox": [ + 104, + 159.66666666666669, + 507, + 235.33333333333337 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 104, + 235.33333333333337, + 507, + 311.00000000000006 + ], + "spans": [], + "index": 2 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 106, + 325, + 506, + 381 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 106, + 325, + 505, + 337 + ], + "spans": [ + { + "bbox": [ + 106, + 325, + 505, + 337 + ], + "score": 1.0, + "content": "Figure 3: VARMISUSE predictions on slots within a snippet of the SEENPROJTEST set for the", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 334, + 505, + 348 + ], + "spans": [ + { + "bbox": [ + 105, + 334, + 505, + 348 + ], + "score": 1.0, + "content": "ServiceStack project. Additional visualizations are available in Appendix B. The underlined tokens", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 106, + 347, + 505, + 359 + ], + "spans": [ + { + "bbox": [ + 106, + 347, + 505, + 359 + ], + "score": 1.0, + "content": "are the correct tokens. The model has to select among a number of string variables at each slot,", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 357, + 506, + 370 + ], + "spans": [ + { + "bbox": [ + 106, + 357, + 506, + 370 + ], + "score": 1.0, + "content": "where all of them represent some kind of path. 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Resilience & Transient Fault Handling Library
quartznetb33e6f86499.6k9.8kScheduler
ravendbDev5523092264778.0k82.7kDocument Database
RestSharp70de357b204.0k4.5kREST and HTTP API Client Library
Rx.NET2d146fe518014.0k21.9kReactive Language Extensions
scriptcsf3cc8bcb182.7k4.3kC# Text Editor
ServiceStack6d59da7523138.0k46.2kWeb Framework
ShareX718dd71112522.3k18.1kSharing Application
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Hangfireffc4912f333.6k6.1kBackground Job Processing Library
Humanizertcclla77e272.4k4.4kString Manipulation and Formatting
Lean†f574bfd719026.4k28.3kAlgorithmic Trading Engine
Nancy72elf614707.5k15.7HTTP Service Framework
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NLog643e326a758.3k11.0kLogging Library
Opserver51b032e7243.7k4.5kMonitoring System
OptiKey7d35c718346.1k3.9kAssistive On-Screen Keyboard
orleans Pollye0d6a150 0afdbc32300 3230.7k 3.8k35.6k 9.1kDistributed Virtual Actor Model
Resilience & Transient Fault Handling Library
quartznetb33e6f86499.6k9.8kScheduler
ravendbDev5523092264778.0k82.7kDocument Database
RestSharp70de357b204.0k4.5kREST and HTTP API Client Library
Rx.NET2d146fe518014.0k21.9kReactive Language Extensions
scriptcsf3cc8bcb182.7k4.3kC# Text Editor
ServiceStack6d59da7523138.0k46.2kWeb Framework
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Ablation DescriptionAccuracy (%) VARMISUSE
VARNAMING
Standard Model (reported in Table 1)85.553.6
Only NextToken, Child, LastUse_LastWrite edges80.631.2
Only semantic edges (all but NextToken, Child) Only syntax edges (NextToken, Child)78.4 55.352.9 34.3
Node Labels: Tokens instead of subtokens
Node Labels:Disabled85.6 84.334.5
31.8
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SEENPROJTESTUNSEENPROJTEST
LocAVGLBLAVGBIRNNGGNNLocAVGLBLAVGBIRNNGGNN
VARMISUSE
Accuracy (%)50.0二73.785.528.960.278.2
PR AUC0.7880.9410.9800.6110.8950.958
VARNAMING Accuracy (%)36.142.9
53.622.723.444.0
F1 (%)44.050.165.830.632.062.0
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Accuracy On SEENPROJTEST (%)91.684.581.878.675.177.5
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However, a concrete theoretical understanding of why deep neural networks work well in", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 429, + 505, + 442 + ], + "spans": [ + { + "bbox": [ + 105, + 429, + 505, + 442 + ], + "score": 1.0, + "content": "practice remains elusive. From the perspective of optimization, a significant barrier is imposed by", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 440, + 505, + 452 + ], + "spans": [ + { + "bbox": [ + 105, + 440, + 505, + 452 + ], + "score": 1.0, + "content": "the nonconvexity of training neural networks. 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Minimizing the loss of a deep linear neural network is a", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 141, + 256, + 470, + 268 + ], + "spans": [ + { + "bbox": [ + 141, + 256, + 470, + 268 + ], + "score": 1.0, + "content": "nonconvex problem, and despite recent progress, our understanding of this loss", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 141, + 267, + 470, + 280 + ], + "spans": [ + { + "bbox": [ + 141, + 267, + 470, + 280 + ], + "score": 1.0, + "content": "surface is still incomplete. 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However, a concrete theoretical understanding of why deep neural networks work well in", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 429, + 505, + 442 + ], + "spans": [ + { + "bbox": [ + 105, + 429, + 505, + 442 + ], + "score": 1.0, + "content": "practice remains elusive. From the perspective of optimization, a significant barrier is imposed by", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 440, + 505, + 452 + ], + "spans": [ + { + "bbox": [ + 105, + 440, + 505, + 452 + ], + "score": 1.0, + "content": "the nonconvexity of training neural networks. Moreover, it was proved by Blum & Rivest (1988)", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 451, + 506, + 463 + ], + "spans": [ + { + "bbox": [ + 105, + 451, + 506, + 463 + ], + "score": 1.0, + "content": "that training even a 3-node neural network to global optimality is NP-Hard in the worst case, so", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 462, + 489, + 474 + ], + "spans": [ + { + "bbox": [ + 105, + 462, + 489, + 474 + ], + "score": 1.0, + "content": "there is little hope that neural networks have properties that make global optimization tractable.", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 21.5, + "bbox_fs": [ + 105, + 383, + 506, + 474 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 478, + 505, + 544 + ], + "lines": [ + { + "bbox": [ + 105, + 478, + 505, + 491 + ], + "spans": [ + { + "bbox": [ + 105, + 478, + 505, + 491 + ], + "score": 1.0, + "content": "Despite the difficulties of optimizing weights in neural networks, the empirical successes suggest", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 488, + 505, + 502 + ], + "spans": [ + { + "bbox": [ + 105, + 488, + 505, + 502 + ], + "score": 1.0, + "content": "that the local minima of their loss surfaces could be close to global minima; and several papers have", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 500, + 505, + 513 + ], + "spans": [ + { + "bbox": [ + 105, + 500, + 505, + 513 + ], + "score": 1.0, + "content": "recently appeared in the literature attempting to provide a theoretical justification for the success", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 511, + 506, + 524 + ], + "spans": [ + { + "bbox": [ + 105, + 511, + 506, + 524 + ], + "score": 1.0, + "content": "of these models. For example, by relating neural networks to spherical spin-glass models from", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 523, + 505, + 534 + ], + "spans": [ + { + "bbox": [ + 106, + 523, + 505, + 534 + ], + "score": 1.0, + "content": "statistical physics, Choromanska et al. (2015) provided some empirical evidence that the increase of", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 533, + 379, + 545 + ], + "spans": [ + { + "bbox": [ + 106, + 533, + 379, + 545 + ], + "score": 1.0, + "content": "size of neural networks makes local minima close to global minima.", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 28.5, + "bbox_fs": [ + 105, + 478, + 506, + 545 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 550, + 505, + 682 + ], + "lines": [ + { + "bbox": [ + 105, + 550, + 505, + 563 + ], + "spans": [ + { + "bbox": [ + 105, + 550, + 505, + 563 + ], + "score": 1.0, + "content": "Another line of results (Yu & Chen, 1995; Soudry & Carmon, 2016; Xie et al., 2016; Nguyen & Hein,", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 560, + 505, + 574 + ], + "spans": [ + { + "bbox": [ + 105, + 560, + 505, + 574 + ], + "score": 1.0, + "content": "2017) provides conditions under which a critical point of the empirical risk is a global minimum.", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 572, + 505, + 585 + ], + "spans": [ + { + "bbox": [ + 105, + 572, + 505, + 585 + ], + "score": 1.0, + "content": "Such results roughly involve proving that if full rank conditions of certain matrices (as well as some", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 104, + 582, + 505, + 596 + ], + "spans": [ + { + "bbox": [ + 104, + 582, + 505, + 596 + ], + "score": 1.0, + "content": "additional technical conditions) are satisfied, derivative of the risk being zero implies loss being", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 594, + 505, + 606 + ], + "spans": [ + { + "bbox": [ + 105, + 594, + 505, + 606 + ], + "score": 1.0, + "content": "zero. However, these results are obtained under restrictive assumptions; for example, Nguyen &", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 104, + 603, + 506, + 619 + ], + "spans": [ + { + "bbox": [ + 104, + 603, + 506, + 619 + ], + "score": 1.0, + "content": "Hein (2017) require the width of one of the hidden layers to be as large as the number of training", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 106, + 617, + 505, + 628 + ], + "spans": [ + { + "bbox": [ + 106, + 617, + 505, + 628 + ], + "score": 1.0, + "content": "examples. Soudry & Carmon (2016) and Xie et al. (2016) require the product of widths of two", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 627, + 506, + 640 + ], + "spans": [ + { + "bbox": [ + 105, + 627, + 506, + 640 + ], + "score": 1.0, + "content": "adjacent layers to be at least as large as the number of training examples, meaning that the number", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 106, + 639, + 504, + 650 + ], + "spans": [ + { + "bbox": [ + 106, + 639, + 504, + 650 + ], + "score": 1.0, + "content": "of parameters in the model must grow rapidly as we have more training data available. Another", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 649, + 504, + 661 + ], + "spans": [ + { + "bbox": [ + 105, + 649, + 504, + 661 + ], + "score": 1.0, + "content": "recent paper (Haeffele & Vidal, 2017) provides a sufficient condition for global optimality when the", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 661, + 505, + 672 + ], + "spans": [ + { + "bbox": [ + 105, + 661, + 505, + 672 + ], + "score": 1.0, + "content": "neural network is composed of subnetworks with identical architectures connected in parallel and a", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 671, + 387, + 684 + ], + "spans": [ + { + "bbox": [ + 105, + 671, + 387, + 684 + ], + "score": 1.0, + "content": "regularizer is designed to control the number of parallel architectures.", + "type": "text" + } + ], + "index": 43 + } + ], + "index": 37.5, + "bbox_fs": [ + 104, + 550, + 506, + 684 + ] + }, + { + "type": "text", + "bbox": [ + 108, + 687, + 504, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 688, + 504, + 700 + ], + "spans": [ + { + "bbox": [ + 106, + 688, + 504, + 700 + ], + "score": 1.0, + "content": "Towards obtaining a more precise characterization of the loss-surfaces, a valuable conceptual simpli-", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 106, + 699, + 505, + 711 + ], + "spans": [ + { + "bbox": [ + 106, + 699, + 505, + 711 + ], + "score": 1.0, + "content": "fication of deep nonlinear networks is deep linear neural networks, in which all activation functions", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 106, + 710, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 106, + 710, + 505, + 722 + ], + "score": 1.0, + "content": "are linear and the output of the entire network is a chained product of weight matrices with the", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 721, + 505, + 733 + ], + "spans": [ + { + "bbox": [ + 105, + 721, + 505, + 733 + ], + "score": 1.0, + "content": "input vector. Although at first sight a deep linear model may appear overly simplistic, even its opti-", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 105, + 81, + 505, + 96 + ], + "spans": [ + { + "bbox": [ + 105, + 81, + 505, + 96 + ], + "score": 1.0, + "content": "mization is nonconvex, and only recently theoretical results on this problem have started emerging.", + "type": "text", + "cross_page": true + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 94, + 505, + 106 + ], + "spans": [ + { + "bbox": [ + 106, + 94, + 505, + 106 + ], + "score": 1.0, + "content": "Interestingly, already in 1989, Baldi & Hornik (1989) showed that some shallow linear neural net-", + "type": "text", + "cross_page": true + } + ], + "index": 1 + }, + { + "bbox": [ + 106, + 105, + 505, + 117 + ], + "spans": [ + { + "bbox": [ + 106, + 105, + 505, + 117 + ], + "score": 1.0, + "content": "works have no local minima. More recently, Kawaguchi (2016) extended this result to deep linear", + "type": "text", + "cross_page": true + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 116, + 505, + 128 + ], + "spans": [ + { + "bbox": [ + 105, + 116, + 505, + 128 + ], + "score": 1.0, + "content": "networks and proved that any local minimum is also global while any other critical point is a saddle", + "type": "text", + "cross_page": true + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 127, + 506, + 139 + ], + "spans": [ + { + "bbox": [ + 105, + 127, + 506, + 139 + ], + "score": 1.0, + "content": "point. Subsequently, Lu & Kawaguchi (2017) provided a simpler proof that any local minimum", + "type": "text", + "cross_page": true + } + ], + "index": 4 + }, + { + "bbox": [ + 104, + 136, + 506, + 151 + ], + "spans": [ + { + "bbox": [ + 104, + 136, + 506, + 151 + ], + "score": 1.0, + "content": "is also global, with fewer assumptions than (Kawaguchi, 2016). Motivated by the success of deep", + "type": "text", + "cross_page": true + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 147, + 505, + 162 + ], + "spans": [ + { + "bbox": [ + 105, + 147, + 505, + 162 + ], + "score": 1.0, + "content": "residual networks (He et al., 2016a;b), Hardt & Ma (2017) investigated loss surfaces of deep linear", + "type": "text", + "cross_page": true + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 158, + 505, + 173 + ], + "spans": [ + { + "bbox": [ + 105, + 158, + 505, + 173 + ], + "score": 1.0, + "content": "residual networks and showed every critical point is a global minimum in a near-identity region;", + "type": "text", + "cross_page": true + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 169, + 474, + 184 + ], + "spans": [ + { + "bbox": [ + 105, + 169, + 474, + 184 + ], + "score": 1.0, + "content": "subsequently, Bartlett et al. (2017) extended this result to a nonlinear function space setting.", + "type": "text", + "cross_page": true + } + ], + "index": 8 + } + ], + "index": 45.5, + "bbox_fs": [ + 105, + 688, + 505, + 733 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 82, + 505, + 182 + ], + "lines": [ + { + "bbox": [ + 105, + 81, + 505, + 96 + ], + "spans": [ + { + "bbox": [ + 105, + 81, + 505, + 96 + ], + "score": 1.0, + "content": "mization is nonconvex, and only recently theoretical results on this problem have started emerging.", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 94, + 505, + 106 + ], + "spans": [ + { + "bbox": [ + 106, + 94, + 505, + 106 + ], + "score": 1.0, + "content": "Interestingly, already in 1989, Baldi & Hornik (1989) showed that some shallow linear neural net-", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 106, + 105, + 505, + 117 + ], + "spans": [ + { + "bbox": [ + 106, + 105, + 505, + 117 + ], + "score": 1.0, + "content": "works have no local minima. More recently, Kawaguchi (2016) extended this result to deep linear", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 116, + 505, + 128 + ], + "spans": [ + { + "bbox": [ + 105, + 116, + 505, + 128 + ], + "score": 1.0, + "content": "networks and proved that any local minimum is also global while any other critical point is a saddle", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 127, + 506, + 139 + ], + "spans": [ + { + "bbox": [ + 105, + 127, + 506, + 139 + ], + "score": 1.0, + "content": "point. Subsequently, Lu & Kawaguchi (2017) provided a simpler proof that any local minimum", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 104, + 136, + 506, + 151 + ], + "spans": [ + { + "bbox": [ + 104, + 136, + 506, + 151 + ], + "score": 1.0, + "content": "is also global, with fewer assumptions than (Kawaguchi, 2016). Motivated by the success of deep", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 147, + 505, + 162 + ], + "spans": [ + { + "bbox": [ + 105, + 147, + 505, + 162 + ], + "score": 1.0, + "content": "residual networks (He et al., 2016a;b), Hardt & Ma (2017) investigated loss surfaces of deep linear", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 158, + 505, + 173 + ], + "spans": [ + { + "bbox": [ + 105, + 158, + 505, + 173 + ], + "score": 1.0, + "content": "residual networks and showed every critical point is a global minimum in a near-identity region;", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 169, + 474, + 184 + ], + "spans": [ + { + "bbox": [ + 105, + 169, + 474, + 184 + ], + "score": 1.0, + "content": "subsequently, Bartlett et al. (2017) extended this result to a nonlinear function space setting.", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 4 + }, + { + "type": "title", + "bbox": [ + 108, + 195, + 224, + 207 + ], + "lines": [ + { + "bbox": [ + 107, + 195, + 226, + 207 + ], + "spans": [ + { + "bbox": [ + 107, + 195, + 226, + 207 + ], + "score": 1.0, + "content": "1.1 OUR CONTRIBUTIONS", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 9 + }, + { + "type": "text", + "bbox": [ + 108, + 213, + 504, + 235 + ], + "lines": [ + { + "bbox": [ + 106, + 213, + 505, + 226 + ], + "spans": [ + { + "bbox": [ + 106, + 213, + 505, + 226 + ], + "score": 1.0, + "content": "Inspired by this recent line of work, we study deep linear and nonlinear networks, in settings either", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 225, + 475, + 236 + ], + "spans": [ + { + "bbox": [ + 105, + 225, + 475, + 236 + ], + "score": 1.0, + "content": "similar to or more general than existing work. We summarize our main contributions below.", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 10.5 + }, + { + "type": "text", + "bbox": [ + 106, + 241, + 506, + 443 + ], + "lines": [ + { + "bbox": [ + 108, + 242, + 506, + 254 + ], + "spans": [ + { + "bbox": [ + 108, + 242, + 506, + 254 + ], + "score": 1.0, + "content": "• We provide both necessary and sufficient conditions for a critical point of the empirical risk to be a", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 115, + 252, + 506, + 266 + ], + "spans": [ + { + "bbox": [ + 115, + 252, + 506, + 266 + ], + "score": 1.0, + "content": "global minimum. Specifically, Theorem 2.1 shows that if the hidden layers are wide enough, then", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 114, + 264, + 506, + 276 + ], + "spans": [ + { + "bbox": [ + 114, + 264, + 506, + 276 + ], + "score": 1.0, + "content": "a critical point of the risk function is a global minimum if and only if the product of all parameter", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 116, + 274, + 506, + 287 + ], + "spans": [ + { + "bbox": [ + 116, + 274, + 506, + 287 + ], + "score": 1.0, + "content": "matrices is full-rank. In Theorem 2.2, we consider the case where some hidden layers have smaller", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 116, + 285, + 506, + 298 + ], + "spans": [ + { + "bbox": [ + 116, + 285, + 506, + 298 + ], + "score": 1.0, + "content": "width than both the input and output layers, and again provide necessary and sufficient conditions", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 116, + 296, + 505, + 309 + ], + "spans": [ + { + "bbox": [ + 116, + 296, + 505, + 309 + ], + "score": 1.0, + "content": "for global optimality. In comparison, Kawaguchi (2016) only proves that every critical point of the", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 115, + 307, + 506, + 320 + ], + "spans": [ + { + "bbox": [ + 115, + 307, + 506, + 320 + ], + "score": 1.0, + "content": "risk is either a global minimum or a saddle; it is an “existence” result without any computational", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 115, + 317, + 506, + 332 + ], + "spans": [ + { + "bbox": [ + 115, + 317, + 506, + 332 + ], + "score": 1.0, + "content": "implication. In contrast, we present efficiently checkable conditions for distinguishing the two", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 115, + 328, + 506, + 342 + ], + "spans": [ + { + "bbox": [ + 115, + 328, + 506, + 342 + ], + "score": 1.0, + "content": "different types of critical points; we can even use these conditions while running optimization to", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 115, + 340, + 506, + 352 + ], + "spans": [ + { + "bbox": [ + 115, + 340, + 506, + 352 + ], + "score": 1.0, + "content": "test whether the critical points we encounter are saddle points or not, if desired. It is also worth", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 115, + 351, + 505, + 364 + ], + "spans": [ + { + "bbox": [ + 115, + 351, + 505, + 364 + ], + "score": 1.0, + "content": "noting that such tests are intractable for general nonconvex optimization (Murty & Kabadi, 1987).", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 113, + 364, + 505, + 377 + ], + "spans": [ + { + "bbox": [ + 113, + 364, + 505, + 377 + ], + "score": 1.0, + "content": "Under the same assumption as (Hardt & Ma, 2017) on the data distribution, namely, a linear model", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 115, + 375, + 505, + 388 + ], + "spans": [ + { + "bbox": [ + 115, + 375, + 505, + 388 + ], + "score": 1.0, + "content": "with Gaussian noise, we can modify Theorem 2.1 to handle the population risk. As a corollary,", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 115, + 386, + 505, + 399 + ], + "spans": [ + { + "bbox": [ + 115, + 386, + 505, + 399 + ], + "score": 1.0, + "content": "we not only recover Theorem 2.2 in (Hardt & Ma, 2017), but also extend it to a strictly larger set,", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 116, + 398, + 505, + 410 + ], + "spans": [ + { + "bbox": [ + 116, + 398, + 505, + 410 + ], + "score": 1.0, + "content": "while removing their assumption that the true underlying linear model has a positive determinant.", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 108, + 411, + 505, + 423 + ], + "spans": [ + { + "bbox": [ + 108, + 411, + 505, + 423 + ], + "score": 1.0, + "content": "• Motivated by (Bartlett et al., 2017), we extend our results on deep linear networks to obtain suf-", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 115, + 421, + 505, + 434 + ], + "spans": [ + { + "bbox": [ + 115, + 421, + 505, + 434 + ], + "score": 1.0, + "content": "ficient conditions for global optimality in deep nonlinear networks, although only via a function", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 115, + 433, + 346, + 444 + ], + "spans": [ + { + "bbox": [ + 115, + 433, + 346, + 444 + ], + "score": 1.0, + "content": "space view; these are presented in Theorems 4.1 and 4.2.", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 20.5 + }, + { + "type": "title", + "bbox": [ + 108, + 460, + 503, + 472 + ], + "lines": [ + { + "bbox": [ + 104, + 458, + 506, + 474 + ], + "spans": [ + { + "bbox": [ + 104, + 458, + 506, + 474 + ], + "score": 1.0, + "content": "2 GLOBAL OPTIMALITY CONDITIONS FOR DEEP LINEAR NEURAL NETWORKS", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 30 + }, + { + "type": "text", + "bbox": [ + 105, + 480, + 504, + 502 + ], + "lines": [ + { + "bbox": [ + 105, + 478, + 505, + 493 + ], + "spans": [ + { + "bbox": [ + 105, + 478, + 505, + 493 + ], + "score": 1.0, + "content": "In this section, we describe the problem formulation and notations for deep linear neural networks,", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 491, + 396, + 503 + ], + "spans": [ + { + "bbox": [ + 105, + 491, + 396, + 503 + ], + "score": 1.0, + "content": "state main results (Theorems 2.1 and 2.2), and explain their implication.", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 31.5 + }, + { + "type": "title", + "bbox": [ + 107, + 511, + 306, + 522 + ], + "lines": [ + { + "bbox": [ + 106, + 511, + 307, + 524 + ], + "spans": [ + { + "bbox": [ + 106, + 511, + 307, + 524 + ], + "score": 1.0, + "content": "2.1 PROBLEM FORMULATION AND NOTATION", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 33 + }, + { + "type": "text", + "bbox": [ + 106, + 531, + 505, + 611 + ], + "lines": [ + { + "bbox": [ + 106, + 532, + 505, + 545 + ], + "spans": [ + { + "bbox": [ + 106, + 532, + 177, + 545 + ], + "score": 1.0, + "content": "Suppose we have", + "type": "text" + }, + { + "bbox": [ + 178, + 534, + 187, + 542 + ], + "score": 0.74, + "content": "m", + "type": "inline_equation" + }, + { + "bbox": [ + 188, + 532, + 402, + 545 + ], + "score": 1.0, + "content": "input-output pairs, where the inputs are of dimension", + "type": "text" + }, + { + "bbox": [ + 403, + 532, + 414, + 543 + ], + "score": 0.89, + "content": "d _ { x }", + "type": "inline_equation" + }, + { + "bbox": [ + 414, + 532, + 505, + 545 + ], + "score": 1.0, + "content": "and outputs of dimen-", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 104, + 541, + 506, + 557 + ], + "spans": [ + { + "bbox": [ + 104, + 541, + 126, + 557 + ], + "score": 1.0, + "content": "sion", + "type": "text" + }, + { + "bbox": [ + 126, + 543, + 137, + 555 + ], + "score": 0.88, + "content": "d _ { y }", + "type": "inline_equation" + }, + { + "bbox": [ + 137, + 541, + 160, + 557 + ], + "score": 1.0, + "content": ". Let", + "type": "text" + }, + { + "bbox": [ + 160, + 542, + 214, + 553 + ], + "score": 0.92, + "content": "\\boldsymbol { X } \\in \\mathbb { R } ^ { d _ { \\boldsymbol { x } } \\times m }", + "type": "inline_equation" + }, + { + "bbox": [ + 214, + 541, + 309, + 557 + ], + "score": 1.0, + "content": "be the data matrix and", + "type": "text" + }, + { + "bbox": [ + 310, + 542, + 362, + 553 + ], + "score": 0.92, + "content": "\\bar { Y } \\in \\mathbb { R } ^ { d _ { y } \\times m }", + "type": "inline_equation" + }, + { + "bbox": [ + 363, + 541, + 506, + 557 + ], + "score": 1.0, + "content": "be the output matrix. Suppose we", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 553, + 505, + 567 + ], + "spans": [ + { + "bbox": [ + 105, + 553, + 128, + 567 + ], + "score": 1.0, + "content": "have", + "type": "text" + }, + { + "bbox": [ + 128, + 555, + 138, + 564 + ], + "score": 0.82, + "content": "H", + "type": "inline_equation" + }, + { + "bbox": [ + 138, + 553, + 337, + 567 + ], + "score": 1.0, + "content": "hidden layers in the network, each having width", + "type": "text" + }, + { + "bbox": [ + 337, + 554, + 383, + 565 + ], + "score": 0.91, + "content": "d _ { 1 } , \\dots , d _ { H }", + "type": "inline_equation" + }, + { + "bbox": [ + 383, + 553, + 505, + 567 + ], + "score": 1.0, + "content": ". For notational simplicity we", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 565, + 505, + 578 + ], + "spans": [ + { + "bbox": [ + 105, + 565, + 120, + 578 + ], + "score": 1.0, + "content": "let", + "type": "text" + }, + { + "bbox": [ + 120, + 565, + 157, + 576 + ], + "score": 0.9, + "content": "d _ { 0 } = d _ { x }", + "type": "inline_equation" + }, + { + "bbox": [ + 157, + 565, + 177, + 578 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 177, + 565, + 227, + 577 + ], + "score": 0.91, + "content": "d _ { H + 1 } = d _ { y }", + "type": "inline_equation" + }, + { + "bbox": [ + 227, + 565, + 505, + 578 + ], + "score": 1.0, + "content": ". The weights between adjacent layers can be represented as matri-", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 104, + 575, + 506, + 590 + ], + "spans": [ + { + "bbox": [ + 104, + 575, + 122, + 590 + ], + "score": 1.0, + "content": "ces", + "type": "text" + }, + { + "bbox": [ + 123, + 577, + 192, + 588 + ], + "score": 0.92, + "content": "W _ { k } \\in \\mathbb { R } ^ { d _ { k } \\times d _ { k - 1 } }", + "type": "inline_equation" + }, + { + "bbox": [ + 192, + 575, + 212, + 590 + ], + "score": 1.0, + "content": ", for", + "type": "text" + }, + { + "bbox": [ + 212, + 577, + 289, + 588 + ], + "score": 0.87, + "content": "k = 1 , \\ldots , H + 1", + "type": "inline_equation" + }, + { + "bbox": [ + 289, + 575, + 506, + 590 + ], + "score": 1.0, + "content": ", and the output of the network can be written as the", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 587, + 506, + 601 + ], + "spans": [ + { + "bbox": [ + 105, + 587, + 218, + 601 + ], + "score": 1.0, + "content": "product of weight matrices", + "type": "text" + }, + { + "bbox": [ + 218, + 589, + 281, + 600 + ], + "score": 0.9, + "content": "W _ { H + 1 } , \\dots , W _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 282, + 587, + 349, + 601 + ], + "score": 1.0, + "content": "and data matrix", + "type": "text" + }, + { + "bbox": [ + 349, + 588, + 359, + 598 + ], + "score": 0.7, + "content": "X", + "type": "inline_equation" + }, + { + "bbox": [ + 360, + 587, + 364, + 601 + ], + "score": 1.0, + "content": ":", + "type": "text" + }, + { + "bbox": [ + 365, + 588, + 447, + 600 + ], + "score": 0.83, + "content": "W _ { H + 1 } W _ { H } \\cdot \\cdot \\cdot W _ { 1 } X", + "type": "inline_equation" + }, + { + "bbox": [ + 448, + 587, + 506, + 601 + ], + "score": 1.0, + "content": ". We consider", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 599, + 460, + 612 + ], + "spans": [ + { + "bbox": [ + 105, + 599, + 460, + 612 + ], + "score": 1.0, + "content": "minimizing the summation of squared error loss over all data points (i.e. empirical risk),", + "type": "text" + } + ], + "index": 40 + } + ], + "index": 37 + }, + { + "type": "interline_equation", + "bbox": [ + 193, + 617, + 417, + 632 + ], + "lines": [ + { + "bbox": [ + 193, + 617, + 417, + 632 + ], + "spans": [ + { + "bbox": [ + 193, + 617, + 417, + 632 + ], + "score": 0.89, + "content": "\\begin{array} { r } { \\mathrm { m i n i m i z e } \\quad L ( W ) : = \\frac { 1 } { 2 } \\left\\| W _ { H + 1 } W _ { H } \\cdot \\cdot \\cdot W _ { 1 } X - Y \\right\\| _ { \\mathrm { F } } ^ { 2 } , } \\end{array}", + "type": "interline_equation", + "image_path": "1e11c803fa4297c7f4a4b82a1691e72fa78ef47bb1675f18a51b9166564e6194.jpg" + } + ] + } + ], + "index": 41, + "virtual_lines": [ + { + "bbox": [ + 193, + 617, + 417, + 632 + ], + "spans": [], + "index": 41 + } + ] + }, + { + "type": "text", + "bbox": [ + 108, + 638, + 366, + 650 + ], + "lines": [ + { + "bbox": [ + 105, + 635, + 366, + 653 + ], + "spans": [ + { + "bbox": [ + 105, + 635, + 133, + 653 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 133, + 639, + 145, + 648 + ], + "score": 0.75, + "content": "W", + "type": "inline_equation" + }, + { + "bbox": [ + 146, + 635, + 291, + 653 + ], + "score": 1.0, + "content": "is a shorthand notation for the tuple", + "type": "text" + }, + { + "bbox": [ + 291, + 638, + 362, + 650 + ], + "score": 0.89, + "content": "( W _ { 1 } , \\dots , W _ { H + 1 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 362, + 635, + 366, + 653 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 42 + } + ], + "index": 42 + }, + { + "type": "text", + "bbox": [ + 107, + 656, + 504, + 702 + ], + "lines": [ + { + "bbox": [ + 105, + 656, + 506, + 670 + ], + "spans": [ + { + "bbox": [ + 105, + 656, + 240, + 669 + ], + "score": 1.0, + "content": "Assumptions. We assume that", + "type": "text" + }, + { + "bbox": [ + 240, + 658, + 275, + 669 + ], + "score": 0.92, + "content": "d _ { x } \\leq m", + "type": "inline_equation" + }, + { + "bbox": [ + 276, + 656, + 294, + 669 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 295, + 658, + 329, + 670 + ], + "score": 0.92, + "content": "d _ { y } \\leq m", + "type": "inline_equation" + }, + { + "bbox": [ + 330, + 656, + 369, + 669 + ], + "score": 1.0, + "content": ", and that", + "type": "text" + }, + { + "bbox": [ + 369, + 657, + 394, + 668 + ], + "score": 0.9, + "content": "X X ^ { T }", + "type": "inline_equation" + }, + { + "bbox": [ + 395, + 656, + 414, + 669 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 414, + 657, + 438, + 668 + ], + "score": 0.93, + "content": "Y X ^ { T }", + "type": "inline_equation" + }, + { + "bbox": [ + 438, + 656, + 506, + 669 + ], + "score": 1.0, + "content": "have full ranks.", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 106, + 669, + 505, + 681 + ], + "spans": [ + { + "bbox": [ + 106, + 669, + 505, + 681 + ], + "score": 1.0, + "content": "These assumptions are common when we consider supervised learning problems with deep neural", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 104, + 678, + 506, + 692 + ], + "spans": [ + { + "bbox": [ + 104, + 678, + 414, + 692 + ], + "score": 1.0, + "content": "networks (e.g. Kawaguchi (2016)). We also assume that the singular values of", + "type": "text" + }, + { + "bbox": [ + 414, + 680, + 489, + 691 + ], + "score": 0.92, + "content": "Y X ^ { T } ( X X ^ { T } ) ^ { - 1 } X", + "type": "inline_equation" + }, + { + "bbox": [ + 489, + 678, + 506, + 692 + ], + "score": 1.0, + "content": "are", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 689, + 503, + 703 + ], + "spans": [ + { + "bbox": [ + 105, + 689, + 503, + 703 + ], + "score": 1.0, + "content": "all distinct, which is made for notational simplicity and can be relaxed without too much difficulty.", + "type": "text" + } + ], + "index": 46 + } + ], + "index": 44.5 + }, + { + "type": "text", + "bbox": [ + 106, + 709, + 504, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 708, + 506, + 723 + ], + "spans": [ + { + "bbox": [ + 105, + 708, + 220, + 723 + ], + "score": 1.0, + "content": "Notation. Given a matrix", + "type": "text" + }, + { + "bbox": [ + 221, + 710, + 229, + 720 + ], + "score": 0.78, + "content": "A", + "type": "inline_equation" + }, + { + "bbox": [ + 229, + 708, + 247, + 723 + ], + "score": 1.0, + "content": ", let", + "type": "text" + }, + { + "bbox": [ + 247, + 710, + 284, + 721 + ], + "score": 0.91, + "content": "\\sigma _ { \\mathrm { m a x } } ( A )", + "type": "inline_equation" + }, + { + "bbox": [ + 284, + 708, + 304, + 723 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 304, + 710, + 339, + 721 + ], + "score": 0.91, + "content": "\\sigma _ { \\mathrm { m i n } } ( A )", + "type": "inline_equation" + }, + { + "bbox": [ + 340, + 708, + 506, + 723 + ], + "score": 1.0, + "content": "denote the largest and smallest singular", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 106, + 720, + 505, + 733 + ], + "spans": [ + { + "bbox": [ + 106, + 720, + 241, + 733 + ], + "score": 1.0, + "content": "values of A, respectively. Let row", + "type": "text" + }, + { + "bbox": [ + 242, + 721, + 257, + 732 + ], + "score": 0.64, + "content": "( A )", + "type": "inline_equation" + }, + { + "bbox": [ + 257, + 720, + 261, + 733 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 261, + 721, + 289, + 732 + ], + "score": 0.57, + "content": "\\operatorname { c o l } ( A )", + "type": "inline_equation" + }, + { + "bbox": [ + 289, + 720, + 293, + 733 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 293, + 721, + 325, + 732 + ], + "score": 0.31, + "content": "\\operatorname { n u l l } ( A )", + "type": "inline_equation" + }, + { + "bbox": [ + 326, + 720, + 329, + 733 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 329, + 721, + 365, + 732 + ], + "score": 0.42, + "content": "\\operatorname { r a n k } ( A )", + "type": "inline_equation" + }, + { + "bbox": [ + 365, + 720, + 386, + 733 + ], + "score": 1.0, + "content": ", and", + "type": "text" + }, + { + "bbox": [ + 386, + 720, + 410, + 733 + ], + "score": 0.92, + "content": "\\Vert A \\Vert _ { \\mathrm { F } }", + "type": "inline_equation" + }, + { + "bbox": [ + 410, + 720, + 505, + 733 + ], + "score": 1.0, + "content": "be respectively the row", + "type": "text" + } + ], + "index": 48 + } + ], + "index": 47.5 + } + ], + "page_idx": 1, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 108, + 27, + 293, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 26, + 294, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 26, + 294, + 38 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2018", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 302, + 751, + 309, + 760 + ], + "lines": [ + { + "bbox": [ + 301, + 750, + 310, + 763 + ], + "spans": [ + { + "bbox": [ + 301, + 750, + 310, + 763 + ], + "score": 1.0, + "content": "2", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 82, + 505, + 182 + ], + "lines": [], + "index": 4, + "bbox_fs": [ + 104, + 81, + 506, + 184 + ], + "lines_deleted": true + }, + { + "type": "title", + "bbox": [ + 108, + 195, + 224, + 207 + ], + "lines": [ + { + "bbox": [ + 107, + 195, + 226, + 207 + ], + "spans": [ + { + "bbox": [ + 107, + 195, + 226, + 207 + ], + "score": 1.0, + "content": "1.1 OUR CONTRIBUTIONS", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 9 + }, + { + "type": "text", + "bbox": [ + 108, + 213, + 504, + 235 + ], + "lines": [ + { + "bbox": [ + 106, + 213, + 505, + 226 + ], + "spans": [ + { + "bbox": [ + 106, + 213, + 505, + 226 + ], + "score": 1.0, + "content": "Inspired by this recent line of work, we study deep linear and nonlinear networks, in settings either", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 225, + 475, + 236 + ], + "spans": [ + { + "bbox": [ + 105, + 225, + 475, + 236 + ], + "score": 1.0, + "content": "similar to or more general than existing work. We summarize our main contributions below.", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 10.5, + "bbox_fs": [ + 105, + 213, + 505, + 236 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 241, + 506, + 443 + ], + "lines": [ + { + "bbox": [ + 108, + 242, + 506, + 254 + ], + "spans": [ + { + "bbox": [ + 108, + 242, + 506, + 254 + ], + "score": 1.0, + "content": "• We provide both necessary and sufficient conditions for a critical point of the empirical risk to be a", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 115, + 252, + 506, + 266 + ], + "spans": [ + { + "bbox": [ + 115, + 252, + 506, + 266 + ], + "score": 1.0, + "content": "global minimum. Specifically, Theorem 2.1 shows that if the hidden layers are wide enough, then", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 114, + 264, + 506, + 276 + ], + "spans": [ + { + "bbox": [ + 114, + 264, + 506, + 276 + ], + "score": 1.0, + "content": "a critical point of the risk function is a global minimum if and only if the product of all parameter", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 116, + 274, + 506, + 287 + ], + "spans": [ + { + "bbox": [ + 116, + 274, + 506, + 287 + ], + "score": 1.0, + "content": "matrices is full-rank. In Theorem 2.2, we consider the case where some hidden layers have smaller", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 116, + 285, + 506, + 298 + ], + "spans": [ + { + "bbox": [ + 116, + 285, + 506, + 298 + ], + "score": 1.0, + "content": "width than both the input and output layers, and again provide necessary and sufficient conditions", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 116, + 296, + 505, + 309 + ], + "spans": [ + { + "bbox": [ + 116, + 296, + 505, + 309 + ], + "score": 1.0, + "content": "for global optimality. In comparison, Kawaguchi (2016) only proves that every critical point of the", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 115, + 307, + 506, + 320 + ], + "spans": [ + { + "bbox": [ + 115, + 307, + 506, + 320 + ], + "score": 1.0, + "content": "risk is either a global minimum or a saddle; it is an “existence” result without any computational", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 115, + 317, + 506, + 332 + ], + "spans": [ + { + "bbox": [ + 115, + 317, + 506, + 332 + ], + "score": 1.0, + "content": "implication. In contrast, we present efficiently checkable conditions for distinguishing the two", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 115, + 328, + 506, + 342 + ], + "spans": [ + { + "bbox": [ + 115, + 328, + 506, + 342 + ], + "score": 1.0, + "content": "different types of critical points; we can even use these conditions while running optimization to", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 115, + 340, + 506, + 352 + ], + "spans": [ + { + "bbox": [ + 115, + 340, + 506, + 352 + ], + "score": 1.0, + "content": "test whether the critical points we encounter are saddle points or not, if desired. It is also worth", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 115, + 351, + 505, + 364 + ], + "spans": [ + { + "bbox": [ + 115, + 351, + 505, + 364 + ], + "score": 1.0, + "content": "noting that such tests are intractable for general nonconvex optimization (Murty & Kabadi, 1987).", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 113, + 364, + 505, + 377 + ], + "spans": [ + { + "bbox": [ + 113, + 364, + 505, + 377 + ], + "score": 1.0, + "content": "Under the same assumption as (Hardt & Ma, 2017) on the data distribution, namely, a linear model", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 115, + 375, + 505, + 388 + ], + "spans": [ + { + "bbox": [ + 115, + 375, + 505, + 388 + ], + "score": 1.0, + "content": "with Gaussian noise, we can modify Theorem 2.1 to handle the population risk. As a corollary,", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 115, + 386, + 505, + 399 + ], + "spans": [ + { + "bbox": [ + 115, + 386, + 505, + 399 + ], + "score": 1.0, + "content": "we not only recover Theorem 2.2 in (Hardt & Ma, 2017), but also extend it to a strictly larger set,", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 116, + 398, + 505, + 410 + ], + "spans": [ + { + "bbox": [ + 116, + 398, + 505, + 410 + ], + "score": 1.0, + "content": "while removing their assumption that the true underlying linear model has a positive determinant.", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 108, + 411, + 505, + 423 + ], + "spans": [ + { + "bbox": [ + 108, + 411, + 505, + 423 + ], + "score": 1.0, + "content": "• Motivated by (Bartlett et al., 2017), we extend our results on deep linear networks to obtain suf-", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 115, + 421, + 505, + 434 + ], + "spans": [ + { + "bbox": [ + 115, + 421, + 505, + 434 + ], + "score": 1.0, + "content": "ficient conditions for global optimality in deep nonlinear networks, although only via a function", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 115, + 433, + 346, + 444 + ], + "spans": [ + { + "bbox": [ + 115, + 433, + 346, + 444 + ], + "score": 1.0, + "content": "space view; these are presented in Theorems 4.1 and 4.2.", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 20.5, + "bbox_fs": [ + 108, + 242, + 506, + 444 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 460, + 503, + 472 + ], + "lines": [ + { + "bbox": [ + 104, + 458, + 506, + 474 + ], + "spans": [ + { + "bbox": [ + 104, + 458, + 506, + 474 + ], + "score": 1.0, + "content": "2 GLOBAL OPTIMALITY CONDITIONS FOR DEEP LINEAR NEURAL NETWORKS", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 30 + }, + { + "type": "text", + "bbox": [ + 105, + 480, + 504, + 502 + ], + "lines": [ + { + "bbox": [ + 105, + 478, + 505, + 493 + ], + "spans": [ + { + "bbox": [ + 105, + 478, + 505, + 493 + ], + "score": 1.0, + "content": "In this section, we describe the problem formulation and notations for deep linear neural networks,", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 491, + 396, + 503 + ], + "spans": [ + { + "bbox": [ + 105, + 491, + 396, + 503 + ], + "score": 1.0, + "content": "state main results (Theorems 2.1 and 2.2), and explain their implication.", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 31.5, + "bbox_fs": [ + 105, + 478, + 505, + 503 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 511, + 306, + 522 + ], + "lines": [ + { + "bbox": [ + 106, + 511, + 307, + 524 + ], + "spans": [ + { + "bbox": [ + 106, + 511, + 307, + 524 + ], + "score": 1.0, + "content": "2.1 PROBLEM FORMULATION AND NOTATION", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 33 + }, + { + "type": "text", + "bbox": [ + 106, + 531, + 505, + 611 + ], + "lines": [ + { + "bbox": [ + 106, + 532, + 505, + 545 + ], + "spans": [ + { + "bbox": [ + 106, + 532, + 177, + 545 + ], + "score": 1.0, + "content": "Suppose we have", + "type": "text" + }, + { + "bbox": [ + 178, + 534, + 187, + 542 + ], + "score": 0.74, + "content": "m", + "type": "inline_equation" + }, + { + "bbox": [ + 188, + 532, + 402, + 545 + ], + "score": 1.0, + "content": "input-output pairs, where the inputs are of dimension", + "type": "text" + }, + { + "bbox": [ + 403, + 532, + 414, + 543 + ], + "score": 0.89, + "content": "d _ { x }", + "type": "inline_equation" + }, + { + "bbox": [ + 414, + 532, + 505, + 545 + ], + "score": 1.0, + "content": "and outputs of dimen-", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 104, + 541, + 506, + 557 + ], + "spans": [ + { + "bbox": [ + 104, + 541, + 126, + 557 + ], + "score": 1.0, + "content": "sion", + "type": "text" + }, + { + "bbox": [ + 126, + 543, + 137, + 555 + ], + "score": 0.88, + "content": "d _ { y }", + "type": "inline_equation" + }, + { + "bbox": [ + 137, + 541, + 160, + 557 + ], + "score": 1.0, + "content": ". Let", + "type": "text" + }, + { + "bbox": [ + 160, + 542, + 214, + 553 + ], + "score": 0.92, + "content": "\\boldsymbol { X } \\in \\mathbb { R } ^ { d _ { \\boldsymbol { x } } \\times m }", + "type": "inline_equation" + }, + { + "bbox": [ + 214, + 541, + 309, + 557 + ], + "score": 1.0, + "content": "be the data matrix and", + "type": "text" + }, + { + "bbox": [ + 310, + 542, + 362, + 553 + ], + "score": 0.92, + "content": "\\bar { Y } \\in \\mathbb { R } ^ { d _ { y } \\times m }", + "type": "inline_equation" + }, + { + "bbox": [ + 363, + 541, + 506, + 557 + ], + "score": 1.0, + "content": "be the output matrix. Suppose we", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 553, + 505, + 567 + ], + "spans": [ + { + "bbox": [ + 105, + 553, + 128, + 567 + ], + "score": 1.0, + "content": "have", + "type": "text" + }, + { + "bbox": [ + 128, + 555, + 138, + 564 + ], + "score": 0.82, + "content": "H", + "type": "inline_equation" + }, + { + "bbox": [ + 138, + 553, + 337, + 567 + ], + "score": 1.0, + "content": "hidden layers in the network, each having width", + "type": "text" + }, + { + "bbox": [ + 337, + 554, + 383, + 565 + ], + "score": 0.91, + "content": "d _ { 1 } , \\dots , d _ { H }", + "type": "inline_equation" + }, + { + "bbox": [ + 383, + 553, + 505, + 567 + ], + "score": 1.0, + "content": ". For notational simplicity we", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 565, + 505, + 578 + ], + "spans": [ + { + "bbox": [ + 105, + 565, + 120, + 578 + ], + "score": 1.0, + "content": "let", + "type": "text" + }, + { + "bbox": [ + 120, + 565, + 157, + 576 + ], + "score": 0.9, + "content": "d _ { 0 } = d _ { x }", + "type": "inline_equation" + }, + { + "bbox": [ + 157, + 565, + 177, + 578 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 177, + 565, + 227, + 577 + ], + "score": 0.91, + "content": "d _ { H + 1 } = d _ { y }", + "type": "inline_equation" + }, + { + "bbox": [ + 227, + 565, + 505, + 578 + ], + "score": 1.0, + "content": ". The weights between adjacent layers can be represented as matri-", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 104, + 575, + 506, + 590 + ], + "spans": [ + { + "bbox": [ + 104, + 575, + 122, + 590 + ], + "score": 1.0, + "content": "ces", + "type": "text" + }, + { + "bbox": [ + 123, + 577, + 192, + 588 + ], + "score": 0.92, + "content": "W _ { k } \\in \\mathbb { R } ^ { d _ { k } \\times d _ { k - 1 } }", + "type": "inline_equation" + }, + { + "bbox": [ + 192, + 575, + 212, + 590 + ], + "score": 1.0, + "content": ", for", + "type": "text" + }, + { + "bbox": [ + 212, + 577, + 289, + 588 + ], + "score": 0.87, + "content": "k = 1 , \\ldots , H + 1", + "type": "inline_equation" + }, + { + "bbox": [ + 289, + 575, + 506, + 590 + ], + "score": 1.0, + "content": ", and the output of the network can be written as the", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 587, + 506, + 601 + ], + "spans": [ + { + "bbox": [ + 105, + 587, + 218, + 601 + ], + "score": 1.0, + "content": "product of weight matrices", + "type": "text" + }, + { + "bbox": [ + 218, + 589, + 281, + 600 + ], + "score": 0.9, + "content": "W _ { H + 1 } , \\dots , W _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 282, + 587, + 349, + 601 + ], + "score": 1.0, + "content": "and data matrix", + "type": "text" + }, + { + "bbox": [ + 349, + 588, + 359, + 598 + ], + "score": 0.7, + "content": "X", + "type": "inline_equation" + }, + { + "bbox": [ + 360, + 587, + 364, + 601 + ], + "score": 1.0, + "content": ":", + "type": "text" + }, + { + "bbox": [ + 365, + 588, + 447, + 600 + ], + "score": 0.83, + "content": "W _ { H + 1 } W _ { H } \\cdot \\cdot \\cdot W _ { 1 } X", + "type": "inline_equation" + }, + { + "bbox": [ + 448, + 587, + 506, + 601 + ], + "score": 1.0, + "content": ". We consider", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 599, + 460, + 612 + ], + "spans": [ + { + "bbox": [ + 105, + 599, + 460, + 612 + ], + "score": 1.0, + "content": "minimizing the summation of squared error loss over all data points (i.e. empirical risk),", + "type": "text" + } + ], + "index": 40 + } + ], + "index": 37, + "bbox_fs": [ + 104, + 532, + 506, + 612 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 193, + 617, + 417, + 632 + ], + "lines": [ + { + "bbox": [ + 193, + 617, + 417, + 632 + ], + "spans": [ + { + "bbox": [ + 193, + 617, + 417, + 632 + ], + "score": 0.89, + "content": "\\begin{array} { r } { \\mathrm { m i n i m i z e } \\quad L ( W ) : = \\frac { 1 } { 2 } \\left\\| W _ { H + 1 } W _ { H } \\cdot \\cdot \\cdot W _ { 1 } X - Y \\right\\| _ { \\mathrm { F } } ^ { 2 } , } \\end{array}", + "type": "interline_equation", + "image_path": "1e11c803fa4297c7f4a4b82a1691e72fa78ef47bb1675f18a51b9166564e6194.jpg" + } + ] + } + ], + "index": 41, + "virtual_lines": [ + { + "bbox": [ + 193, + 617, + 417, + 632 + ], + "spans": [], + "index": 41 + } + ] + }, + { + "type": "text", + "bbox": [ + 108, + 638, + 366, + 650 + ], + "lines": [ + { + "bbox": [ + 105, + 635, + 366, + 653 + ], + "spans": [ + { + "bbox": [ + 105, + 635, + 133, + 653 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 133, + 639, + 145, + 648 + ], + "score": 0.75, + "content": "W", + "type": "inline_equation" + }, + { + "bbox": [ + 146, + 635, + 291, + 653 + ], + "score": 1.0, + "content": "is a shorthand notation for the tuple", + "type": "text" + }, + { + "bbox": [ + 291, + 638, + 362, + 650 + ], + "score": 0.89, + "content": "( W _ { 1 } , \\dots , W _ { H + 1 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 362, + 635, + 366, + 653 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 42 + } + ], + "index": 42, + "bbox_fs": [ + 105, + 635, + 366, + 653 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 656, + 504, + 702 + ], + "lines": [ + { + "bbox": [ + 105, + 656, + 506, + 670 + ], + "spans": [ + { + "bbox": [ + 105, + 656, + 240, + 669 + ], + "score": 1.0, + "content": "Assumptions. We assume that", + "type": "text" + }, + { + "bbox": [ + 240, + 658, + 275, + 669 + ], + "score": 0.92, + "content": "d _ { x } \\leq m", + "type": "inline_equation" + }, + { + "bbox": [ + 276, + 656, + 294, + 669 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 295, + 658, + 329, + 670 + ], + "score": 0.92, + "content": "d _ { y } \\leq m", + "type": "inline_equation" + }, + { + "bbox": [ + 330, + 656, + 369, + 669 + ], + "score": 1.0, + "content": ", and that", + "type": "text" + }, + { + "bbox": [ + 369, + 657, + 394, + 668 + ], + "score": 0.9, + "content": "X X ^ { T }", + "type": "inline_equation" + }, + { + "bbox": [ + 395, + 656, + 414, + 669 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 414, + 657, + 438, + 668 + ], + "score": 0.93, + "content": "Y X ^ { T }", + "type": "inline_equation" + }, + { + "bbox": [ + 438, + 656, + 506, + 669 + ], + "score": 1.0, + "content": "have full ranks.", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 106, + 669, + 505, + 681 + ], + "spans": [ + { + "bbox": [ + 106, + 669, + 505, + 681 + ], + "score": 1.0, + "content": "These assumptions are common when we consider supervised learning problems with deep neural", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 104, + 678, + 506, + 692 + ], + "spans": [ + { + "bbox": [ + 104, + 678, + 414, + 692 + ], + "score": 1.0, + "content": "networks (e.g. Kawaguchi (2016)). 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Given a matrix", + "type": "text" + }, + { + "bbox": [ + 221, + 710, + 229, + 720 + ], + "score": 0.78, + "content": "A", + "type": "inline_equation" + }, + { + "bbox": [ + 229, + 708, + 247, + 723 + ], + "score": 1.0, + "content": ", let", + "type": "text" + }, + { + "bbox": [ + 247, + 710, + 284, + 721 + ], + "score": 0.91, + "content": "\\sigma _ { \\mathrm { m a x } } ( A )", + "type": "inline_equation" + }, + { + "bbox": [ + 284, + 708, + 304, + 723 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 304, + 710, + 339, + 721 + ], + "score": 0.91, + "content": "\\sigma _ { \\mathrm { m i n } } ( A )", + "type": "inline_equation" + }, + { + "bbox": [ + 340, + 708, + 506, + 723 + ], + "score": 1.0, + "content": "denote the largest and smallest singular", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 106, + 720, + 505, + 733 + ], + "spans": [ + { + "bbox": [ + 106, + 720, + 241, + 733 + ], + "score": 1.0, + "content": "values of A, respectively. 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Given a set", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 337, + 94, + 345, + 104 + ], + "score": 0.76, + "content": "\\nu", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 345, + 93, + 361, + 106 + ], + "score": 1.0, + "content": ", let", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 361, + 94, + 374, + 104 + ], + "score": 0.86, + "content": "\\mathcal { V } ^ { c }", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 374, + 93, + 481, + 106 + ], + "score": 1.0, + "content": "denote the complement of", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 482, + 94, + 489, + 104 + ], + "score": 0.83, + "content": "\\nu", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 489, + 93, + 493, + 106 + ], + "score": 1.0, + "content": ".", + "type": "text", + "cross_page": true + } + ], + "index": 1 + } + ], + "index": 47.5, + "bbox_fs": [ + 105, + 708, + 506, + 733 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 105, + 82, + 505, + 105 + ], + "lines": [ + { + "bbox": [ + 105, + 82, + 505, + 95 + ], + "spans": [ + { + "bbox": [ + 105, + 82, + 382, + 95 + ], + "score": 1.0, + "content": "space, column space, null space, rank, and Frobenius norm of matrix", + "type": "text" + }, + { + "bbox": [ + 382, + 83, + 390, + 92 + ], + "score": 0.71, + "content": "A", + "type": "inline_equation" + }, + { + "bbox": [ + 391, + 82, + 466, + 95 + ], + "score": 1.0, + "content": ". Given a subspace", + "type": "text" + }, + { + "bbox": [ + 466, + 83, + 476, + 92 + ], + "score": 0.8, + "content": "V", + "type": "inline_equation" + }, + { + "bbox": [ + 476, + 82, + 487, + 95 + ], + "score": 1.0, + "content": "of", + "type": "text" + }, + { + "bbox": [ + 488, + 83, + 501, + 93 + ], + "score": 0.9, + "content": "\\mathbb { R } ^ { n }", + "type": "inline_equation" + }, + { + "bbox": [ + 501, + 82, + 505, + 95 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 93, + 493, + 106 + ], + "spans": [ + { + "bbox": [ + 106, + 93, + 149, + 106 + ], + "score": 1.0, + "content": "we denote", + "type": "text" + }, + { + "bbox": [ + 150, + 93, + 165, + 104 + ], + "score": 0.89, + "content": "V ^ { \\perp }", + "type": "inline_equation" + }, + { + "bbox": [ + 166, + 93, + 336, + 106 + ], + "score": 1.0, + "content": "as its orthogonal complement. Given a set", + "type": "text" + }, + { + "bbox": [ + 337, + 94, + 345, + 104 + ], + "score": 0.76, + "content": "\\nu", + "type": "inline_equation" + }, + { + "bbox": [ + 345, + 93, + 361, + 106 + ], + "score": 1.0, + "content": ", let", + "type": "text" + }, + { + "bbox": [ + 361, + 94, + 374, + 104 + ], + "score": 0.86, + "content": "\\mathcal { V } ^ { c }", + "type": "inline_equation" + }, + { + "bbox": [ + 374, + 93, + 481, + 106 + ], + "score": 1.0, + "content": "denote the complement of", + "type": "text" + }, + { + "bbox": [ + 482, + 94, + 489, + 104 + ], + "score": 0.83, + "content": "\\nu", + "type": "inline_equation" + }, + { + "bbox": [ + 489, + 93, + 493, + 106 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 1 + } + ], + "index": 0.5 + }, + { + "type": "text", + "bbox": [ + 107, + 110, + 505, + 169 + ], + "lines": [ + { + "bbox": [ + 104, + 109, + 505, + 125 + ], + "spans": [ + { + "bbox": [ + 104, + 109, + 165, + 125 + ], + "score": 1.0, + "content": "Let us denote", + "type": "text" + }, + { + "bbox": [ + 165, + 111, + 267, + 123 + ], + "score": 0.91, + "content": "k : = \\mathrm { m i n } _ { i \\in \\{ 0 , \\dots , H + 1 \\} } d _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 268, + 109, + 317, + 125 + ], + "score": 1.0, + "content": ", and define", + "type": "text" + }, + { + "bbox": [ + 317, + 111, + 429, + 124 + ], + "score": 0.92, + "content": "p \\in { \\mathrm { ~ a r g m i n } } _ { i \\in \\{ 0 , . . . , H + 1 \\} } d _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 429, + 109, + 470, + 125 + ], + "score": 1.0, + "content": ". That is,", + "type": "text" + }, + { + "bbox": [ + 470, + 113, + 477, + 122 + ], + "score": 0.79, + "content": "p", + "type": "inline_equation" + }, + { + "bbox": [ + 477, + 109, + 505, + 125 + ], + "score": 1.0, + "content": "is any", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 121, + 505, + 135 + ], + "spans": [ + { + "bbox": [ + 105, + 121, + 246, + 135 + ], + "score": 1.0, + "content": "layer with the smallest width, and", + "type": "text" + }, + { + "bbox": [ + 246, + 123, + 278, + 135 + ], + "score": 0.92, + "content": "k = d _ { p }", + "type": "inline_equation" + }, + { + "bbox": [ + 278, + 121, + 410, + 135 + ], + "score": 1.0, + "content": "is the width of that layer. Here,", + "type": "text" + }, + { + "bbox": [ + 411, + 125, + 417, + 134 + ], + "score": 0.8, + "content": "p", + "type": "inline_equation" + }, + { + "bbox": [ + 417, + 121, + 505, + 135 + ], + "score": 1.0, + "content": "might not be unique,", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 133, + 504, + 146 + ], + "spans": [ + { + "bbox": [ + 105, + 133, + 240, + 146 + ], + "score": 1.0, + "content": "but our results hold for any layer", + "type": "text" + }, + { + "bbox": [ + 240, + 136, + 247, + 144 + ], + "score": 0.8, + "content": "p", + "type": "inline_equation" + }, + { + "bbox": [ + 248, + 133, + 447, + 146 + ], + "score": 1.0, + "content": "with smallest width. Notice also that the product", + "type": "text" + }, + { + "bbox": [ + 447, + 134, + 504, + 145 + ], + "score": 0.91, + "content": "W _ { H + 1 } \\cdots W _ { 1 }", + "type": "inline_equation" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 144, + 506, + 156 + ], + "spans": [ + { + "bbox": [ + 105, + 144, + 197, + 156 + ], + "score": 1.0, + "content": "can have rank at most", + "type": "text" + }, + { + "bbox": [ + 197, + 145, + 204, + 154 + ], + "score": 0.79, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 204, + 144, + 225, + 156 + ], + "score": 1.0, + "content": ". Let", + "type": "text" + }, + { + "bbox": [ + 225, + 144, + 345, + 156 + ], + "score": 0.92, + "content": "\\mathbf { \\dot { \\boldsymbol { Y } } } \\bar { X ^ { T } } ( \\bar { X } \\bar { X ^ { T } } ) ^ { - 1 } \\boldsymbol { X } = \\boldsymbol { U } \\Sigma \\boldsymbol { V } ^ { T }", + "type": "inline_equation" + }, + { + "bbox": [ + 345, + 144, + 506, + 156 + ], + "score": 1.0, + "content": "be the singular value decomposition of", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 155, + 498, + 171 + ], + "spans": [ + { + "bbox": [ + 106, + 156, + 226, + 169 + ], + "score": 0.91, + "content": "Y X ^ { T } ( X X ^ { T } ) ^ { - 1 } X \\in \\mathbb { R } ^ { d _ { y } \\times d _ { x } }", + "type": "inline_equation" + }, + { + "bbox": [ + 226, + 155, + 246, + 171 + ], + "score": 1.0, + "content": ". Let", + "type": "text" + }, + { + "bbox": [ + 246, + 156, + 294, + 168 + ], + "score": 0.91, + "content": "\\hat { U } \\in \\mathbb { R } ^ { d _ { y } \\times k }", + "type": "inline_equation" + }, + { + "bbox": [ + 294, + 155, + 429, + 171 + ], + "score": 1.0, + "content": "be a matrix consisting of the first", + "type": "text" + }, + { + "bbox": [ + 429, + 158, + 436, + 167 + ], + "score": 0.79, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 436, + 155, + 484, + 171 + ], + "score": 1.0, + "content": "columns of", + "type": "text" + }, + { + "bbox": [ + 485, + 158, + 493, + 167 + ], + "score": 0.82, + "content": "U", + "type": "inline_equation" + }, + { + "bbox": [ + 493, + 155, + 498, + 171 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 4 + }, + { + "type": "title", + "bbox": [ + 107, + 176, + 426, + 187 + ], + "lines": [ + { + "bbox": [ + 105, + 176, + 427, + 189 + ], + "spans": [ + { + "bbox": [ + 105, + 176, + 427, + 189 + ], + "score": 1.0, + "content": "2.2 NECESSARY AND SUFFICIENT CONDITIONS FOR GLOBAL OPTIMALITY", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 7 + }, + { + "type": "text", + "bbox": [ + 106, + 191, + 506, + 280 + ], + "lines": [ + { + "bbox": [ + 106, + 191, + 506, + 204 + ], + "spans": [ + { + "bbox": [ + 106, + 191, + 506, + 204 + ], + "score": 1.0, + "content": "We now present two main theorems for deep linear neural networks. The theorems describe two", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 203, + 506, + 216 + ], + "spans": [ + { + "bbox": [ + 105, + 203, + 197, + 216 + ], + "score": 1.0, + "content": "sets, one for the case", + "type": "text" + }, + { + "bbox": [ + 197, + 203, + 271, + 215 + ], + "score": 0.92, + "content": "k = \\operatorname* { m i n } \\{ d _ { x } , d _ { y } \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 272, + 203, + 346, + 216 + ], + "score": 1.0, + "content": "and the other for", + "type": "text" + }, + { + "bbox": [ + 346, + 203, + 421, + 216 + ], + "score": 0.93, + "content": "k < \\operatorname* { m i n } \\{ d _ { x } , d _ { y } \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 421, + 203, + 506, + 216 + ], + "score": 1.0, + "content": ", inside which every", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 213, + 505, + 227 + ], + "spans": [ + { + "bbox": [ + 105, + 213, + 172, + 227 + ], + "score": 1.0, + "content": "critical point of", + "type": "text" + }, + { + "bbox": [ + 172, + 214, + 198, + 226 + ], + "score": 0.92, + "content": "L ( W )", + "type": "inline_equation" + }, + { + "bbox": [ + 198, + 213, + 505, + 227 + ], + "score": 1.0, + "content": "is a global minimum. Moreover, the sets have another remarkable property", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 225, + 506, + 237 + ], + "spans": [ + { + "bbox": [ + 106, + 225, + 506, + 237 + ], + "score": 1.0, + "content": "that every critical point outside of these sets is a saddle point. Previous works (Kawaguchi, 2016;", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 235, + 505, + 249 + ], + "spans": [ + { + "bbox": [ + 105, + 235, + 505, + 249 + ], + "score": 1.0, + "content": "Lu & Kawaguchi, 2017) showed that any critical point is either a global minimum or a saddle point,", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 247, + 506, + 259 + ], + "spans": [ + { + "bbox": [ + 106, + 247, + 506, + 259 + ], + "score": 1.0, + "content": "without providing any condition to distinguish between the two; here, we take a step further and", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 258, + 506, + 271 + ], + "spans": [ + { + "bbox": [ + 105, + 258, + 202, + 271 + ], + "score": 1.0, + "content": "partition the domain of", + "type": "text" + }, + { + "bbox": [ + 203, + 258, + 229, + 270 + ], + "score": 0.93, + "content": "L ( W )", + "type": "inline_equation" + }, + { + "bbox": [ + 229, + 258, + 506, + 271 + ], + "score": 1.0, + "content": "into two sets clearly delineating one set which only contains global", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 269, + 305, + 281 + ], + "spans": [ + { + "bbox": [ + 106, + 269, + 305, + 281 + ], + "score": 1.0, + "content": "minima and the other set with only saddle points.", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 11.5 + }, + { + "type": "text", + "bbox": [ + 107, + 281, + 347, + 294 + ], + "lines": [ + { + "bbox": [ + 106, + 280, + 346, + 295 + ], + "spans": [ + { + "bbox": [ + 106, + 280, + 176, + 295 + ], + "score": 1.0, + "content": "Theorem 2.1. If", + "type": "text" + }, + { + "bbox": [ + 176, + 281, + 246, + 294 + ], + "score": 0.89, + "content": "k = \\operatorname* { m i n } \\{ d _ { x } , d _ { y } \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 247, + 280, + 346, + 295 + ], + "score": 1.0, + "content": ", define the following set", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 16 + }, + { + "type": "interline_equation", + "bbox": [ + 193, + 294, + 416, + 307 + ], + "lines": [ + { + "bbox": [ + 193, + 294, + 416, + 307 + ], + "spans": [ + { + "bbox": [ + 193, + 294, + 416, + 307 + ], + "score": 0.81, + "content": "\\mathcal { V } _ { 1 } : = \\left\\{ ( W _ { 1 } , \\ldots , W _ { H + 1 } ) : \\operatorname { r a n k } ( W _ { H + 1 } \\cdot \\cdot \\cdot W _ { 1 } ) = k \\right\\} .", + "type": "interline_equation", + "image_path": "a189d278b1afc096aec77cd729e759795ff3cea05faf7d136d0f4a1380835b3f.jpg" + } + ] + } + ], + "index": 17, + "virtual_lines": [ + { + "bbox": [ + 193, + 294, + 416, + 307 + ], + "spans": [], + "index": 17 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 308, + 504, + 330 + ], + "lines": [ + { + "bbox": [ + 105, + 306, + 506, + 321 + ], + "spans": [ + { + "bbox": [ + 105, + 306, + 224, + 321 + ], + "score": 1.0, + "content": "Then, every critical point of", + "type": "text" + }, + { + "bbox": [ + 225, + 308, + 251, + 320 + ], + "score": 0.9, + "content": "L ( W )", + "type": "inline_equation" + }, + { + "bbox": [ + 251, + 306, + 263, + 321 + ], + "score": 1.0, + "content": "in", + "type": "text" + }, + { + "bbox": [ + 264, + 308, + 276, + 319 + ], + "score": 0.85, + "content": "\\mathcal { V } _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 276, + 306, + 506, + 321 + ], + "score": 1.0, + "content": "is a global minimum. Moreover, every critical point of", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 107, + 318, + 228, + 331 + ], + "spans": [ + { + "bbox": [ + 107, + 318, + 133, + 331 + ], + "score": 0.91, + "content": "L ( W )", + "type": "inline_equation" + }, + { + "bbox": [ + 133, + 318, + 144, + 331 + ], + "score": 1.0, + "content": "in", + "type": "text" + }, + { + "bbox": [ + 145, + 319, + 157, + 331 + ], + "score": 0.88, + "content": "\\mathcal { V } _ { 1 } ^ { c }", + "type": "inline_equation" + }, + { + "bbox": [ + 157, + 318, + 228, + 331 + ], + "score": 1.0, + "content": "is a saddle point.", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 18.5 + }, + { + "type": "text", + "bbox": [ + 106, + 331, + 351, + 344 + ], + "lines": [ + { + "bbox": [ + 106, + 330, + 347, + 345 + ], + "spans": [ + { + "bbox": [ + 106, + 330, + 176, + 345 + ], + "score": 1.0, + "content": "Theorem 2.2. If", + "type": "text" + }, + { + "bbox": [ + 177, + 331, + 247, + 344 + ], + "score": 0.91, + "content": "k < \\operatorname* { m i n } \\{ d _ { x } , d _ { y } \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 247, + 330, + 347, + 345 + ], + "score": 1.0, + "content": ", define the following set", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 20 + }, + { + "type": "interline_equation", + "bbox": [ + 125, + 344, + 484, + 367 + ], + "lines": [ + { + "bbox": [ + 125, + 344, + 484, + 367 + ], + "spans": [ + { + "bbox": [ + 125, + 344, + 484, + 367 + ], + "score": 0.87, + "content": "\\mathcal { V } _ { 2 } : = \\Big \\{ ( W _ { 1 } , \\ldots , W _ { H + 1 } ) : \\mathrm { r a n k } \\big ( W _ { H + 1 } \\cdot \\cdot \\cdot W _ { 1 } \\big ) = k , \\mathrm { c o l } \\big ( W _ { H + 1 } \\cdot \\cdot \\cdot W _ { p + 1 } \\big ) = \\mathrm { c o l } ( \\hat { U } ) \\Big \\} .", + "type": "interline_equation", + "image_path": "1922a19506601003a8b5233eed3ed80b08b57f1c19e06ecf984efdad1f41d622.jpg" + } + ] + } + ], + "index": 21, + "virtual_lines": [ + { + "bbox": [ + 125, + 344, + 484, + 367 + ], + "spans": [], + "index": 21 + } + ] + }, + { + "type": "text", + "bbox": [ + 109, + 366, + 504, + 388 + ], + "lines": [ + { + "bbox": [ + 106, + 365, + 506, + 379 + ], + "spans": [ + { + "bbox": [ + 106, + 365, + 224, + 379 + ], + "score": 1.0, + "content": "Then, every critical point of", + "type": "text" + }, + { + "bbox": [ + 225, + 366, + 251, + 378 + ], + "score": 0.89, + "content": "L ( W )", + "type": "inline_equation" + }, + { + "bbox": [ + 251, + 365, + 264, + 379 + ], + "score": 1.0, + "content": "in", + "type": "text" + }, + { + "bbox": [ + 264, + 366, + 276, + 377 + ], + "score": 0.88, + "content": "\\nu _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 276, + 365, + 506, + 379 + ], + "score": 1.0, + "content": "is a global minimum. Moreover, every critical point of", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 107, + 377, + 228, + 389 + ], + "spans": [ + { + "bbox": [ + 107, + 377, + 133, + 389 + ], + "score": 0.92, + "content": "L ( W )", + "type": "inline_equation" + }, + { + "bbox": [ + 133, + 377, + 145, + 389 + ], + "score": 1.0, + "content": "in", + "type": "text" + }, + { + "bbox": [ + 145, + 378, + 157, + 389 + ], + "score": 0.89, + "content": "\\mathcal { V } _ { 2 } ^ { c }", + "type": "inline_equation" + }, + { + "bbox": [ + 157, + 377, + 228, + 389 + ], + "score": 1.0, + "content": "is a saddle point.", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 22.5 + }, + { + "type": "text", + "bbox": [ + 107, + 390, + 505, + 447 + ], + "lines": [ + { + "bbox": [ + 106, + 390, + 505, + 403 + ], + "spans": [ + { + "bbox": [ + 106, + 390, + 455, + 403 + ], + "score": 1.0, + "content": "Theorems 2.1 and 2.2 provide necessary and sufficient conditions for a critical point of", + "type": "text" + }, + { + "bbox": [ + 455, + 391, + 481, + 403 + ], + "score": 0.93, + "content": "L ( W )", + "type": "inline_equation" + }, + { + "bbox": [ + 482, + 390, + 505, + 403 + ], + "score": 1.0, + "content": "to be", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 401, + 505, + 414 + ], + "spans": [ + { + "bbox": [ + 105, + 401, + 505, + 414 + ], + "score": 1.0, + "content": "globally optimal. From an algorithmic perspective, they provide easily checkable conditions, which", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 413, + 505, + 425 + ], + "spans": [ + { + "bbox": [ + 106, + 413, + 505, + 425 + ], + "score": 1.0, + "content": "we can use to determine if the critical point the algorithm encountered is a global optimum or not.", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 421, + 506, + 438 + ], + "spans": [ + { + "bbox": [ + 105, + 421, + 152, + 438 + ], + "score": 1.0, + "content": "Given that", + "type": "text" + }, + { + "bbox": [ + 153, + 424, + 178, + 436 + ], + "score": 0.92, + "content": "L ( W )", + "type": "inline_equation" + }, + { + "bbox": [ + 179, + 421, + 506, + 438 + ], + "score": 1.0, + "content": "is nonconvex, it is interesting to have such efficient tests for global optimality,", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 435, + 338, + 447 + ], + "spans": [ + { + "bbox": [ + 105, + 435, + 338, + 447 + ], + "score": 1.0, + "content": "which is not possible in general (Murty & Kabadi, 1987).", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 26 + }, + { + "type": "text", + "bbox": [ + 108, + 451, + 501, + 464 + ], + "lines": [ + { + "bbox": [ + 106, + 451, + 502, + 464 + ], + "spans": [ + { + "bbox": [ + 106, + 451, + 502, + 464 + ], + "score": 1.0, + "content": "In Hardt & Ma (2017), the authors consider minimizing population risk of linear residual networks:", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 29 + }, + { + "type": "interline_equation", + "bbox": [ + 186, + 464, + 423, + 484 + ], + "lines": [ + { + "bbox": [ + 186, + 464, + 423, + 484 + ], + "spans": [ + { + "bbox": [ + 186, + 464, + 423, + 484 + ], + "score": 0.85, + "content": "\\begin{array} { r l } { \\mathrm { m i n i m i z e } } & { { } \\frac { 1 } { 2 } \\mathbb { E } _ { \\boldsymbol { x } , \\boldsymbol { y } } \\left[ \\| ( I + W _ { H + 1 } ) \\cdot \\cdot \\cdot ( I + W _ { 1 } ) \\boldsymbol { x } - \\boldsymbol { y } \\| _ { \\mathrm { F } } ^ { 2 } \\right] , } \\end{array}", + "type": "interline_equation", + "image_path": "b81cf7458ff00c67c3830645ed75f18fdc89a5d93c908d1a6dd94f1019bdb064.jpg" + } + ] + } + ], + "index": 30, + "virtual_lines": [ + { + "bbox": [ + 186, + 464, + 423, + 484 + ], + "spans": [], + "index": 30 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 485, + 506, + 529 + ], + "lines": [ + { + "bbox": [ + 106, + 485, + 505, + 497 + ], + "spans": [ + { + "bbox": [ + 106, + 485, + 133, + 497 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 133, + 485, + 262, + 497 + ], + "score": 0.89, + "content": "d _ { x } = d _ { 1 } = \\cdot \\cdot \\cdot = d _ { H } = d _ { y } = d", + "type": "inline_equation" + }, + { + "bbox": [ + 262, + 485, + 338, + 497 + ], + "score": 1.0, + "content": ". They assume that", + "type": "text" + }, + { + "bbox": [ + 339, + 488, + 345, + 495 + ], + "score": 0.78, + "content": "x", + "type": "inline_equation" + }, + { + "bbox": [ + 346, + 485, + 505, + 497 + ], + "score": 1.0, + "content": "is drawn from a zero-mean distribution", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 496, + 506, + 508 + ], + "spans": [ + { + "bbox": [ + 106, + 496, + 253, + 508 + ], + "score": 1.0, + "content": "with a fixed covariance matrix, and", + "type": "text" + }, + { + "bbox": [ + 253, + 496, + 307, + 507 + ], + "score": 0.93, + "content": "y = R x + \\xi", + "type": "inline_equation" + }, + { + "bbox": [ + 307, + 496, + 335, + 508 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 335, + 496, + 342, + 507 + ], + "score": 0.84, + "content": "\\xi", + "type": "inline_equation" + }, + { + "bbox": [ + 343, + 496, + 485, + 508 + ], + "score": 1.0, + "content": "is iid standard Gaussian noise and", + "type": "text" + }, + { + "bbox": [ + 485, + 496, + 494, + 506 + ], + "score": 0.8, + "content": "R", + "type": "inline_equation" + }, + { + "bbox": [ + 495, + 496, + 506, + 508 + ], + "score": 1.0, + "content": "is", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 507, + 506, + 519 + ], + "spans": [ + { + "bbox": [ + 106, + 507, + 239, + 519 + ], + "score": 1.0, + "content": "the true underlying matrix with", + "type": "text" + }, + { + "bbox": [ + 239, + 507, + 291, + 519 + ], + "score": 0.91, + "content": "\\operatorname* { d e t } ( R ) > 0", + "type": "inline_equation" + }, + { + "bbox": [ + 292, + 507, + 506, + 519 + ], + "score": 1.0, + "content": ". With these assumptions they prove that whenever", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 518, + 500, + 530 + ], + "spans": [ + { + "bbox": [ + 106, + 518, + 167, + 530 + ], + "score": 0.92, + "content": "\\sigma _ { \\operatorname* { m a x } } ( W _ { i } ) < 1", + "type": "inline_equation" + }, + { + "bbox": [ + 168, + 518, + 195, + 530 + ], + "score": 1.0, + "content": "for all", + "type": "text" + }, + { + "bbox": [ + 195, + 519, + 200, + 528 + ], + "score": 0.69, + "content": "i", + "type": "inline_equation" + }, + { + "bbox": [ + 200, + 518, + 500, + 530 + ], + "score": 1.0, + "content": ", any critical point is a global minimum (Hardt & Ma, 2017, Theorem 2.2).", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 32.5 + }, + { + "type": "text", + "bbox": [ + 106, + 534, + 505, + 590 + ], + "lines": [ + { + "bbox": [ + 106, + 535, + 505, + 547 + ], + "spans": [ + { + "bbox": [ + 106, + 535, + 505, + 547 + ], + "score": 1.0, + "content": "Under the same assumptions on data distribution, we can slightly modify Theorem 2.1 to derive", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 546, + 506, + 558 + ], + "spans": [ + { + "bbox": [ + 105, + 546, + 506, + 558 + ], + "score": 1.0, + "content": "a population risk counterpart, and in fact notice that the result proved in Hardt & Ma (2017) is a", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 556, + 506, + 570 + ], + "spans": [ + { + "bbox": [ + 105, + 556, + 293, + 570 + ], + "score": 1.0, + "content": "corollary of this modification because having", + "type": "text" + }, + { + "bbox": [ + 293, + 556, + 356, + 569 + ], + "score": 0.93, + "content": "\\sigma _ { \\operatorname* { m a x } } ( W _ { i } ) < 1", + "type": "inline_equation" + }, + { + "bbox": [ + 357, + 556, + 386, + 570 + ], + "score": 1.0, + "content": "for all", + "type": "text" + }, + { + "bbox": [ + 386, + 558, + 391, + 567 + ], + "score": 0.73, + "content": "i", + "type": "inline_equation" + }, + { + "bbox": [ + 391, + 556, + 506, + 570 + ], + "score": 1.0, + "content": "is a sufficient condition for", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 107, + 567, + 505, + 581 + ], + "spans": [ + { + "bbox": [ + 107, + 568, + 215, + 579 + ], + "score": 0.9, + "content": "( I + W _ { H + 1 } ) \\cdot \\cdot \\cdot ( I + W _ { 1 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 216, + 567, + 505, + 581 + ], + "score": 1.0, + "content": "having full rank. Moreover, notice that we can remove the assumption", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 107, + 577, + 495, + 592 + ], + "spans": [ + { + "bbox": [ + 107, + 579, + 155, + 591 + ], + "score": 0.89, + "content": "\\operatorname* { d e t } ( R ) > 0", + "type": "inline_equation" + }, + { + "bbox": [ + 156, + 577, + 495, + 592 + ], + "score": 1.0, + "content": "which was required by Hardt & Ma (2017). We state this special case as a corollary:", + "type": "text" + } + ], + "index": 39 + } + ], + "index": 37 + }, + { + "type": "text", + "bbox": [ + 106, + 591, + 505, + 633 + ], + "lines": [ + { + "bbox": [ + 106, + 590, + 505, + 604 + ], + "spans": [ + { + "bbox": [ + 106, + 590, + 505, + 604 + ], + "score": 1.0, + "content": "Corollary 2.3 (Theorem 2.2 of Hardt & Ma (2017)). Under assumptions on data distribution as", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 104, + 601, + 504, + 623 + ], + "spans": [ + { + "bbox": [ + 104, + 601, + 257, + 622 + ], + "score": 1.0, + "content": "described above, any critical point of", + "type": "text" + }, + { + "bbox": [ + 258, + 603, + 439, + 623 + ], + "score": 0.91, + "content": "\\begin{array} { r } { \\frac { 1 } { 2 } \\mathbb { E } _ { x , y } \\left[ \\left\\| ( I + W _ { H + 1 } ) \\cdot \\cdot \\cdot ( I + W _ { 1 } ) x - y \\right\\| _ { \\mathrm { F } } ^ { 2 } \\right] } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 439, + 601, + 504, + 622 + ], + "score": 1.0, + "content": "is a global min-", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 106, + 620, + 237, + 634 + ], + "spans": [ + { + "bbox": [ + 106, + 620, + 139, + 634 + ], + "score": 1.0, + "content": "imum if", + "type": "text" + }, + { + "bbox": [ + 140, + 621, + 200, + 633 + ], + "score": 0.9, + "content": "\\sigma _ { \\operatorname* { m a x } } ( W _ { i } ) < 1", + "type": "inline_equation" + }, + { + "bbox": [ + 200, + 620, + 228, + 634 + ], + "score": 1.0, + "content": "for all", + "type": "text" + }, + { + "bbox": [ + 228, + 622, + 233, + 631 + ], + "score": 0.32, + "content": "i", + "type": "inline_equation" + }, + { + "bbox": [ + 233, + 620, + 237, + 634 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 42 + } + ], + "index": 41 + }, + { + "type": "text", + "bbox": [ + 106, + 636, + 504, + 661 + ], + "lines": [ + { + "bbox": [ + 104, + 632, + 508, + 655 + ], + "spans": [ + { + "bbox": [ + 104, + 632, + 401, + 655 + ], + "score": 1.0, + "content": "We also note in passing that the classical problem of matrix factorization", + "type": "text" + }, + { + "bbox": [ + 402, + 635, + 494, + 651 + ], + "score": 0.91, + "content": "\\begin{array} { r } { \\operatorname* { m i n } _ { U , V } \\left\\| U V ^ { T } - Y \\right\\| _ { \\mathrm { F } } ^ { 2 } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 495, + 632, + 508, + 655 + ], + "score": 1.0, + "content": "is", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 648, + 470, + 662 + ], + "spans": [ + { + "bbox": [ + 105, + 648, + 470, + 662 + ], + "score": 1.0, + "content": "a special case of deep linear neural networks, so our theorems can also be directly applied.", + "type": "text" + } + ], + "index": 44 + } + ], + "index": 43.5 + }, + { + "type": "text", + "bbox": [ + 106, + 665, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 664, + 505, + 679 + ], + "spans": [ + { + "bbox": [ + 105, + 664, + 357, + 679 + ], + "score": 1.0, + "content": "Remarks. The previous result (Kawaguchi, 2016) assumed", + "type": "text" + }, + { + "bbox": [ + 357, + 666, + 394, + 678 + ], + "score": 0.92, + "content": "d _ { y } \\leq d _ { x }", + "type": "inline_equation" + }, + { + "bbox": [ + 394, + 664, + 505, + 679 + ], + "score": 1.0, + "content": "and showed that: 1) every", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 106, + 677, + 505, + 689 + ], + "spans": [ + { + "bbox": [ + 106, + 677, + 505, + 689 + ], + "score": 1.0, + "content": "local minimum is a global minimum, and 2) any other critical point is a saddle point. A subsequent", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 104, + 687, + 506, + 700 + ], + "spans": [ + { + "bbox": [ + 104, + 687, + 392, + 700 + ], + "score": 1.0, + "content": "paper by Lu & Kawaguchi (2017) proved 1) without the assumption", + "type": "text" + }, + { + "bbox": [ + 393, + 688, + 431, + 700 + ], + "score": 0.92, + "content": "d _ { y } \\ \\leq \\ d _ { x }", + "type": "inline_equation" + }, + { + "bbox": [ + 431, + 687, + 506, + 700 + ], + "score": 1.0, + "content": ", but as far as we", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 105, + 699, + 505, + 712 + ], + "spans": [ + { + "bbox": [ + 105, + 699, + 314, + 712 + ], + "score": 1.0, + "content": "know there is no result showing 2) in the case of", + "type": "text" + }, + { + "bbox": [ + 314, + 699, + 353, + 711 + ], + "score": 0.91, + "content": "d _ { y } \\ > \\ d _ { x }", + "type": "inline_equation" + }, + { + "bbox": [ + 353, + 699, + 505, + 712 + ], + "score": 1.0, + "content": ". We provide the proof for this case", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 105, + 708, + 505, + 723 + ], + "spans": [ + { + "bbox": [ + 105, + 708, + 505, + 723 + ], + "score": 1.0, + "content": "in Lemma B.1. In fact, we propose an alternative proof technique for handling degenerate critical", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 105, + 720, + 437, + 733 + ], + "spans": [ + { + "bbox": [ + 105, + 720, + 437, + 733 + ], + "score": 1.0, + "content": "points, which is much simpler than the technique presented by Kawaguchi (2016).", + "type": "text" + } + ], + "index": 50 + } + ], + "index": 47.5 + } + ], + "page_idx": 2, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 108, + 27, + 293, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 26, + 294, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 26, + 294, + 38 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2018", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 302, + 751, + 309, + 760 + ], + "lines": [ + { + "bbox": [ + 301, + 750, + 310, + 762 + ], + "spans": [ + { + "bbox": [ + 301, + 750, + 310, + 762 + ], + "score": 1.0, + "content": "3", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 105, + 82, + 505, + 105 + ], + "lines": [], + "index": 0.5, + "bbox_fs": [ + 105, + 82, + 505, + 106 + ], + "lines_deleted": true + }, + { + "type": "text", + "bbox": [ + 107, + 110, + 505, + 169 + ], + "lines": [ + { + "bbox": [ + 104, + 109, + 505, + 125 + ], + "spans": [ + { + "bbox": [ + 104, + 109, + 165, + 125 + ], + "score": 1.0, + "content": "Let us denote", + "type": "text" + }, + { + "bbox": [ + 165, + 111, + 267, + 123 + ], + "score": 0.91, + "content": "k : = \\mathrm { m i n } _ { i \\in \\{ 0 , \\dots , H + 1 \\} } d _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 268, + 109, + 317, + 125 + ], + "score": 1.0, + "content": ", and define", + "type": "text" + }, + { + "bbox": [ + 317, + 111, + 429, + 124 + ], + "score": 0.92, + "content": "p \\in { \\mathrm { ~ a r g m i n } } _ { i \\in \\{ 0 , . . . , H + 1 \\} } d _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 429, + 109, + 470, + 125 + ], + "score": 1.0, + "content": ". That is,", + "type": "text" + }, + { + "bbox": [ + 470, + 113, + 477, + 122 + ], + "score": 0.79, + "content": "p", + "type": "inline_equation" + }, + { + "bbox": [ + 477, + 109, + 505, + 125 + ], + "score": 1.0, + "content": "is any", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 121, + 505, + 135 + ], + "spans": [ + { + "bbox": [ + 105, + 121, + 246, + 135 + ], + "score": 1.0, + "content": "layer with the smallest width, and", + "type": "text" + }, + { + "bbox": [ + 246, + 123, + 278, + 135 + ], + "score": 0.92, + "content": "k = d _ { p }", + "type": "inline_equation" + }, + { + "bbox": [ + 278, + 121, + 410, + 135 + ], + "score": 1.0, + "content": "is the width of that layer. Here,", + "type": "text" + }, + { + "bbox": [ + 411, + 125, + 417, + 134 + ], + "score": 0.8, + "content": "p", + "type": "inline_equation" + }, + { + "bbox": [ + 417, + 121, + 505, + 135 + ], + "score": 1.0, + "content": "might not be unique,", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 133, + 504, + 146 + ], + "spans": [ + { + "bbox": [ + 105, + 133, + 240, + 146 + ], + "score": 1.0, + "content": "but our results hold for any layer", + "type": "text" + }, + { + "bbox": [ + 240, + 136, + 247, + 144 + ], + "score": 0.8, + "content": "p", + "type": "inline_equation" + }, + { + "bbox": [ + 248, + 133, + 447, + 146 + ], + "score": 1.0, + "content": "with smallest width. Notice also that the product", + "type": "text" + }, + { + "bbox": [ + 447, + 134, + 504, + 145 + ], + "score": 0.91, + "content": "W _ { H + 1 } \\cdots W _ { 1 }", + "type": "inline_equation" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 144, + 506, + 156 + ], + "spans": [ + { + "bbox": [ + 105, + 144, + 197, + 156 + ], + "score": 1.0, + "content": "can have rank at most", + "type": "text" + }, + { + "bbox": [ + 197, + 145, + 204, + 154 + ], + "score": 0.79, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 204, + 144, + 225, + 156 + ], + "score": 1.0, + "content": ". Let", + "type": "text" + }, + { + "bbox": [ + 225, + 144, + 345, + 156 + ], + "score": 0.92, + "content": "\\mathbf { \\dot { \\boldsymbol { Y } } } \\bar { X ^ { T } } ( \\bar { X } \\bar { X ^ { T } } ) ^ { - 1 } \\boldsymbol { X } = \\boldsymbol { U } \\Sigma \\boldsymbol { V } ^ { T }", + "type": "inline_equation" + }, + { + "bbox": [ + 345, + 144, + 506, + 156 + ], + "score": 1.0, + "content": "be the singular value decomposition of", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 155, + 498, + 171 + ], + "spans": [ + { + "bbox": [ + 106, + 156, + 226, + 169 + ], + "score": 0.91, + "content": "Y X ^ { T } ( X X ^ { T } ) ^ { - 1 } X \\in \\mathbb { R } ^ { d _ { y } \\times d _ { x } }", + "type": "inline_equation" + }, + { + "bbox": [ + 226, + 155, + 246, + 171 + ], + "score": 1.0, + "content": ". Let", + "type": "text" + }, + { + "bbox": [ + 246, + 156, + 294, + 168 + ], + "score": 0.91, + "content": "\\hat { U } \\in \\mathbb { R } ^ { d _ { y } \\times k }", + "type": "inline_equation" + }, + { + "bbox": [ + 294, + 155, + 429, + 171 + ], + "score": 1.0, + "content": "be a matrix consisting of the first", + "type": "text" + }, + { + "bbox": [ + 429, + 158, + 436, + 167 + ], + "score": 0.79, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 436, + 155, + 484, + 171 + ], + "score": 1.0, + "content": "columns of", + "type": "text" + }, + { + "bbox": [ + 485, + 158, + 493, + 167 + ], + "score": 0.82, + "content": "U", + "type": "inline_equation" + }, + { + "bbox": [ + 493, + 155, + 498, + 171 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 4, + "bbox_fs": [ + 104, + 109, + 506, + 171 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 176, + 426, + 187 + ], + "lines": [ + { + "bbox": [ + 105, + 176, + 427, + 189 + ], + "spans": [ + { + "bbox": [ + 105, + 176, + 427, + 189 + ], + "score": 1.0, + "content": "2.2 NECESSARY AND SUFFICIENT CONDITIONS FOR GLOBAL OPTIMALITY", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 7 + }, + { + "type": "text", + "bbox": [ + 106, + 191, + 506, + 280 + ], + "lines": [ + { + "bbox": [ + 106, + 191, + 506, + 204 + ], + "spans": [ + { + "bbox": [ + 106, + 191, + 506, + 204 + ], + "score": 1.0, + "content": "We now present two main theorems for deep linear neural networks. The theorems describe two", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 203, + 506, + 216 + ], + "spans": [ + { + "bbox": [ + 105, + 203, + 197, + 216 + ], + "score": 1.0, + "content": "sets, one for the case", + "type": "text" + }, + { + "bbox": [ + 197, + 203, + 271, + 215 + ], + "score": 0.92, + "content": "k = \\operatorname* { m i n } \\{ d _ { x } , d _ { y } \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 272, + 203, + 346, + 216 + ], + "score": 1.0, + "content": "and the other for", + "type": "text" + }, + { + "bbox": [ + 346, + 203, + 421, + 216 + ], + "score": 0.93, + "content": "k < \\operatorname* { m i n } \\{ d _ { x } , d _ { y } \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 421, + 203, + 506, + 216 + ], + "score": 1.0, + "content": ", inside which every", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 213, + 505, + 227 + ], + "spans": [ + { + "bbox": [ + 105, + 213, + 172, + 227 + ], + "score": 1.0, + "content": "critical point of", + "type": "text" + }, + { + "bbox": [ + 172, + 214, + 198, + 226 + ], + "score": 0.92, + "content": "L ( W )", + "type": "inline_equation" + }, + { + "bbox": [ + 198, + 213, + 505, + 227 + ], + "score": 1.0, + "content": "is a global minimum. Moreover, the sets have another remarkable property", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 225, + 506, + 237 + ], + "spans": [ + { + "bbox": [ + 106, + 225, + 506, + 237 + ], + "score": 1.0, + "content": "that every critical point outside of these sets is a saddle point. Previous works (Kawaguchi, 2016;", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 235, + 505, + 249 + ], + "spans": [ + { + "bbox": [ + 105, + 235, + 505, + 249 + ], + "score": 1.0, + "content": "Lu & Kawaguchi, 2017) showed that any critical point is either a global minimum or a saddle point,", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 247, + 506, + 259 + ], + "spans": [ + { + "bbox": [ + 106, + 247, + 506, + 259 + ], + "score": 1.0, + "content": "without providing any condition to distinguish between the two; here, we take a step further and", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 258, + 506, + 271 + ], + "spans": [ + { + "bbox": [ + 105, + 258, + 202, + 271 + ], + "score": 1.0, + "content": "partition the domain of", + "type": "text" + }, + { + "bbox": [ + 203, + 258, + 229, + 270 + ], + "score": 0.93, + "content": "L ( W )", + "type": "inline_equation" + }, + { + "bbox": [ + 229, + 258, + 506, + 271 + ], + "score": 1.0, + "content": "into two sets clearly delineating one set which only contains global", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 269, + 305, + 281 + ], + "spans": [ + { + "bbox": [ + 106, + 269, + 305, + 281 + ], + "score": 1.0, + "content": "minima and the other set with only saddle points.", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 11.5, + "bbox_fs": [ + 105, + 191, + 506, + 281 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 281, + 347, + 294 + ], + "lines": [ + { + "bbox": [ + 106, + 280, + 346, + 295 + ], + "spans": [ + { + "bbox": [ + 106, + 280, + 176, + 295 + ], + "score": 1.0, + "content": "Theorem 2.1. If", + "type": "text" + }, + { + "bbox": [ + 176, + 281, + 246, + 294 + ], + "score": 0.89, + "content": "k = \\operatorname* { m i n } \\{ d _ { x } , d _ { y } \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 247, + 280, + 346, + 295 + ], + "score": 1.0, + "content": ", define the following set", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 16, + "bbox_fs": [ + 106, + 280, + 346, + 295 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 193, + 294, + 416, + 307 + ], + "lines": [ + { + "bbox": [ + 193, + 294, + 416, + 307 + ], + "spans": [ + { + "bbox": [ + 193, + 294, + 416, + 307 + ], + "score": 0.81, + "content": "\\mathcal { V } _ { 1 } : = \\left\\{ ( W _ { 1 } , \\ldots , W _ { H + 1 } ) : \\operatorname { r a n k } ( W _ { H + 1 } \\cdot \\cdot \\cdot W _ { 1 } ) = k \\right\\} .", + "type": "interline_equation", + "image_path": "a189d278b1afc096aec77cd729e759795ff3cea05faf7d136d0f4a1380835b3f.jpg" + } + ] + } + ], + "index": 17, + "virtual_lines": [ + { + "bbox": [ + 193, + 294, + 416, + 307 + ], + "spans": [], + "index": 17 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 308, + 504, + 330 + ], + "lines": [ + { + "bbox": [ + 105, + 306, + 506, + 321 + ], + "spans": [ + { + "bbox": [ + 105, + 306, + 224, + 321 + ], + "score": 1.0, + "content": "Then, every critical point of", + "type": "text" + }, + { + "bbox": [ + 225, + 308, + 251, + 320 + ], + "score": 0.9, + "content": "L ( W )", + "type": "inline_equation" + }, + { + "bbox": [ + 251, + 306, + 263, + 321 + ], + "score": 1.0, + "content": "in", + "type": "text" + }, + { + "bbox": [ + 264, + 308, + 276, + 319 + ], + "score": 0.85, + "content": "\\mathcal { V } _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 276, + 306, + 506, + 321 + ], + "score": 1.0, + "content": "is a global minimum. Moreover, every critical point of", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 107, + 318, + 228, + 331 + ], + "spans": [ + { + "bbox": [ + 107, + 318, + 133, + 331 + ], + "score": 0.91, + "content": "L ( W )", + "type": "inline_equation" + }, + { + "bbox": [ + 133, + 318, + 144, + 331 + ], + "score": 1.0, + "content": "in", + "type": "text" + }, + { + "bbox": [ + 145, + 319, + 157, + 331 + ], + "score": 0.88, + "content": "\\mathcal { V } _ { 1 } ^ { c }", + "type": "inline_equation" + }, + { + "bbox": [ + 157, + 318, + 228, + 331 + ], + "score": 1.0, + "content": "is a saddle point.", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 18.5, + "bbox_fs": [ + 105, + 306, + 506, + 331 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 331, + 351, + 344 + ], + "lines": [ + { + "bbox": [ + 106, + 330, + 347, + 345 + ], + "spans": [ + { + "bbox": [ + 106, + 330, + 176, + 345 + ], + "score": 1.0, + "content": "Theorem 2.2. If", + "type": "text" + }, + { + "bbox": [ + 177, + 331, + 247, + 344 + ], + "score": 0.91, + "content": "k < \\operatorname* { m i n } \\{ d _ { x } , d _ { y } \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 247, + 330, + 347, + 345 + ], + "score": 1.0, + "content": ", define the following set", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 20, + "bbox_fs": [ + 106, + 330, + 347, + 345 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 125, + 344, + 484, + 367 + ], + "lines": [ + { + "bbox": [ + 125, + 344, + 484, + 367 + ], + "spans": [ + { + "bbox": [ + 125, + 344, + 484, + 367 + ], + "score": 0.87, + "content": "\\mathcal { V } _ { 2 } : = \\Big \\{ ( W _ { 1 } , \\ldots , W _ { H + 1 } ) : \\mathrm { r a n k } \\big ( W _ { H + 1 } \\cdot \\cdot \\cdot W _ { 1 } \\big ) = k , \\mathrm { c o l } \\big ( W _ { H + 1 } \\cdot \\cdot \\cdot W _ { p + 1 } \\big ) = \\mathrm { c o l } ( \\hat { U } ) \\Big \\} .", + "type": "interline_equation", + "image_path": "1922a19506601003a8b5233eed3ed80b08b57f1c19e06ecf984efdad1f41d622.jpg" + } + ] + } + ], + "index": 21, + "virtual_lines": [ + { + "bbox": [ + 125, + 344, + 484, + 367 + ], + "spans": [], + "index": 21 + } + ] + }, + { + "type": "text", + "bbox": [ + 109, + 366, + 504, + 388 + ], + "lines": [ + { + "bbox": [ + 106, + 365, + 506, + 379 + ], + "spans": [ + { + "bbox": [ + 106, + 365, + 224, + 379 + ], + "score": 1.0, + "content": "Then, every critical point of", + "type": "text" + }, + { + "bbox": [ + 225, + 366, + 251, + 378 + ], + "score": 0.89, + "content": "L ( W )", + "type": "inline_equation" + }, + { + "bbox": [ + 251, + 365, + 264, + 379 + ], + "score": 1.0, + "content": "in", + "type": "text" + }, + { + "bbox": [ + 264, + 366, + 276, + 377 + ], + "score": 0.88, + "content": "\\nu _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 276, + 365, + 506, + 379 + ], + "score": 1.0, + "content": "is a global minimum. Moreover, every critical point of", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 107, + 377, + 228, + 389 + ], + "spans": [ + { + "bbox": [ + 107, + 377, + 133, + 389 + ], + "score": 0.92, + "content": "L ( W )", + "type": "inline_equation" + }, + { + "bbox": [ + 133, + 377, + 145, + 389 + ], + "score": 1.0, + "content": "in", + "type": "text" + }, + { + "bbox": [ + 145, + 378, + 157, + 389 + ], + "score": 0.89, + "content": "\\mathcal { V } _ { 2 } ^ { c }", + "type": "inline_equation" + }, + { + "bbox": [ + 157, + 377, + 228, + 389 + ], + "score": 1.0, + "content": "is a saddle point.", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 22.5, + "bbox_fs": [ + 106, + 365, + 506, + 389 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 390, + 505, + 447 + ], + "lines": [ + { + "bbox": [ + 106, + 390, + 505, + 403 + ], + "spans": [ + { + "bbox": [ + 106, + 390, + 455, + 403 + ], + "score": 1.0, + "content": "Theorems 2.1 and 2.2 provide necessary and sufficient conditions for a critical point of", + "type": "text" + }, + { + "bbox": [ + 455, + 391, + 481, + 403 + ], + "score": 0.93, + "content": "L ( W )", + "type": "inline_equation" + }, + { + "bbox": [ + 482, + 390, + 505, + 403 + ], + "score": 1.0, + "content": "to be", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 401, + 505, + 414 + ], + "spans": [ + { + "bbox": [ + 105, + 401, + 505, + 414 + ], + "score": 1.0, + "content": "globally optimal. From an algorithmic perspective, they provide easily checkable conditions, which", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 413, + 505, + 425 + ], + "spans": [ + { + "bbox": [ + 106, + 413, + 505, + 425 + ], + "score": 1.0, + "content": "we can use to determine if the critical point the algorithm encountered is a global optimum or not.", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 421, + 506, + 438 + ], + "spans": [ + { + "bbox": [ + 105, + 421, + 152, + 438 + ], + "score": 1.0, + "content": "Given that", + "type": "text" + }, + { + "bbox": [ + 153, + 424, + 178, + 436 + ], + "score": 0.92, + "content": "L ( W )", + "type": "inline_equation" + }, + { + "bbox": [ + 179, + 421, + 506, + 438 + ], + "score": 1.0, + "content": "is nonconvex, it is interesting to have such efficient tests for global optimality,", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 435, + 338, + 447 + ], + "spans": [ + { + "bbox": [ + 105, + 435, + 338, + 447 + ], + "score": 1.0, + "content": "which is not possible in general (Murty & Kabadi, 1987).", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 26, + "bbox_fs": [ + 105, + 390, + 506, + 447 + ] + }, + { + "type": "text", + "bbox": [ + 108, + 451, + 501, + 464 + ], + "lines": [ + { + "bbox": [ + 106, + 451, + 502, + 464 + ], + "spans": [ + { + "bbox": [ + 106, + 451, + 502, + 464 + ], + "score": 1.0, + "content": "In Hardt & Ma (2017), the authors consider minimizing population risk of linear residual networks:", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 29, + "bbox_fs": [ + 106, + 451, + 502, + 464 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 186, + 464, + 423, + 484 + ], + "lines": [ + { + "bbox": [ + 186, + 464, + 423, + 484 + ], + "spans": [ + { + "bbox": [ + 186, + 464, + 423, + 484 + ], + "score": 0.85, + "content": "\\begin{array} { r l } { \\mathrm { m i n i m i z e } } & { { } \\frac { 1 } { 2 } \\mathbb { E } _ { \\boldsymbol { x } , \\boldsymbol { y } } \\left[ \\| ( I + W _ { H + 1 } ) \\cdot \\cdot \\cdot ( I + W _ { 1 } ) \\boldsymbol { x } - \\boldsymbol { y } \\| _ { \\mathrm { F } } ^ { 2 } \\right] , } \\end{array}", + "type": "interline_equation", + "image_path": "b81cf7458ff00c67c3830645ed75f18fdc89a5d93c908d1a6dd94f1019bdb064.jpg" + } + ] + } + ], + "index": 30, + "virtual_lines": [ + { + "bbox": [ + 186, + 464, + 423, + 484 + ], + "spans": [], + "index": 30 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 485, + 506, + 529 + ], + "lines": [ + { + "bbox": [ + 106, + 485, + 505, + 497 + ], + "spans": [ + { + "bbox": [ + 106, + 485, + 133, + 497 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 133, + 485, + 262, + 497 + ], + "score": 0.89, + "content": "d _ { x } = d _ { 1 } = \\cdot \\cdot \\cdot = d _ { H } = d _ { y } = d", + "type": "inline_equation" + }, + { + "bbox": [ + 262, + 485, + 338, + 497 + ], + "score": 1.0, + "content": ". They assume that", + "type": "text" + }, + { + "bbox": [ + 339, + 488, + 345, + 495 + ], + "score": 0.78, + "content": "x", + "type": "inline_equation" + }, + { + "bbox": [ + 346, + 485, + 505, + 497 + ], + "score": 1.0, + "content": "is drawn from a zero-mean distribution", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 496, + 506, + 508 + ], + "spans": [ + { + "bbox": [ + 106, + 496, + 253, + 508 + ], + "score": 1.0, + "content": "with a fixed covariance matrix, and", + "type": "text" + }, + { + "bbox": [ + 253, + 496, + 307, + 507 + ], + "score": 0.93, + "content": "y = R x + \\xi", + "type": "inline_equation" + }, + { + "bbox": [ + 307, + 496, + 335, + 508 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 335, + 496, + 342, + 507 + ], + "score": 0.84, + "content": "\\xi", + "type": "inline_equation" + }, + { + "bbox": [ + 343, + 496, + 485, + 508 + ], + "score": 1.0, + "content": "is iid standard Gaussian noise and", + "type": "text" + }, + { + "bbox": [ + 485, + 496, + 494, + 506 + ], + "score": 0.8, + "content": "R", + "type": "inline_equation" + }, + { + "bbox": [ + 495, + 496, + 506, + 508 + ], + "score": 1.0, + "content": "is", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 507, + 506, + 519 + ], + "spans": [ + { + "bbox": [ + 106, + 507, + 239, + 519 + ], + "score": 1.0, + "content": "the true underlying matrix with", + "type": "text" + }, + { + "bbox": [ + 239, + 507, + 291, + 519 + ], + "score": 0.91, + "content": "\\operatorname* { d e t } ( R ) > 0", + "type": "inline_equation" + }, + { + "bbox": [ + 292, + 507, + 506, + 519 + ], + "score": 1.0, + "content": ". With these assumptions they prove that whenever", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 518, + 500, + 530 + ], + "spans": [ + { + "bbox": [ + 106, + 518, + 167, + 530 + ], + "score": 0.92, + "content": "\\sigma _ { \\operatorname* { m a x } } ( W _ { i } ) < 1", + "type": "inline_equation" + }, + { + "bbox": [ + 168, + 518, + 195, + 530 + ], + "score": 1.0, + "content": "for all", + "type": "text" + }, + { + "bbox": [ + 195, + 519, + 200, + 528 + ], + "score": 0.69, + "content": "i", + "type": "inline_equation" + }, + { + "bbox": [ + 200, + 518, + 500, + 530 + ], + "score": 1.0, + "content": ", any critical point is a global minimum (Hardt & Ma, 2017, Theorem 2.2).", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 32.5, + "bbox_fs": [ + 106, + 485, + 506, + 530 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 534, + 505, + 590 + ], + "lines": [ + { + "bbox": [ + 106, + 535, + 505, + 547 + ], + "spans": [ + { + "bbox": [ + 106, + 535, + 505, + 547 + ], + "score": 1.0, + "content": "Under the same assumptions on data distribution, we can slightly modify Theorem 2.1 to derive", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 546, + 506, + 558 + ], + "spans": [ + { + "bbox": [ + 105, + 546, + 506, + 558 + ], + "score": 1.0, + "content": "a population risk counterpart, and in fact notice that the result proved in Hardt & Ma (2017) is a", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 556, + 506, + 570 + ], + "spans": [ + { + "bbox": [ + 105, + 556, + 293, + 570 + ], + "score": 1.0, + "content": "corollary of this modification because having", + "type": "text" + }, + { + "bbox": [ + 293, + 556, + 356, + 569 + ], + "score": 0.93, + "content": "\\sigma _ { \\operatorname* { m a x } } ( W _ { i } ) < 1", + "type": "inline_equation" + }, + { + "bbox": [ + 357, + 556, + 386, + 570 + ], + "score": 1.0, + "content": "for all", + "type": "text" + }, + { + "bbox": [ + 386, + 558, + 391, + 567 + ], + "score": 0.73, + "content": "i", + "type": "inline_equation" + }, + { + "bbox": [ + 391, + 556, + 506, + 570 + ], + "score": 1.0, + "content": "is a sufficient condition for", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 107, + 567, + 505, + 581 + ], + "spans": [ + { + "bbox": [ + 107, + 568, + 215, + 579 + ], + "score": 0.9, + "content": "( I + W _ { H + 1 } ) \\cdot \\cdot \\cdot ( I + W _ { 1 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 216, + 567, + 505, + 581 + ], + "score": 1.0, + "content": "having full rank. Moreover, notice that we can remove the assumption", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 107, + 577, + 495, + 592 + ], + "spans": [ + { + "bbox": [ + 107, + 579, + 155, + 591 + ], + "score": 0.89, + "content": "\\operatorname* { d e t } ( R ) > 0", + "type": "inline_equation" + }, + { + "bbox": [ + 156, + 577, + 495, + 592 + ], + "score": 1.0, + "content": "which was required by Hardt & Ma (2017). We state this special case as a corollary:", + "type": "text" + } + ], + "index": 39 + } + ], + "index": 37, + "bbox_fs": [ + 105, + 535, + 506, + 592 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 591, + 505, + 633 + ], + "lines": [ + { + "bbox": [ + 106, + 590, + 505, + 604 + ], + "spans": [ + { + "bbox": [ + 106, + 590, + 505, + 604 + ], + "score": 1.0, + "content": "Corollary 2.3 (Theorem 2.2 of Hardt & Ma (2017)). Under assumptions on data distribution as", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 104, + 601, + 504, + 623 + ], + "spans": [ + { + "bbox": [ + 104, + 601, + 257, + 622 + ], + "score": 1.0, + "content": "described above, any critical point of", + "type": "text" + }, + { + "bbox": [ + 258, + 603, + 439, + 623 + ], + "score": 0.91, + "content": "\\begin{array} { r } { \\frac { 1 } { 2 } \\mathbb { E } _ { x , y } \\left[ \\left\\| ( I + W _ { H + 1 } ) \\cdot \\cdot \\cdot ( I + W _ { 1 } ) x - y \\right\\| _ { \\mathrm { F } } ^ { 2 } \\right] } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 439, + 601, + 504, + 622 + ], + "score": 1.0, + "content": "is a global min-", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 106, + 620, + 237, + 634 + ], + "spans": [ + { + "bbox": [ + 106, + 620, + 139, + 634 + ], + "score": 1.0, + "content": "imum if", + "type": "text" + }, + { + "bbox": [ + 140, + 621, + 200, + 633 + ], + "score": 0.9, + "content": "\\sigma _ { \\operatorname* { m a x } } ( W _ { i } ) < 1", + "type": "inline_equation" + }, + { + "bbox": [ + 200, + 620, + 228, + 634 + ], + "score": 1.0, + "content": "for all", + "type": "text" + }, + { + "bbox": [ + 228, + 622, + 233, + 631 + ], + "score": 0.32, + "content": "i", + "type": "inline_equation" + }, + { + "bbox": [ + 233, + 620, + 237, + 634 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 42 + } + ], + "index": 41, + "bbox_fs": [ + 104, + 590, + 505, + 634 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 636, + 504, + 661 + ], + "lines": [ + { + "bbox": [ + 104, + 632, + 508, + 655 + ], + "spans": [ + { + "bbox": [ + 104, + 632, + 401, + 655 + ], + "score": 1.0, + "content": "We also note in passing that the classical problem of matrix factorization", + "type": "text" + }, + { + "bbox": [ + 402, + 635, + 494, + 651 + ], + "score": 0.91, + "content": "\\begin{array} { r } { \\operatorname* { m i n } _ { U , V } \\left\\| U V ^ { T } - Y \\right\\| _ { \\mathrm { F } } ^ { 2 } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 495, + 632, + 508, + 655 + ], + "score": 1.0, + "content": "is", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 648, + 470, + 662 + ], + "spans": [ + { + "bbox": [ + 105, + 648, + 470, + 662 + ], + "score": 1.0, + "content": "a special case of deep linear neural networks, so our theorems can also be directly applied.", + "type": "text" + } + ], + "index": 44 + } + ], + "index": 43.5, + "bbox_fs": [ + 104, + 632, + 508, + 662 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 665, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 664, + 505, + 679 + ], + "spans": [ + { + "bbox": [ + 105, + 664, + 357, + 679 + ], + "score": 1.0, + "content": "Remarks. The previous result (Kawaguchi, 2016) assumed", + "type": "text" + }, + { + "bbox": [ + 357, + 666, + 394, + 678 + ], + "score": 0.92, + "content": "d _ { y } \\leq d _ { x }", + "type": "inline_equation" + }, + { + "bbox": [ + 394, + 664, + 505, + 679 + ], + "score": 1.0, + "content": "and showed that: 1) every", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 106, + 677, + 505, + 689 + ], + "spans": [ + { + "bbox": [ + 106, + 677, + 505, + 689 + ], + "score": 1.0, + "content": "local minimum is a global minimum, and 2) any other critical point is a saddle point. A subsequent", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 104, + 687, + 506, + 700 + ], + "spans": [ + { + "bbox": [ + 104, + 687, + 392, + 700 + ], + "score": 1.0, + "content": "paper by Lu & Kawaguchi (2017) proved 1) without the assumption", + "type": "text" + }, + { + "bbox": [ + 393, + 688, + 431, + 700 + ], + "score": 0.92, + "content": "d _ { y } \\ \\leq \\ d _ { x }", + "type": "inline_equation" + }, + { + "bbox": [ + 431, + 687, + 506, + 700 + ], + "score": 1.0, + "content": ", but as far as we", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 105, + 699, + 505, + 712 + ], + "spans": [ + { + "bbox": [ + 105, + 699, + 314, + 712 + ], + "score": 1.0, + "content": "know there is no result showing 2) in the case of", + "type": "text" + }, + { + "bbox": [ + 314, + 699, + 353, + 711 + ], + "score": 0.91, + "content": "d _ { y } \\ > \\ d _ { x }", + "type": "inline_equation" + }, + { + "bbox": [ + 353, + 699, + 505, + 712 + ], + "score": 1.0, + "content": ". We provide the proof for this case", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 105, + 708, + 505, + 723 + ], + "spans": [ + { + "bbox": [ + 105, + 708, + 505, + 723 + ], + "score": 1.0, + "content": "in Lemma B.1. In fact, we propose an alternative proof technique for handling degenerate critical", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 105, + 720, + 437, + 733 + ], + "spans": [ + { + "bbox": [ + 105, + 720, + 437, + 733 + ], + "score": 1.0, + "content": "points, which is much simpler than the technique presented by Kawaguchi (2016).", + "type": "text" + } + ], + "index": 50 + } + ], + "index": 47.5, + "bbox_fs": [ + 104, + 664, + 506, + 733 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "title", + "bbox": [ + 107, + 81, + 327, + 94 + ], + "lines": [ + { + "bbox": [ + 105, + 80, + 327, + 96 + ], + "spans": [ + { + "bbox": [ + 105, + 80, + 327, + 96 + ], + "score": 1.0, + "content": "3 ANALYSIS OF DEEP LINEAR NETWORKS", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "text", + "bbox": [ + 106, + 105, + 347, + 118 + ], + "lines": [ + { + "bbox": [ + 106, + 105, + 347, + 118 + ], + "spans": [ + { + "bbox": [ + 106, + 105, + 347, + 118 + ], + "score": 1.0, + "content": "In this section, we provide proofs for Theorems 2.1 and 2.2.", + "type": "text" + } + ], + "index": 1 + } + ], + "index": 1 + }, + { + "type": "title", + "bbox": [ + 107, + 129, + 302, + 141 + ], + "lines": [ + { + "bbox": [ + 106, + 129, + 303, + 142 + ], + "spans": [ + { + "bbox": [ + 106, + 129, + 303, + 142 + ], + "score": 1.0, + "content": "3.1 SOLUTIONS OF THE RELAXED PROBLEM", + "type": "text" + } + ], + "index": 2 + } + ], + "index": 2 + }, + { + "type": "text", + "bbox": [ + 106, + 150, + 504, + 172 + ], + "lines": [ + { + "bbox": [ + 105, + 149, + 505, + 164 + ], + "spans": [ + { + "bbox": [ + 105, + 149, + 369, + 164 + ], + "score": 1.0, + "content": "We first analyze the globally optimal solution of a “relaxation” of", + "type": "text" + }, + { + "bbox": [ + 369, + 150, + 395, + 162 + ], + "score": 0.92, + "content": "L ( W )", + "type": "inline_equation" + }, + { + "bbox": [ + 395, + 149, + 505, + 164 + ], + "score": 1.0, + "content": ", which turns out to be very", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 106, + 161, + 422, + 172 + ], + "spans": [ + { + "bbox": [ + 106, + 161, + 422, + 172 + ], + "score": 1.0, + "content": "useful while proving Theorems 2.1 and 2.2. Consider the relaxed risk function", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 3.5 + }, + { + "type": "interline_equation", + "bbox": [ + 251, + 173, + 359, + 197 + ], + "lines": [ + { + "bbox": [ + 251, + 173, + 359, + 197 + ], + "spans": [ + { + "bbox": [ + 251, + 173, + 359, + 197 + ], + "score": 0.94, + "content": "L _ { 0 } ( R ) = { \\frac { 1 } { 2 } } \\left\\| R X - Y \\right\\| _ { \\mathrm { F } } ^ { 2 } ,", + "type": "interline_equation", + "image_path": "04315d046488d0ddc0ed06e75421bc708c93696eac88b8c6060ace9ecf40115c.jpg" + } + ] + } + ], + "index": 5, + "virtual_lines": [ + { + "bbox": [ + 251, + 173, + 359, + 197 + ], + "spans": [], + "index": 5 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 199, + 506, + 232 + ], + "lines": [ + { + "bbox": [ + 106, + 197, + 506, + 212 + ], + "spans": [ + { + "bbox": [ + 106, + 197, + 133, + 212 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 134, + 198, + 185, + 209 + ], + "score": 0.92, + "content": "R \\in \\mathbb { R } ^ { d _ { y } \\times d _ { x } }", + "type": "inline_equation" + }, + { + "bbox": [ + 186, + 197, + 205, + 212 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 205, + 199, + 260, + 211 + ], + "score": 0.92, + "content": "\\operatorname { r a n k } ( R ) \\leq k", + "type": "inline_equation" + }, + { + "bbox": [ + 260, + 197, + 298, + 212 + ], + "score": 1.0, + "content": ". For any", + "type": "text" + }, + { + "bbox": [ + 298, + 199, + 309, + 209 + ], + "score": 0.82, + "content": "W", + "type": "inline_equation" + }, + { + "bbox": [ + 310, + 197, + 362, + 212 + ], + "score": 1.0, + "content": ", the product", + "type": "text" + }, + { + "bbox": [ + 362, + 199, + 435, + 210 + ], + "score": 0.88, + "content": "W _ { H + 1 } W _ { H } \\cdot \\cdot \\cdot W _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 435, + 197, + 506, + 212 + ], + "score": 1.0, + "content": "has rank at most", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 209, + 505, + 223 + ], + "spans": [ + { + "bbox": [ + 106, + 210, + 114, + 220 + ], + "score": 0.8, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 114, + 209, + 163, + 223 + ], + "score": 1.0, + "content": "and setting", + "type": "text" + }, + { + "bbox": [ + 164, + 210, + 172, + 220 + ], + "score": 0.8, + "content": "R", + "type": "inline_equation" + }, + { + "bbox": [ + 173, + 209, + 366, + 223 + ], + "score": 1.0, + "content": "to be this product gives the same loss values:", + "type": "text" + }, + { + "bbox": [ + 366, + 210, + 501, + 222 + ], + "score": 0.9, + "content": "L _ { 0 } ( W _ { H + 1 } W _ { H } \\cdot \\cdot \\cdot W _ { 1 } ) = L ( W )", + "type": "inline_equation" + }, + { + "bbox": [ + 502, + 209, + 505, + 223 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 221, + 261, + 233 + ], + "spans": [ + { + "bbox": [ + 106, + 221, + 150, + 233 + ], + "score": 1.0, + "content": "Therefore,", + "type": "text" + }, + { + "bbox": [ + 151, + 222, + 163, + 232 + ], + "score": 0.89, + "content": "L _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 163, + 221, + 234, + 233 + ], + "score": 1.0, + "content": "is a relaxation of", + "type": "text" + }, + { + "bbox": [ + 234, + 222, + 242, + 231 + ], + "score": 0.84, + "content": "L", + "type": "inline_equation" + }, + { + "bbox": [ + 242, + 221, + 261, + 233 + ], + "score": 1.0, + "content": "and", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 7 + }, + { + "type": "interline_equation", + "bbox": [ + 239, + 234, + 371, + 253 + ], + "lines": [ + { + "bbox": [ + 239, + 234, + 371, + 253 + ], + "spans": [ + { + "bbox": [ + 239, + 234, + 371, + 253 + ], + "score": 0.92, + "content": "\\operatorname* { i n f } _ { R : \\operatorname { r a n k } ( R ) \\leq k } L _ { 0 } ( R ) \\leq \\operatorname* { i n f } _ { W } L ( W ) .", + "type": "interline_equation", + "image_path": "50506c1c363c0a274afb18bbe41357afb02081abe14b32a2e94471e7cb64b3d4.jpg" + } + ] + } + ], + "index": 9, + "virtual_lines": [ + { + "bbox": [ + 239, + 234, + 371, + 253 + ], + "spans": [], + "index": 9 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 254, + 505, + 311 + ], + "lines": [ + { + "bbox": [ + 105, + 254, + 506, + 269 + ], + "spans": [ + { + "bbox": [ + 105, + 254, + 234, + 269 + ], + "score": 1.0, + "content": "This means that if there exists", + "type": "text" + }, + { + "bbox": [ + 234, + 255, + 246, + 266 + ], + "score": 0.79, + "content": "W", + "type": "inline_equation" + }, + { + "bbox": [ + 246, + 254, + 288, + 269 + ], + "score": 1.0, + "content": "such that", + "type": "text" + }, + { + "bbox": [ + 289, + 255, + 420, + 268 + ], + "score": 0.9, + "content": "L ( W ) = \\operatorname* { i n f } _ { R : \\operatorname { r a n k } ( R ) \\leq k } L _ { 0 } ( R )", + "type": "inline_equation" + }, + { + "bbox": [ + 420, + 254, + 445, + 269 + ], + "score": 1.0, + "content": ", then", + "type": "text" + }, + { + "bbox": [ + 446, + 255, + 457, + 266 + ], + "score": 0.77, + "content": "W", + "type": "inline_equation" + }, + { + "bbox": [ + 458, + 254, + 506, + 269 + ], + "score": 1.0, + "content": "is a global", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 266, + 506, + 279 + ], + "spans": [ + { + "bbox": [ + 105, + 266, + 210, + 279 + ], + "score": 1.0, + "content": "minimum of the function", + "type": "text" + }, + { + "bbox": [ + 210, + 267, + 218, + 276 + ], + "score": 0.79, + "content": "L", + "type": "inline_equation" + }, + { + "bbox": [ + 218, + 266, + 506, + 279 + ], + "score": 1.0, + "content": ". This observation is very important in proofs; we will show that inside", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 276, + 505, + 290 + ], + "spans": [ + { + "bbox": [ + 105, + 276, + 227, + 290 + ], + "score": 1.0, + "content": "certain sets, any critical point", + "type": "text" + }, + { + "bbox": [ + 227, + 278, + 240, + 288 + ], + "score": 0.76, + "content": "W", + "type": "inline_equation" + }, + { + "bbox": [ + 240, + 276, + 252, + 290 + ], + "score": 1.0, + "content": "of", + "type": "text" + }, + { + "bbox": [ + 252, + 277, + 278, + 289 + ], + "score": 0.91, + "content": "L ( W )", + "type": "inline_equation" + }, + { + "bbox": [ + 279, + 276, + 331, + 290 + ], + "score": 1.0, + "content": "must satisfy", + "type": "text" + }, + { + "bbox": [ + 331, + 277, + 415, + 289 + ], + "score": 0.91, + "content": "R ^ { * } = W _ { H + 1 } \\cdot \\cdot \\cdot W _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 415, + 276, + 446, + 290 + ], + "score": 1.0, + "content": ", where", + "type": "text" + }, + { + "bbox": [ + 446, + 278, + 459, + 288 + ], + "score": 0.88, + "content": "R ^ { * }", + "type": "inline_equation" + }, + { + "bbox": [ + 460, + 276, + 505, + 290 + ], + "score": 1.0, + "content": "is a global", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 287, + 506, + 303 + ], + "spans": [ + { + "bbox": [ + 105, + 287, + 156, + 303 + ], + "score": 1.0, + "content": "optimum of", + "type": "text" + }, + { + "bbox": [ + 156, + 289, + 183, + 300 + ], + "score": 0.92, + "content": "L _ { 0 } ( R )", + "type": "inline_equation" + }, + { + "bbox": [ + 184, + 287, + 255, + 303 + ], + "score": 1.0, + "content": ". This proves that", + "type": "text" + }, + { + "bbox": [ + 255, + 289, + 428, + 301 + ], + "score": 0.89, + "content": "L ( W ) = L _ { 0 } ( R ^ { * } ) = \\operatorname* { i n f } _ { R : \\operatorname { r a n k } ( R ) \\leq k } L _ { 0 } ( R )", + "type": "inline_equation" + }, + { + "bbox": [ + 428, + 287, + 506, + 303 + ], + "score": 1.0, + "content": ", thus showing that", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 107, + 299, + 227, + 311 + ], + "spans": [ + { + "bbox": [ + 107, + 300, + 118, + 309 + ], + "score": 0.77, + "content": "W", + "type": "inline_equation" + }, + { + "bbox": [ + 119, + 299, + 215, + 311 + ], + "score": 1.0, + "content": "is a global minimum of", + "type": "text" + }, + { + "bbox": [ + 216, + 300, + 223, + 309 + ], + "score": 0.84, + "content": "L", + "type": "inline_equation" + }, + { + "bbox": [ + 223, + 299, + 227, + 311 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 12 + }, + { + "type": "text", + "bbox": [ + 107, + 316, + 504, + 339 + ], + "lines": [ + { + "bbox": [ + 105, + 316, + 505, + 329 + ], + "spans": [ + { + "bbox": [ + 105, + 316, + 505, + 329 + ], + "score": 1.0, + "content": "By restating this observation as an optimization problem, the solution of problem in (1) is bounded", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 326, + 294, + 340 + ], + "spans": [ + { + "bbox": [ + 105, + 326, + 294, + 340 + ], + "score": 1.0, + "content": "below by the minimum value of the following:", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 15.5 + }, + { + "type": "interline_equation", + "bbox": [ + 246, + 339, + 363, + 366 + ], + "lines": [ + { + "bbox": [ + 246, + 339, + 363, + 366 + ], + "spans": [ + { + "bbox": [ + 246, + 339, + 363, + 366 + ], + "score": 0.93, + "content": "\\begin{array} { l l } { \\mathrm { m i n i m i z e } } & { \\frac { 1 } { 2 } \\left\\| R X - Y \\right\\| _ { \\mathrm { F } } ^ { 2 } } \\\\ { \\mathrm { s u b j e c t ~ t o } } & { \\mathrm { r a n k } ( R ) \\leq k . } \\end{array}", + "type": "interline_equation", + "image_path": "e7bde960e32fb52bf9df8235decbef75805c10070abfa6748e338ee2ff49ab97.jpg" + } + ] + } + ], + "index": 17, + "virtual_lines": [ + { + "bbox": [ + 246, + 339, + 363, + 366 + ], + "spans": [], + "index": 17 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 368, + 506, + 402 + ], + "lines": [ + { + "bbox": [ + 105, + 367, + 506, + 382 + ], + "spans": [ + { + "bbox": [ + 105, + 367, + 165, + 382 + ], + "score": 1.0, + "content": "In case where", + "type": "text" + }, + { + "bbox": [ + 165, + 368, + 238, + 380 + ], + "score": 0.93, + "content": "k = \\operatorname* { m i n } \\{ d _ { x } , d _ { y } \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 239, + 367, + 506, + 382 + ], + "score": 1.0, + "content": ", (2) is actually an unconstrained optimization problem. Note that", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 107, + 379, + 505, + 391 + ], + "spans": [ + { + "bbox": [ + 107, + 380, + 119, + 390 + ], + "score": 0.88, + "content": "L _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 119, + 379, + 219, + 391 + ], + "score": 1.0, + "content": "is a convex function of", + "type": "text" + }, + { + "bbox": [ + 220, + 380, + 228, + 389 + ], + "score": 0.82, + "content": "R", + "type": "inline_equation" + }, + { + "bbox": [ + 228, + 379, + 505, + 391 + ], + "score": 1.0, + "content": ", so any critical point is a global minimum. By differentiating and", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 390, + 442, + 402 + ], + "spans": [ + { + "bbox": [ + 105, + 390, + 442, + 402 + ], + "score": 1.0, + "content": "setting the derivative to zero, we can easily get the unique globally optimal solution", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 19 + }, + { + "type": "interline_equation", + "bbox": [ + 257, + 402, + 353, + 417 + ], + "lines": [ + { + "bbox": [ + 257, + 402, + 353, + 417 + ], + "spans": [ + { + "bbox": [ + 257, + 402, + 353, + 417 + ], + "score": 0.92, + "content": "R ^ { * } = Y X ^ { T } ( X X ^ { T } ) ^ { - 1 } .", + "type": "interline_equation", + "image_path": "c8accd26cf813fed54b87edb2a46f770a126cb471396d3f7a0fdc6898698d916.jpg" + } + ] + } + ], + "index": 21, + "virtual_lines": [ + { + "bbox": [ + 257, + 402, + 353, + 417 + ], + "spans": [], + "index": 21 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 418, + 506, + 452 + ], + "lines": [ + { + "bbox": [ + 105, + 417, + 506, + 432 + ], + "spans": [ + { + "bbox": [ + 105, + 417, + 149, + 432 + ], + "score": 1.0, + "content": "In case of", + "type": "text" + }, + { + "bbox": [ + 149, + 418, + 221, + 430 + ], + "score": 0.93, + "content": "k < \\operatorname* { m i n } \\{ d _ { x } , d _ { y } \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 222, + 417, + 506, + 432 + ], + "score": 1.0, + "content": ", the problem becomes non-convex because of the rank constraint, but", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 429, + 505, + 442 + ], + "spans": [ + { + "bbox": [ + 106, + 429, + 505, + 442 + ], + "score": 1.0, + "content": "its exact solution can still be computed easily. We present the solution of this case as a proposition", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 440, + 343, + 452 + ], + "spans": [ + { + "bbox": [ + 105, + 440, + 343, + 452 + ], + "score": 1.0, + "content": "and defer the proof to Appendix C due to its technicalities.", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 23 + }, + { + "type": "text", + "bbox": [ + 107, + 453, + 427, + 465 + ], + "lines": [ + { + "bbox": [ + 105, + 452, + 428, + 468 + ], + "spans": [ + { + "bbox": [ + 105, + 452, + 214, + 468 + ], + "score": 1.0, + "content": "Proposition 3.1. Suppose", + "type": "text" + }, + { + "bbox": [ + 214, + 453, + 284, + 466 + ], + "score": 0.93, + "content": "k < \\operatorname* { m i n } \\{ d _ { x } , d _ { y } \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 285, + 452, + 428, + 468 + ], + "score": 1.0, + "content": ". Then the optimal solution to (2) is", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 25 + }, + { + "type": "interline_equation", + "bbox": [ + 246, + 467, + 364, + 482 + ], + "lines": [ + { + "bbox": [ + 246, + 467, + 364, + 482 + ], + "spans": [ + { + "bbox": [ + 246, + 467, + 364, + 482 + ], + "score": 0.91, + "content": "R ^ { * } = \\hat { U } \\hat { U } ^ { T } Y X ^ { T } ( X X ^ { T } ) ^ { - 1 } ,", + "type": "interline_equation", + "image_path": "d8b7753f67fc0ce5146b4ab402bcc5a916cbf5efe766f1c3764be20d2739de88.jpg" + } + ] + } + ], + "index": 26, + "virtual_lines": [ + { + "bbox": [ + 246, + 467, + 364, + 482 + ], + "spans": [], + "index": 26 + } + ] + }, + { + "type": "text", + "bbox": [ + 108, + 483, + 438, + 497 + ], + "lines": [ + { + "bbox": [ + 105, + 482, + 436, + 499 + ], + "spans": [ + { + "bbox": [ + 105, + 482, + 256, + 499 + ], + "score": 1.0, + "content": "which is the orthogonal projection of", + "type": "text" + }, + { + "bbox": [ + 257, + 484, + 324, + 497 + ], + "score": 0.93, + "content": "Y X ^ { T } ( X X ^ { T } ) ^ { - 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 324, + 482, + 427, + 499 + ], + "score": 1.0, + "content": "onto the column space of", + "type": "text" + }, + { + "bbox": [ + 428, + 483, + 436, + 495 + ], + "score": 0.81, + "content": "\\hat { U }", + "type": "inline_equation" + } + ], + "index": 27 + } + ], + "index": 27 + }, + { + "type": "title", + "bbox": [ + 107, + 509, + 269, + 521 + ], + "lines": [ + { + "bbox": [ + 105, + 508, + 268, + 523 + ], + "spans": [ + { + "bbox": [ + 105, + 508, + 242, + 523 + ], + "score": 1.0, + "content": "3.2 PARTIAL DERIVATIVES OF", + "type": "text" + }, + { + "bbox": [ + 242, + 509, + 268, + 522 + ], + "score": 0.89, + "content": "L ( W )", + "type": "inline_equation" + } + ], + "index": 28 + } + ], + "index": 28 + }, + { + "type": "text", + "bbox": [ + 105, + 529, + 505, + 552 + ], + "lines": [ + { + "bbox": [ + 106, + 529, + 505, + 542 + ], + "spans": [ + { + "bbox": [ + 106, + 529, + 362, + 542 + ], + "score": 1.0, + "content": "By simple matrix calculus, we can calculate the derivatives of", + "type": "text" + }, + { + "bbox": [ + 362, + 530, + 388, + 542 + ], + "score": 0.92, + "content": "L ( W )", + "type": "inline_equation" + }, + { + "bbox": [ + 389, + 529, + 453, + 542 + ], + "score": 1.0, + "content": "with respect to", + "type": "text" + }, + { + "bbox": [ + 454, + 530, + 468, + 541 + ], + "score": 0.88, + "content": "W _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 468, + 529, + 487, + 542 + ], + "score": 1.0, + "content": ", for", + "type": "text" + }, + { + "bbox": [ + 487, + 531, + 505, + 541 + ], + "score": 0.84, + "content": "i =", + "type": "inline_equation" + } + ], + "index": 29 + }, + { + "bbox": [ + 107, + 541, + 497, + 554 + ], + "spans": [ + { + "bbox": [ + 107, + 541, + 161, + 552 + ], + "score": 0.91, + "content": "1 , \\ldots , H + 1", + "type": "inline_equation" + }, + { + "bbox": [ + 162, + 541, + 497, + 554 + ], + "score": 1.0, + "content": ". We present the result as the following lemma, and defer the details to Appendix C.", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 29.5 + }, + { + "type": "text", + "bbox": [ + 108, + 554, + 412, + 566 + ], + "lines": [ + { + "bbox": [ + 105, + 552, + 410, + 568 + ], + "spans": [ + { + "bbox": [ + 105, + 552, + 261, + 568 + ], + "score": 1.0, + "content": "Lemma 3.2. The partial derivative of", + "type": "text" + }, + { + "bbox": [ + 262, + 554, + 288, + 566 + ], + "score": 0.9, + "content": "L ( W )", + "type": "inline_equation" + }, + { + "bbox": [ + 288, + 552, + 349, + 568 + ], + "score": 1.0, + "content": "with respect to", + "type": "text" + }, + { + "bbox": [ + 350, + 554, + 363, + 565 + ], + "score": 0.89, + "content": "W _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 364, + 552, + 410, + 568 + ], + "score": 1.0, + "content": "is given as", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 31 + }, + { + "type": "interline_equation", + "bbox": [ + 163, + 567, + 447, + 592 + ], + "lines": [ + { + "bbox": [ + 163, + 567, + 447, + 592 + ], + "spans": [ + { + "bbox": [ + 163, + 567, + 447, + 592 + ], + "score": 0.87, + "content": "\\frac { \\partial L } { \\partial W _ { i } } = W _ { i + 1 } ^ { T } \\cdot \\cdot \\cdot W _ { H + 1 } ^ { T } ( W _ { H + 1 } W _ { H } \\cdot \\cdot \\cdot W _ { 1 } X - Y ) X ^ { T } W _ { 1 } ^ { T } \\cdot \\cdot \\cdot W _ { i - 1 } ^ { T } ,", + "type": "interline_equation", + "image_path": "fe5cff211b7e77242b6d945873020a79ecf87627040fc811c8b70a9b4862375a.jpg" + } + ] + } + ], + "index": 32, + "virtual_lines": [ + { + "bbox": [ + 163, + 567, + 447, + 592 + ], + "spans": [], + "index": 32 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 592, + 194, + 605 + ], + "lines": [ + { + "bbox": [ + 105, + 591, + 192, + 606 + ], + "spans": [ + { + "bbox": [ + 105, + 591, + 120, + 606 + ], + "score": 1.0, + "content": "for", + "type": "text" + }, + { + "bbox": [ + 121, + 593, + 192, + 604 + ], + "score": 0.83, + "content": "i = 1 , \\ldots , H + 1", + "type": "inline_equation" + } + ], + "index": 33 + } + ], + "index": 33 + }, + { + "type": "text", + "bbox": [ + 106, + 612, + 504, + 646 + ], + "lines": [ + { + "bbox": [ + 105, + 610, + 506, + 625 + ], + "spans": [ + { + "bbox": [ + 105, + 610, + 506, + 625 + ], + "score": 1.0, + "content": "This result will be used throughout the proof of Theorems 2.1 and 2.2. For clarity in notation, note", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 150, + 620, + 501, + 652 + ], + "spans": [ + { + "bbox": [ + 150, + 623, + 175, + 633 + ], + "score": 0.84, + "content": "i = 1", + "type": "inline_equation" + }, + { + "bbox": [ + 181, + 622, + 230, + 635 + ], + "score": 0.87, + "content": "W _ { 1 } ^ { T } \\cdots W _ { 0 } ^ { \\breve { T } }", + "type": "inline_equation" + }, + { + "bbox": [ + 231, + 620, + 271, + 652 + ], + "score": 1.0, + "content": "is just an matrix in", + "type": "text" + }, + { + "bbox": [ + 303, + 620, + 347, + 652 + ], + "score": 1.0, + "content": "matrix in .", + "type": "text" + }, + { + "bbox": [ + 347, + 622, + 378, + 633 + ], + "score": 0.91, + "content": "\\mathbb { R } ^ { d _ { x } \\times d _ { x } }", + "type": "inline_equation" + }, + { + "bbox": [ + 379, + 620, + 453, + 652 + ], + "score": 1.0, + "content": ". Similarly, when", + "type": "text" + }, + { + "bbox": [ + 453, + 623, + 501, + 633 + ], + "score": 0.9, + "content": "i = H + 1", + "type": "inline_equation" + } + ], + "index": 36 + }, + { + "bbox": [ + 107, + 633, + 302, + 647 + ], + "spans": [ + { + "bbox": [ + 107, + 633, + 177, + 647 + ], + "score": 0.93, + "content": "W _ { H + 2 } ^ { T } \\cdot \\cdot \\cdot W _ { H + 1 } ^ { T }", + "type": "inline_equation" + }, + { + "bbox": [ + 271, + 633, + 302, + 644 + ], + "score": 0.91, + "content": "\\mathbb { R } ^ { d _ { y } \\times d _ { y } ^ { * } }", + "type": "inline_equation" + } + ], + "index": 35 + } + ], + "index": 35 + }, + { + "type": "text", + "bbox": [ + 106, + 650, + 504, + 673 + ], + "lines": [ + { + "bbox": [ + 106, + 649, + 506, + 663 + ], + "spans": [ + { + "bbox": [ + 106, + 649, + 506, + 663 + ], + "score": 1.0, + "content": "We also state an elementary lemma which proves useful in our proofs, whose proof we defer to", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 660, + 161, + 675 + ], + "spans": [ + { + "bbox": [ + 105, + 660, + 161, + 675 + ], + "score": 1.0, + "content": "Appendix C.", + "type": "text" + } + ], + "index": 38 + } + ], + "index": 37.5 + }, + { + "type": "text", + "bbox": [ + 106, + 674, + 376, + 686 + ], + "lines": [ + { + "bbox": [ + 104, + 671, + 376, + 690 + ], + "spans": [ + { + "bbox": [ + 104, + 671, + 205, + 690 + ], + "score": 1.0, + "content": "Lemma 3.3. 1. For any", + "type": "text" + }, + { + "bbox": [ + 206, + 675, + 252, + 685 + ], + "score": 0.91, + "content": "A \\in \\mathbb { R } ^ { m \\times n }", + "type": "inline_equation" + }, + { + "bbox": [ + 252, + 671, + 271, + 690 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 271, + 674, + 314, + 685 + ], + "score": 0.83, + "content": "\\boldsymbol { B } \\in \\mathbb { R } ^ { n \\times l }", + "type": "inline_equation" + }, + { + "bbox": [ + 314, + 671, + 342, + 690 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 342, + 675, + 371, + 686 + ], + "score": 0.4, + "content": "m \\geq n ,", + "type": "inline_equation" + }, + { + "bbox": [ + 372, + 671, + 376, + 690 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 39 + } + ], + "index": 39 + }, + { + "type": "interline_equation", + "bbox": [ + 250, + 688, + 371, + 703 + ], + "lines": [ + { + "bbox": [ + 250, + 688, + 371, + 703 + ], + "spans": [ + { + "bbox": [ + 250, + 688, + 371, + 703 + ], + "score": 0.64, + "content": "\\left\\| A B \\right\\| _ { \\mathrm { F } } ^ { 2 } \\geq \\sigma _ { \\operatorname* { m i n } } ^ { 2 } ( A ) \\left\\| B \\right\\| _ { \\mathrm { F } } ^ { 2 } .", + "type": "interline_equation", + "image_path": "1aabd83d43aae35f20466257db4d0b1ad4b21ea7821b44dd69c28e82b8b4ee6e.jpg" + } + ] + } + ], + "index": 40, + "virtual_lines": [ + { + "bbox": [ + 250, + 688, + 371, + 703 + ], + "spans": [], + "index": 40 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 705, + 313, + 717 + ], + "lines": [ + { + "bbox": [ + 104, + 703, + 310, + 719 + ], + "spans": [ + { + "bbox": [ + 104, + 703, + 150, + 719 + ], + "score": 1.0, + "content": "2. For any", + "type": "text" + }, + { + "bbox": [ + 151, + 705, + 197, + 716 + ], + "score": 0.91, + "content": "A \\in \\mathbb { R } ^ { m \\times n }", + "type": "inline_equation" + }, + { + "bbox": [ + 197, + 703, + 216, + 719 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 216, + 704, + 258, + 716 + ], + "score": 0.88, + "content": "\\boldsymbol { B } \\in \\mathbb { R } ^ { n \\times l }", + "type": "inline_equation" + }, + { + "bbox": [ + 259, + 703, + 286, + 719 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 287, + 705, + 310, + 716 + ], + "score": 0.81, + "content": "n \\leq l", + "type": "inline_equation" + } + ], + "index": 41 + } + ], + "index": 41 + }, + { + "type": "interline_equation", + "bbox": [ + 250, + 718, + 360, + 734 + ], + "lines": [ + { + "bbox": [ + 250, + 718, + 360, + 734 + ], + "spans": [ + { + "bbox": [ + 250, + 718, + 360, + 734 + ], + "score": 0.82, + "content": "\\left\\| A B \\right\\| _ { \\mathrm { F } } ^ { 2 } \\geq \\sigma _ { \\operatorname* { m i n } } ^ { 2 } ( B ) \\left\\| A \\right\\| _ { \\mathrm { F } } ^ { 2 } .", + "type": "interline_equation", + "image_path": "3736bab224c123ec71b9d5071d84030d73c45e344f6de9daa941150abcc92504.jpg" + } + ] + } + ], + "index": 42, + "virtual_lines": [ + { + "bbox": [ + 250, + 718, + 360, + 734 + ], + "spans": [], + "index": 42 + } + ] + } + ], + "page_idx": 3, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 107, + 26, + 293, + 38 + ], + "lines": [ + { + "bbox": [ + 106, + 25, + 294, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 25, + 294, + 38 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2018", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 302, + 751, + 308, + 759 + ], + "lines": [ + { + "bbox": [ + 302, + 750, + 310, + 762 + ], + "spans": [ + { + "bbox": [ + 302, + 750, + 310, + 762 + ], + "score": 1.0, + "content": "", + "type": "text", + "height": 12, + "width": 8 + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "title", + "bbox": [ + 107, + 81, + 327, + 94 + ], + "lines": [ + { + "bbox": [ + 105, + 80, + 327, + 96 + ], + "spans": [ + { + "bbox": [ + 105, + 80, + 327, + 96 + ], + "score": 1.0, + "content": "3 ANALYSIS OF DEEP LINEAR NETWORKS", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "text", + "bbox": [ + 106, + 105, + 347, + 118 + ], + "lines": [ + { + "bbox": [ + 106, + 105, + 347, + 118 + ], + "spans": [ + { + "bbox": [ + 106, + 105, + 347, + 118 + ], + "score": 1.0, + "content": "In this section, we provide proofs for Theorems 2.1 and 2.2.", + "type": "text" + } + ], + "index": 1 + } + ], + "index": 1, + "bbox_fs": [ + 106, + 105, + 347, + 118 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 129, + 302, + 141 + ], + "lines": [ + { + "bbox": [ + 106, + 129, + 303, + 142 + ], + "spans": [ + { + "bbox": [ + 106, + 129, + 303, + 142 + ], + "score": 1.0, + "content": "3.1 SOLUTIONS OF THE RELAXED PROBLEM", + "type": "text" + } + ], + "index": 2 + } + ], + "index": 2 + }, + { + "type": "text", + "bbox": [ + 106, + 150, + 504, + 172 + ], + "lines": [ + { + "bbox": [ + 105, + 149, + 505, + 164 + ], + "spans": [ + { + "bbox": [ + 105, + 149, + 369, + 164 + ], + "score": 1.0, + "content": "We first analyze the globally optimal solution of a “relaxation” of", + "type": "text" + }, + { + "bbox": [ + 369, + 150, + 395, + 162 + ], + "score": 0.92, + "content": "L ( W )", + "type": "inline_equation" + }, + { + "bbox": [ + 395, + 149, + 505, + 164 + ], + "score": 1.0, + "content": ", which turns out to be very", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 106, + 161, + 422, + 172 + ], + "spans": [ + { + "bbox": [ + 106, + 161, + 422, + 172 + ], + "score": 1.0, + "content": "useful while proving Theorems 2.1 and 2.2. Consider the relaxed risk function", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 3.5, + "bbox_fs": [ + 105, + 149, + 505, + 172 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 251, + 173, + 359, + 197 + ], + "lines": [ + { + "bbox": [ + 251, + 173, + 359, + 197 + ], + "spans": [ + { + "bbox": [ + 251, + 173, + 359, + 197 + ], + "score": 0.94, + "content": "L _ { 0 } ( R ) = { \\frac { 1 } { 2 } } \\left\\| R X - Y \\right\\| _ { \\mathrm { F } } ^ { 2 } ,", + "type": "interline_equation", + "image_path": "04315d046488d0ddc0ed06e75421bc708c93696eac88b8c6060ace9ecf40115c.jpg" + } + ] + } + ], + "index": 5, + "virtual_lines": [ + { + "bbox": [ + 251, + 173, + 359, + 197 + ], + "spans": [], + "index": 5 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 199, + 506, + 232 + ], + "lines": [ + { + "bbox": [ + 106, + 197, + 506, + 212 + ], + "spans": [ + { + "bbox": [ + 106, + 197, + 133, + 212 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 134, + 198, + 185, + 209 + ], + "score": 0.92, + "content": "R \\in \\mathbb { R } ^ { d _ { y } \\times d _ { x } }", + "type": "inline_equation" + }, + { + "bbox": [ + 186, + 197, + 205, + 212 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 205, + 199, + 260, + 211 + ], + "score": 0.92, + "content": "\\operatorname { r a n k } ( R ) \\leq k", + "type": "inline_equation" + }, + { + "bbox": [ + 260, + 197, + 298, + 212 + ], + "score": 1.0, + "content": ". For any", + "type": "text" + }, + { + "bbox": [ + 298, + 199, + 309, + 209 + ], + "score": 0.82, + "content": "W", + "type": "inline_equation" + }, + { + "bbox": [ + 310, + 197, + 362, + 212 + ], + "score": 1.0, + "content": ", the product", + "type": "text" + }, + { + "bbox": [ + 362, + 199, + 435, + 210 + ], + "score": 0.88, + "content": "W _ { H + 1 } W _ { H } \\cdot \\cdot \\cdot W _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 435, + 197, + 506, + 212 + ], + "score": 1.0, + "content": "has rank at most", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 209, + 505, + 223 + ], + "spans": [ + { + "bbox": [ + 106, + 210, + 114, + 220 + ], + "score": 0.8, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 114, + 209, + 163, + 223 + ], + "score": 1.0, + "content": "and setting", + "type": "text" + }, + { + "bbox": [ + 164, + 210, + 172, + 220 + ], + "score": 0.8, + "content": "R", + "type": "inline_equation" + }, + { + "bbox": [ + 173, + 209, + 366, + 223 + ], + "score": 1.0, + "content": "to be this product gives the same loss values:", + "type": "text" + }, + { + "bbox": [ + 366, + 210, + 501, + 222 + ], + "score": 0.9, + "content": "L _ { 0 } ( W _ { H + 1 } W _ { H } \\cdot \\cdot \\cdot W _ { 1 } ) = L ( W )", + "type": "inline_equation" + }, + { + "bbox": [ + 502, + 209, + 505, + 223 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 221, + 261, + 233 + ], + "spans": [ + { + "bbox": [ + 106, + 221, + 150, + 233 + ], + "score": 1.0, + "content": "Therefore,", + "type": "text" + }, + { + "bbox": [ + 151, + 222, + 163, + 232 + ], + "score": 0.89, + "content": "L _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 163, + 221, + 234, + 233 + ], + "score": 1.0, + "content": "is a relaxation of", + "type": "text" + }, + { + "bbox": [ + 234, + 222, + 242, + 231 + ], + "score": 0.84, + "content": "L", + "type": "inline_equation" + }, + { + "bbox": [ + 242, + 221, + 261, + 233 + ], + "score": 1.0, + "content": "and", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 7, + "bbox_fs": [ + 106, + 197, + 506, + 233 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 239, + 234, + 371, + 253 + ], + "lines": [ + { + "bbox": [ + 239, + 234, + 371, + 253 + ], + "spans": [ + { + "bbox": [ + 239, + 234, + 371, + 253 + ], + "score": 0.92, + "content": "\\operatorname* { i n f } _ { R : \\operatorname { r a n k } ( R ) \\leq k } L _ { 0 } ( R ) \\leq \\operatorname* { i n f } _ { W } L ( W ) .", + "type": "interline_equation", + "image_path": "50506c1c363c0a274afb18bbe41357afb02081abe14b32a2e94471e7cb64b3d4.jpg" + } + ] + } + ], + "index": 9, + "virtual_lines": [ + { + "bbox": [ + 239, + 234, + 371, + 253 + ], + "spans": [], + "index": 9 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 254, + 505, + 311 + ], + "lines": [ + { + "bbox": [ + 105, + 254, + 506, + 269 + ], + "spans": [ + { + "bbox": [ + 105, + 254, + 234, + 269 + ], + "score": 1.0, + "content": "This means that if there exists", + "type": "text" + }, + { + "bbox": [ + 234, + 255, + 246, + 266 + ], + "score": 0.79, + "content": "W", + "type": "inline_equation" + }, + { + "bbox": [ + 246, + 254, + 288, + 269 + ], + "score": 1.0, + "content": "such that", + "type": "text" + }, + { + "bbox": [ + 289, + 255, + 420, + 268 + ], + "score": 0.9, + "content": "L ( W ) = \\operatorname* { i n f } _ { R : \\operatorname { r a n k } ( R ) \\leq k } L _ { 0 } ( R )", + "type": "inline_equation" + }, + { + "bbox": [ + 420, + 254, + 445, + 269 + ], + "score": 1.0, + "content": ", then", + "type": "text" + }, + { + "bbox": [ + 446, + 255, + 457, + 266 + ], + "score": 0.77, + "content": "W", + "type": "inline_equation" + }, + { + "bbox": [ + 458, + 254, + 506, + 269 + ], + "score": 1.0, + "content": "is a global", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 266, + 506, + 279 + ], + "spans": [ + { + "bbox": [ + 105, + 266, + 210, + 279 + ], + "score": 1.0, + "content": "minimum of the function", + "type": "text" + }, + { + "bbox": [ + 210, + 267, + 218, + 276 + ], + "score": 0.79, + "content": "L", + "type": "inline_equation" + }, + { + "bbox": [ + 218, + 266, + 506, + 279 + ], + "score": 1.0, + "content": ". This observation is very important in proofs; we will show that inside", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 276, + 505, + 290 + ], + "spans": [ + { + "bbox": [ + 105, + 276, + 227, + 290 + ], + "score": 1.0, + "content": "certain sets, any critical point", + "type": "text" + }, + { + "bbox": [ + 227, + 278, + 240, + 288 + ], + "score": 0.76, + "content": "W", + "type": "inline_equation" + }, + { + "bbox": [ + 240, + 276, + 252, + 290 + ], + "score": 1.0, + "content": "of", + "type": "text" + }, + { + "bbox": [ + 252, + 277, + 278, + 289 + ], + "score": 0.91, + "content": "L ( W )", + "type": "inline_equation" + }, + { + "bbox": [ + 279, + 276, + 331, + 290 + ], + "score": 1.0, + "content": "must satisfy", + "type": "text" + }, + { + "bbox": [ + 331, + 277, + 415, + 289 + ], + "score": 0.91, + "content": "R ^ { * } = W _ { H + 1 } \\cdot \\cdot \\cdot W _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 415, + 276, + 446, + 290 + ], + "score": 1.0, + "content": ", where", + "type": "text" + }, + { + "bbox": [ + 446, + 278, + 459, + 288 + ], + "score": 0.88, + "content": "R ^ { * }", + "type": "inline_equation" + }, + { + "bbox": [ + 460, + 276, + 505, + 290 + ], + "score": 1.0, + "content": "is a global", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 287, + 506, + 303 + ], + "spans": [ + { + "bbox": [ + 105, + 287, + 156, + 303 + ], + "score": 1.0, + "content": "optimum of", + "type": "text" + }, + { + "bbox": [ + 156, + 289, + 183, + 300 + ], + "score": 0.92, + "content": "L _ { 0 } ( R )", + "type": "inline_equation" + }, + { + "bbox": [ + 184, + 287, + 255, + 303 + ], + "score": 1.0, + "content": ". This proves that", + "type": "text" + }, + { + "bbox": [ + 255, + 289, + 428, + 301 + ], + "score": 0.89, + "content": "L ( W ) = L _ { 0 } ( R ^ { * } ) = \\operatorname* { i n f } _ { R : \\operatorname { r a n k } ( R ) \\leq k } L _ { 0 } ( R )", + "type": "inline_equation" + }, + { + "bbox": [ + 428, + 287, + 506, + 303 + ], + "score": 1.0, + "content": ", thus showing that", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 107, + 299, + 227, + 311 + ], + "spans": [ + { + "bbox": [ + 107, + 300, + 118, + 309 + ], + "score": 0.77, + "content": "W", + "type": "inline_equation" + }, + { + "bbox": [ + 119, + 299, + 215, + 311 + ], + "score": 1.0, + "content": "is a global minimum of", + "type": "text" + }, + { + "bbox": [ + 216, + 300, + 223, + 309 + ], + "score": 0.84, + "content": "L", + "type": "inline_equation" + }, + { + "bbox": [ + 223, + 299, + 227, + 311 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 12, + "bbox_fs": [ + 105, + 254, + 506, + 311 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 316, + 504, + 339 + ], + "lines": [ + { + "bbox": [ + 105, + 316, + 505, + 329 + ], + "spans": [ + { + "bbox": [ + 105, + 316, + 505, + 329 + ], + "score": 1.0, + "content": "By restating this observation as an optimization problem, the solution of problem in (1) is bounded", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 326, + 294, + 340 + ], + "spans": [ + { + "bbox": [ + 105, + 326, + 294, + 340 + ], + "score": 1.0, + "content": "below by the minimum value of the following:", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 15.5, + "bbox_fs": [ + 105, + 316, + 505, + 340 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 246, + 339, + 363, + 366 + ], + "lines": [ + { + "bbox": [ + 246, + 339, + 363, + 366 + ], + "spans": [ + { + "bbox": [ + 246, + 339, + 363, + 366 + ], + "score": 0.93, + "content": "\\begin{array} { l l } { \\mathrm { m i n i m i z e } } & { \\frac { 1 } { 2 } \\left\\| R X - Y \\right\\| _ { \\mathrm { F } } ^ { 2 } } \\\\ { \\mathrm { s u b j e c t ~ t o } } & { \\mathrm { r a n k } ( R ) \\leq k . } \\end{array}", + "type": "interline_equation", + "image_path": "e7bde960e32fb52bf9df8235decbef75805c10070abfa6748e338ee2ff49ab97.jpg" + } + ] + } + ], + "index": 17, + "virtual_lines": [ + { + "bbox": [ + 246, + 339, + 363, + 366 + ], + "spans": [], + "index": 17 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 368, + 506, + 402 + ], + "lines": [ + { + "bbox": [ + 105, + 367, + 506, + 382 + ], + "spans": [ + { + "bbox": [ + 105, + 367, + 165, + 382 + ], + "score": 1.0, + "content": "In case where", + "type": "text" + }, + { + "bbox": [ + 165, + 368, + 238, + 380 + ], + "score": 0.93, + "content": "k = \\operatorname* { m i n } \\{ d _ { x } , d _ { y } \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 239, + 367, + 506, + 382 + ], + "score": 1.0, + "content": ", (2) is actually an unconstrained optimization problem. Note that", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 107, + 379, + 505, + 391 + ], + "spans": [ + { + "bbox": [ + 107, + 380, + 119, + 390 + ], + "score": 0.88, + "content": "L _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 119, + 379, + 219, + 391 + ], + "score": 1.0, + "content": "is a convex function of", + "type": "text" + }, + { + "bbox": [ + 220, + 380, + 228, + 389 + ], + "score": 0.82, + "content": "R", + "type": "inline_equation" + }, + { + "bbox": [ + 228, + 379, + 505, + 391 + ], + "score": 1.0, + "content": ", so any critical point is a global minimum. By differentiating and", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 390, + 442, + 402 + ], + "spans": [ + { + "bbox": [ + 105, + 390, + 442, + 402 + ], + "score": 1.0, + "content": "setting the derivative to zero, we can easily get the unique globally optimal solution", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 19, + "bbox_fs": [ + 105, + 367, + 506, + 402 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 257, + 402, + 353, + 417 + ], + "lines": [ + { + "bbox": [ + 257, + 402, + 353, + 417 + ], + "spans": [ + { + "bbox": [ + 257, + 402, + 353, + 417 + ], + "score": 0.92, + "content": "R ^ { * } = Y X ^ { T } ( X X ^ { T } ) ^ { - 1 } .", + "type": "interline_equation", + "image_path": "c8accd26cf813fed54b87edb2a46f770a126cb471396d3f7a0fdc6898698d916.jpg" + } + ] + } + ], + "index": 21, + "virtual_lines": [ + { + "bbox": [ + 257, + 402, + 353, + 417 + ], + "spans": [], + "index": 21 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 418, + 506, + 452 + ], + "lines": [ + { + "bbox": [ + 105, + 417, + 506, + 432 + ], + "spans": [ + { + "bbox": [ + 105, + 417, + 149, + 432 + ], + "score": 1.0, + "content": "In case of", + "type": "text" + }, + { + "bbox": [ + 149, + 418, + 221, + 430 + ], + "score": 0.93, + "content": "k < \\operatorname* { m i n } \\{ d _ { x } , d _ { y } \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 222, + 417, + 506, + 432 + ], + "score": 1.0, + "content": ", the problem becomes non-convex because of the rank constraint, but", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 429, + 505, + 442 + ], + "spans": [ + { + "bbox": [ + 106, + 429, + 505, + 442 + ], + "score": 1.0, + "content": "its exact solution can still be computed easily. We present the solution of this case as a proposition", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 440, + 343, + 452 + ], + "spans": [ + { + "bbox": [ + 105, + 440, + 343, + 452 + ], + "score": 1.0, + "content": "and defer the proof to Appendix C due to its technicalities.", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 23, + "bbox_fs": [ + 105, + 417, + 506, + 452 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 453, + 427, + 465 + ], + "lines": [ + { + "bbox": [ + 105, + 452, + 428, + 468 + ], + "spans": [ + { + "bbox": [ + 105, + 452, + 214, + 468 + ], + "score": 1.0, + "content": "Proposition 3.1. Suppose", + "type": "text" + }, + { + "bbox": [ + 214, + 453, + 284, + 466 + ], + "score": 0.93, + "content": "k < \\operatorname* { m i n } \\{ d _ { x } , d _ { y } \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 285, + 452, + 428, + 468 + ], + "score": 1.0, + "content": ". Then the optimal solution to (2) is", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 25, + "bbox_fs": [ + 105, + 452, + 428, + 468 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 246, + 467, + 364, + 482 + ], + "lines": [ + { + "bbox": [ + 246, + 467, + 364, + 482 + ], + "spans": [ + { + "bbox": [ + 246, + 467, + 364, + 482 + ], + "score": 0.91, + "content": "R ^ { * } = \\hat { U } \\hat { U } ^ { T } Y X ^ { T } ( X X ^ { T } ) ^ { - 1 } ,", + "type": "interline_equation", + "image_path": "d8b7753f67fc0ce5146b4ab402bcc5a916cbf5efe766f1c3764be20d2739de88.jpg" + } + ] + } + ], + "index": 26, + "virtual_lines": [ + { + "bbox": [ + 246, + 467, + 364, + 482 + ], + "spans": [], + "index": 26 + } + ] + }, + { + "type": "text", + "bbox": [ + 108, + 483, + 438, + 497 + ], + "lines": [ + { + "bbox": [ + 105, + 482, + 436, + 499 + ], + "spans": [ + { + "bbox": [ + 105, + 482, + 256, + 499 + ], + "score": 1.0, + "content": "which is the orthogonal projection of", + "type": "text" + }, + { + "bbox": [ + 257, + 484, + 324, + 497 + ], + "score": 0.93, + "content": "Y X ^ { T } ( X X ^ { T } ) ^ { - 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 324, + 482, + 427, + 499 + ], + "score": 1.0, + "content": "onto the column space of", + "type": "text" + }, + { + "bbox": [ + 428, + 483, + 436, + 495 + ], + "score": 0.81, + "content": "\\hat { U }", + "type": "inline_equation" + } + ], + "index": 27 + } + ], + "index": 27, + "bbox_fs": [ + 105, + 482, + 436, + 499 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 509, + 269, + 521 + ], + "lines": [ + { + "bbox": [ + 105, + 508, + 268, + 523 + ], + "spans": [ + { + "bbox": [ + 105, + 508, + 242, + 523 + ], + "score": 1.0, + "content": "3.2 PARTIAL DERIVATIVES OF", + "type": "text" + }, + { + "bbox": [ + 242, + 509, + 268, + 522 + ], + "score": 0.89, + "content": "L ( W )", + "type": "inline_equation" + } + ], + "index": 28 + } + ], + "index": 28 + }, + { + "type": "text", + "bbox": [ + 105, + 529, + 505, + 552 + ], + "lines": [ + { + "bbox": [ + 106, + 529, + 505, + 542 + ], + "spans": [ + { + "bbox": [ + 106, + 529, + 362, + 542 + ], + "score": 1.0, + "content": "By simple matrix calculus, we can calculate the derivatives of", + "type": "text" + }, + { + "bbox": [ + 362, + 530, + 388, + 542 + ], + "score": 0.92, + "content": "L ( W )", + "type": "inline_equation" + }, + { + "bbox": [ + 389, + 529, + 453, + 542 + ], + "score": 1.0, + "content": "with respect to", + "type": "text" + }, + { + "bbox": [ + 454, + 530, + 468, + 541 + ], + "score": 0.88, + "content": "W _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 468, + 529, + 487, + 542 + ], + "score": 1.0, + "content": ", for", + "type": "text" + }, + { + "bbox": [ + 487, + 531, + 505, + 541 + ], + "score": 0.84, + "content": "i =", + "type": "inline_equation" + } + ], + "index": 29 + }, + { + "bbox": [ + 107, + 541, + 497, + 554 + ], + "spans": [ + { + "bbox": [ + 107, + 541, + 161, + 552 + ], + "score": 0.91, + "content": "1 , \\ldots , H + 1", + "type": "inline_equation" + }, + { + "bbox": [ + 162, + 541, + 497, + 554 + ], + "score": 1.0, + "content": ". We present the result as the following lemma, and defer the details to Appendix C.", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 29.5, + "bbox_fs": [ + 106, + 529, + 505, + 554 + ] + }, + { + "type": "text", + "bbox": [ + 108, + 554, + 412, + 566 + ], + "lines": [ + { + "bbox": [ + 105, + 552, + 410, + 568 + ], + "spans": [ + { + "bbox": [ + 105, + 552, + 261, + 568 + ], + "score": 1.0, + "content": "Lemma 3.2. The partial derivative of", + "type": "text" + }, + { + "bbox": [ + 262, + 554, + 288, + 566 + ], + "score": 0.9, + "content": "L ( W )", + "type": "inline_equation" + }, + { + "bbox": [ + 288, + 552, + 349, + 568 + ], + "score": 1.0, + "content": "with respect to", + "type": "text" + }, + { + "bbox": [ + 350, + 554, + 363, + 565 + ], + "score": 0.89, + "content": "W _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 364, + 552, + 410, + 568 + ], + "score": 1.0, + "content": "is given as", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 31, + "bbox_fs": [ + 105, + 552, + 410, + 568 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 163, + 567, + 447, + 592 + ], + "lines": [ + { + "bbox": [ + 163, + 567, + 447, + 592 + ], + "spans": [ + { + "bbox": [ + 163, + 567, + 447, + 592 + ], + "score": 0.87, + "content": "\\frac { \\partial L } { \\partial W _ { i } } = W _ { i + 1 } ^ { T } \\cdot \\cdot \\cdot W _ { H + 1 } ^ { T } ( W _ { H + 1 } W _ { H } \\cdot \\cdot \\cdot W _ { 1 } X - Y ) X ^ { T } W _ { 1 } ^ { T } \\cdot \\cdot \\cdot W _ { i - 1 } ^ { T } ,", + "type": "interline_equation", + "image_path": "fe5cff211b7e77242b6d945873020a79ecf87627040fc811c8b70a9b4862375a.jpg" + } + ] + } + ], + "index": 32, + "virtual_lines": [ + { + "bbox": [ + 163, + 567, + 447, + 592 + ], + "spans": [], + "index": 32 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 592, + 194, + 605 + ], + "lines": [ + { + "bbox": [ + 105, + 591, + 192, + 606 + ], + "spans": [ + { + "bbox": [ + 105, + 591, + 120, + 606 + ], + "score": 1.0, + "content": "for", + "type": "text" + }, + { + "bbox": [ + 121, + 593, + 192, + 604 + ], + "score": 0.83, + "content": "i = 1 , \\ldots , H + 1", + "type": "inline_equation" + } + ], + "index": 33 + } + ], + "index": 33, + "bbox_fs": [ + 105, + 591, + 192, + 606 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 612, + 504, + 646 + ], + "lines": [ + { + "bbox": [ + 105, + 610, + 506, + 625 + ], + "spans": [ + { + "bbox": [ + 105, + 610, + 506, + 625 + ], + "score": 1.0, + "content": "This result will be used throughout the proof of Theorems 2.1 and 2.2. For clarity in notation, note", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 150, + 620, + 501, + 652 + ], + "spans": [ + { + "bbox": [ + 150, + 623, + 175, + 633 + ], + "score": 0.84, + "content": "i = 1", + "type": "inline_equation" + }, + { + "bbox": [ + 181, + 622, + 230, + 635 + ], + "score": 0.87, + "content": "W _ { 1 } ^ { T } \\cdots W _ { 0 } ^ { \\breve { T } }", + "type": "inline_equation" + }, + { + "bbox": [ + 231, + 620, + 271, + 652 + ], + "score": 1.0, + "content": "is just an matrix in", + "type": "text" + }, + { + "bbox": [ + 303, + 620, + 347, + 652 + ], + "score": 1.0, + "content": "matrix in .", + "type": "text" + }, + { + "bbox": [ + 347, + 622, + 378, + 633 + ], + "score": 0.91, + "content": "\\mathbb { R } ^ { d _ { x } \\times d _ { x } }", + "type": "inline_equation" + }, + { + "bbox": [ + 379, + 620, + 453, + 652 + ], + "score": 1.0, + "content": ". Similarly, when", + "type": "text" + }, + { + "bbox": [ + 453, + 623, + 501, + 633 + ], + "score": 0.9, + "content": "i = H + 1", + "type": "inline_equation" + } + ], + "index": 36 + }, + { + "bbox": [ + 107, + 633, + 302, + 647 + ], + "spans": [ + { + "bbox": [ + 107, + 633, + 177, + 647 + ], + "score": 0.93, + "content": "W _ { H + 2 } ^ { T } \\cdot \\cdot \\cdot W _ { H + 1 } ^ { T }", + "type": "inline_equation" + }, + { + "bbox": [ + 271, + 633, + 302, + 644 + ], + "score": 0.91, + "content": "\\mathbb { R } ^ { d _ { y } \\times d _ { y } ^ { * } }", + "type": "inline_equation" + } + ], + "index": 35 + } + ], + "index": 35, + "bbox_fs": [ + 105, + 610, + 506, + 652 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 650, + 504, + 673 + ], + "lines": [ + { + "bbox": [ + 106, + 649, + 506, + 663 + ], + "spans": [ + { + "bbox": [ + 106, + 649, + 506, + 663 + ], + "score": 1.0, + "content": "We also state an elementary lemma which proves useful in our proofs, whose proof we defer to", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 660, + 161, + 675 + ], + "spans": [ + { + "bbox": [ + 105, + 660, + 161, + 675 + ], + "score": 1.0, + "content": "Appendix C.", + "type": "text" + } + ], + "index": 38 + } + ], + "index": 37.5, + "bbox_fs": [ + 105, + 649, + 506, + 675 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 674, + 376, + 686 + ], + "lines": [ + { + "bbox": [ + 104, + 671, + 376, + 690 + ], + "spans": [ + { + "bbox": [ + 104, + 671, + 205, + 690 + ], + "score": 1.0, + "content": "Lemma 3.3. 1. For any", + "type": "text" + }, + { + "bbox": [ + 206, + 675, + 252, + 685 + ], + "score": 0.91, + "content": "A \\in \\mathbb { R } ^ { m \\times n }", + "type": "inline_equation" + }, + { + "bbox": [ + 252, + 671, + 271, + 690 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 271, + 674, + 314, + 685 + ], + "score": 0.83, + "content": "\\boldsymbol { B } \\in \\mathbb { R } ^ { n \\times l }", + "type": "inline_equation" + }, + { + "bbox": [ + 314, + 671, + 342, + 690 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 342, + 675, + 371, + 686 + ], + "score": 0.4, + "content": "m \\geq n ,", + "type": "inline_equation" + }, + { + "bbox": [ + 372, + 671, + 376, + 690 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 39 + } + ], + "index": 39, + "bbox_fs": [ + 104, + 671, + 376, + 690 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 250, + 688, + 371, + 703 + ], + "lines": [ + { + "bbox": [ + 250, + 688, + 371, + 703 + ], + "spans": [ + { + "bbox": [ + 250, + 688, + 371, + 703 + ], + "score": 0.64, + "content": "\\left\\| A B \\right\\| _ { \\mathrm { F } } ^ { 2 } \\geq \\sigma _ { \\operatorname* { m i n } } ^ { 2 } ( A ) \\left\\| B \\right\\| _ { \\mathrm { F } } ^ { 2 } .", + "type": "interline_equation", + "image_path": "1aabd83d43aae35f20466257db4d0b1ad4b21ea7821b44dd69c28e82b8b4ee6e.jpg" + } + ] + } + ], + "index": 40, + "virtual_lines": [ + { + "bbox": [ + 250, + 688, + 371, + 703 + ], + "spans": [], + "index": 40 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 705, + 313, + 717 + ], + "lines": [ + { + "bbox": [ + 104, + 703, + 310, + 719 + ], + "spans": [ + { + "bbox": [ + 104, + 703, + 150, + 719 + ], + "score": 1.0, + "content": "2. For any", + "type": "text" + }, + { + "bbox": [ + 151, + 705, + 197, + 716 + ], + "score": 0.91, + "content": "A \\in \\mathbb { R } ^ { m \\times n }", + "type": "inline_equation" + }, + { + "bbox": [ + 197, + 703, + 216, + 719 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 216, + 704, + 258, + 716 + ], + "score": 0.88, + "content": "\\boldsymbol { B } \\in \\mathbb { R } ^ { n \\times l }", + "type": "inline_equation" + }, + { + "bbox": [ + 259, + 703, + 286, + 719 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 287, + 705, + 310, + 716 + ], + "score": 0.81, + "content": "n \\leq l", + "type": "inline_equation" + } + ], + "index": 41 + } + ], + "index": 41, + "bbox_fs": [ + 104, + 703, + 310, + 719 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 250, + 718, + 360, + 734 + ], + "lines": [ + { + "bbox": [ + 250, + 718, + 360, + 734 + ], + "spans": [ + { + "bbox": [ + 250, + 718, + 360, + 734 + ], + "score": 0.82, + "content": "\\left\\| A B \\right\\| _ { \\mathrm { F } } ^ { 2 } \\geq \\sigma _ { \\operatorname* { m i n } } ^ { 2 } ( B ) \\left\\| A \\right\\| _ { \\mathrm { F } } ^ { 2 } .", + "type": "interline_equation", + "image_path": "3736bab224c123ec71b9d5071d84030d73c45e344f6de9daa941150abcc92504.jpg" + } + ] + } + ], + "index": 42, + "virtual_lines": [ + { + "bbox": [ + 250, + 718, + 360, + 734 + ], + "spans": [], + "index": 42 + } + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "title", + "bbox": [ + 107, + 82, + 236, + 93 + ], + "lines": [ + { + "bbox": [ + 106, + 81, + 237, + 95 + ], + "spans": [ + { + "bbox": [ + 106, + 81, + 237, + 95 + ], + "score": 1.0, + "content": "3.3 PROOF OF THEOREM 2.1", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "text", + "bbox": [ + 105, + 102, + 505, + 126 + ], + "lines": [ + { + "bbox": [ + 105, + 101, + 506, + 116 + ], + "spans": [ + { + "bbox": [ + 105, + 101, + 304, + 116 + ], + "score": 1.0, + "content": "We prove Theorem 2.1, which addresses the case", + "type": "text" + }, + { + "bbox": [ + 304, + 103, + 375, + 115 + ], + "score": 0.93, + "content": "k = \\operatorname* { m i n } \\{ d _ { x } , d _ { y } \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 376, + 101, + 506, + 116 + ], + "score": 1.0, + "content": ". First, recall that the set defined", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 114, + 180, + 126 + ], + "spans": [ + { + "bbox": [ + 105, + 114, + 180, + 126 + ], + "score": 1.0, + "content": "in Theorem 2.1 is", + "type": "text" + } + ], + "index": 2 + } + ], + "index": 1.5 + }, + { + "type": "interline_equation", + "bbox": [ + 192, + 126, + 416, + 140 + ], + "lines": [ + { + "bbox": [ + 192, + 126, + 416, + 140 + ], + "spans": [ + { + "bbox": [ + 192, + 126, + 416, + 140 + ], + "score": 0.79, + "content": "\\mathcal { V } _ { 1 } : = \\left\\{ ( W _ { 1 } , \\ldots , W _ { H + 1 } ) : \\operatorname { r a n k } ( W _ { H + 1 } \\cdot \\cdot \\cdot W _ { 1 } ) = k \\right\\} .", + "type": "interline_equation", + "image_path": "22f37a06b5e57e58268a3726f1418e0c705e79519fba482e28d44631f77f280a.jpg" + } + ] + } + ], + "index": 3, + "virtual_lines": [ + { + "bbox": [ + 192, + 126, + 416, + 140 + ], + "spans": [], + "index": 3 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 141, + 504, + 174 + ], + "lines": [ + { + "bbox": [ + 105, + 140, + 505, + 153 + ], + "spans": [ + { + "bbox": [ + 105, + 140, + 289, + 153 + ], + "score": 1.0, + "content": "As seen in (3), the unique minimum point of", + "type": "text" + }, + { + "bbox": [ + 289, + 141, + 302, + 152 + ], + "score": 0.87, + "content": "L _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 302, + 140, + 339, + 153 + ], + "score": 1.0, + "content": "has rank", + "type": "text" + }, + { + "bbox": [ + 340, + 141, + 346, + 151 + ], + "score": 0.78, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 347, + 140, + 403, + 153 + ], + "score": 1.0, + "content": ". So, no point", + "type": "text" + }, + { + "bbox": [ + 404, + 141, + 440, + 153 + ], + "score": 0.91, + "content": "W \\in \\mathcal { V } _ { 1 } ^ { c }", + "type": "inline_equation" + }, + { + "bbox": [ + 441, + 140, + 505, + 153 + ], + "score": 1.0, + "content": "can be a global", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 151, + 505, + 164 + ], + "spans": [ + { + "bbox": [ + 105, + 151, + 160, + 164 + ], + "score": 1.0, + "content": "minimum of", + "type": "text" + }, + { + "bbox": [ + 160, + 153, + 168, + 162 + ], + "score": 0.82, + "content": "L", + "type": "inline_equation" + }, + { + "bbox": [ + 168, + 151, + 505, + 164 + ], + "score": 1.0, + "content": ". Therefore, by Kawaguchi (2016, Theorem 2.3.(iii)) and Lemma B.1, any critical", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 162, + 247, + 176 + ], + "spans": [ + { + "bbox": [ + 105, + 162, + 140, + 176 + ], + "score": 1.0, + "content": "point in", + "type": "text" + }, + { + "bbox": [ + 140, + 163, + 152, + 174 + ], + "score": 0.89, + "content": "\\mathcal { V } _ { 1 } ^ { c }", + "type": "inline_equation" + }, + { + "bbox": [ + 153, + 162, + 247, + 176 + ], + "score": 1.0, + "content": "must be a saddle point.", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 5 + }, + { + "type": "text", + "bbox": [ + 106, + 179, + 505, + 235 + ], + "lines": [ + { + "bbox": [ + 105, + 179, + 505, + 192 + ], + "spans": [ + { + "bbox": [ + 105, + 179, + 338, + 192 + ], + "score": 1.0, + "content": "For the rest of our proof, we need to consider two cases:", + "type": "text" + }, + { + "bbox": [ + 338, + 180, + 374, + 192 + ], + "score": 0.93, + "content": "d _ { y } \\leq d _ { x }", + "type": "inline_equation" + }, + { + "bbox": [ + 375, + 179, + 393, + 192 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 393, + 180, + 429, + 192 + ], + "score": 0.91, + "content": "d _ { x } \\leq d _ { y }", + "type": "inline_equation" + }, + { + "bbox": [ + 430, + 179, + 444, + 192 + ], + "score": 1.0, + "content": ". If", + "type": "text" + }, + { + "bbox": [ + 444, + 180, + 480, + 192 + ], + "score": 0.92, + "content": "d _ { x } = d _ { y }", + "type": "inline_equation" + }, + { + "bbox": [ + 480, + 179, + 505, + 192 + ], + "score": 1.0, + "content": ", both", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 191, + 505, + 203 + ], + "spans": [ + { + "bbox": [ + 105, + 191, + 397, + 203 + ], + "score": 1.0, + "content": "cases work. The outline of the proof is as follows: we define a new set", + "type": "text" + }, + { + "bbox": [ + 397, + 191, + 412, + 201 + ], + "score": 0.89, + "content": "\\mathcal { W } _ { \\epsilon }", + "type": "inline_equation" + }, + { + "bbox": [ + 412, + 191, + 505, + 203 + ], + "score": 1.0, + "content": ", show that any critical", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 200, + 506, + 215 + ], + "spans": [ + { + "bbox": [ + 105, + 200, + 168, + 215 + ], + "score": 1.0, + "content": "point in the set", + "type": "text" + }, + { + "bbox": [ + 169, + 202, + 183, + 213 + ], + "score": 0.9, + "content": "\\mathcal { W } _ { \\epsilon }", + "type": "inline_equation" + }, + { + "bbox": [ + 184, + 200, + 375, + 215 + ], + "score": 1.0, + "content": "is a global minimum, and then show that every", + "type": "text" + }, + { + "bbox": [ + 376, + 202, + 411, + 213 + ], + "score": 0.92, + "content": "W \\in \\mathcal { V } _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 411, + 200, + 451, + 215 + ], + "score": 1.0, + "content": "is also in", + "type": "text" + }, + { + "bbox": [ + 451, + 202, + 465, + 213 + ], + "score": 0.9, + "content": "\\mathcal { W } _ { \\epsilon }", + "type": "inline_equation" + }, + { + "bbox": [ + 466, + 200, + 506, + 215 + ], + "score": 1.0, + "content": "for some", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 106, + 212, + 505, + 226 + ], + "spans": [ + { + "bbox": [ + 106, + 213, + 133, + 223 + ], + "score": 0.88, + "content": "\\epsilon > 0", + "type": "inline_equation" + }, + { + "bbox": [ + 133, + 212, + 293, + 226 + ], + "score": 1.0, + "content": ". This proves that any critical point of", + "type": "text" + }, + { + "bbox": [ + 293, + 212, + 319, + 225 + ], + "score": 0.93, + "content": "L ( W )", + "type": "inline_equation" + }, + { + "bbox": [ + 320, + 212, + 332, + 226 + ], + "score": 1.0, + "content": "in", + "type": "text" + }, + { + "bbox": [ + 333, + 213, + 344, + 223 + ], + "score": 0.89, + "content": "\\nu _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 345, + 212, + 449, + 226 + ], + "score": 1.0, + "content": "is also a critical point in", + "type": "text" + }, + { + "bbox": [ + 450, + 213, + 464, + 223 + ], + "score": 0.89, + "content": "\\mathcal { W } _ { \\epsilon }", + "type": "inline_equation" + }, + { + "bbox": [ + 465, + 212, + 505, + 226 + ], + "score": 1.0, + "content": "for some", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 224, + 237, + 236 + ], + "spans": [ + { + "bbox": [ + 106, + 224, + 130, + 234 + ], + "score": 0.86, + "content": "\\epsilon > 0", + "type": "inline_equation" + }, + { + "bbox": [ + 130, + 224, + 237, + 236 + ], + "score": 1.0, + "content": ", hence a global minimum.", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 9 + }, + { + "type": "text", + "bbox": [ + 108, + 240, + 294, + 252 + ], + "lines": [ + { + "bbox": [ + 105, + 238, + 296, + 254 + ], + "spans": [ + { + "bbox": [ + 105, + 238, + 296, + 254 + ], + "score": 1.0, + "content": "The following proposition proves the first step:", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 12 + }, + { + "type": "text", + "bbox": [ + 106, + 253, + 462, + 266 + ], + "lines": [ + { + "bbox": [ + 105, + 252, + 463, + 268 + ], + "spans": [ + { + "bbox": [ + 105, + 252, + 229, + 268 + ], + "score": 1.0, + "content": "Proposition 3.4. Assume that", + "type": "text" + }, + { + "bbox": [ + 229, + 253, + 300, + 266 + ], + "score": 0.93, + "content": "k = \\operatorname* { m i n } \\{ d _ { x } , d _ { y } \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 300, + 252, + 338, + 268 + ], + "score": 1.0, + "content": ". For any", + "type": "text" + }, + { + "bbox": [ + 338, + 254, + 361, + 264 + ], + "score": 0.88, + "content": "\\epsilon > 0", + "type": "inline_equation" + }, + { + "bbox": [ + 362, + 252, + 463, + 268 + ], + "score": 1.0, + "content": ", define the following set:", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 13 + }, + { + "type": "interline_equation", + "bbox": [ + 160, + 268, + 449, + 295 + ], + "lines": [ + { + "bbox": [ + 160, + 268, + 449, + 295 + ], + "spans": [ + { + "bbox": [ + 160, + 268, + 449, + 295 + ], + "score": 0.91, + "content": "\\mathcal { W } _ { \\epsilon } : = \\left\\{ \\begin{array} { l l } { \\{ ( W _ { 1 } , \\dots , W _ { H + 1 } ) : \\sigma _ { \\operatorname* { m i n } } ( W _ { H + 1 } \\cdot \\cdot \\cdot W _ { 2 } ) \\geq \\epsilon \\} , } & { \\mathrm { i f ~ } d _ { y } \\leq d _ { x } , } \\\\ { \\{ ( W _ { 1 } , \\dots , W _ { H + 1 } ) : \\sigma _ { \\operatorname* { m i n } } ( W _ { H } \\cdot \\cdot \\cdot W _ { 1 } ) \\geq \\epsilon \\} , } & { \\mathrm { i f ~ } d _ { x } \\leq d _ { y } . } \\end{array} \\right.", + "type": "interline_equation", + "image_path": "a5c01d59a18b54b64d8e3533acef12e3053e575a03c0a3b181ca09627ef55889.jpg" + } + ] + } + ], + "index": 14, + "virtual_lines": [ + { + "bbox": [ + 160, + 268, + 449, + 295 + ], + "spans": [], + "index": 14 + } + ] + }, + { + "type": "text", + "bbox": [ + 108, + 296, + 372, + 309 + ], + "lines": [ + { + "bbox": [ + 106, + 295, + 374, + 310 + ], + "spans": [ + { + "bbox": [ + 106, + 295, + 210, + 310 + ], + "score": 1.0, + "content": "Then any critical point of", + "type": "text" + }, + { + "bbox": [ + 210, + 297, + 236, + 309 + ], + "score": 0.9, + "content": "L ( W )", + "type": "inline_equation" + }, + { + "bbox": [ + 237, + 295, + 248, + 310 + ], + "score": 1.0, + "content": "in", + "type": "text" + }, + { + "bbox": [ + 248, + 297, + 263, + 308 + ], + "score": 0.87, + "content": "\\mathcal { W } _ { \\epsilon }", + "type": "inline_equation" + }, + { + "bbox": [ + 263, + 295, + 374, + 310 + ], + "score": 1.0, + "content": "is a global minimum point.", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 15 + }, + { + "type": "text", + "bbox": [ + 106, + 319, + 503, + 343 + ], + "lines": [ + { + "bbox": [ + 102, + 316, + 503, + 336 + ], + "spans": [ + { + "bbox": [ + 102, + 316, + 148, + 336 + ], + "score": 1.0, + "content": "Proof. (If", + "type": "text" + }, + { + "bbox": [ + 149, + 320, + 184, + 332 + ], + "score": 0.89, + "content": "d _ { y } \\leq d _ { x } ,", + "type": "inline_equation" + }, + { + "bbox": [ + 184, + 316, + 293, + 336 + ], + "score": 1.0, + "content": ") Consider (5) in the case of", + "type": "text" + }, + { + "bbox": [ + 293, + 321, + 316, + 330 + ], + "score": 0.89, + "content": "i = 1", + "type": "inline_equation" + }, + { + "bbox": [ + 316, + 316, + 401, + 336 + ], + "score": 1.0, + "content": ". We can observe that", + "type": "text" + }, + { + "bbox": [ + 401, + 319, + 503, + 333 + ], + "score": 0.92, + "content": "W _ { 2 } ^ { T } \\cdot \\cdot \\cdot W _ { H + 1 } ^ { T } \\in \\mathbb { R } ^ { d _ { 1 } \\times d _ { y } }", + "type": "inline_equation" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 331, + 273, + 344 + ], + "spans": [ + { + "bbox": [ + 106, + 331, + 141, + 344 + ], + "score": 1.0, + "content": "and that", + "type": "text" + }, + { + "bbox": [ + 141, + 332, + 174, + 344 + ], + "score": 0.92, + "content": "d _ { 1 } \\geq d _ { y }", + "type": "inline_equation" + }, + { + "bbox": [ + 175, + 331, + 273, + 344 + ], + "score": 1.0, + "content": ". Then by Lemma 3.3.1,", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 16.5 + }, + { + "type": "interline_equation", + "bbox": [ + 163, + 344, + 447, + 393 + ], + "lines": [ + { + "bbox": [ + 163, + 344, + 447, + 393 + ], + "spans": [ + { + "bbox": [ + 163, + 344, + 447, + 393 + ], + "score": 0.93, + "content": "\\begin{array} { r l } { { \\| \\frac { \\partial L } { \\partial W _ { 1 } } \\| _ { \\mathrm { F } } ^ { 2 } \\geq \\sigma _ { \\operatorname* { m i n } } ^ { 2 } ( W _ { H + 1 } \\cdot \\cdot \\cdot W _ { 2 } ) \\| ( W _ { H + 1 } W _ { H } \\cdot \\cdot \\cdot W _ { 1 } X - Y ) X ^ { T } \\| _ { \\mathrm { F } } ^ { 2 } } } \\\\ & { \\geq \\epsilon ^ { 2 } \\| ( W _ { H + 1 } W _ { H } \\cdot \\cdot \\cdot W _ { 1 } X - Y ) X ^ { T } \\| _ { \\mathrm { F } } ^ { 2 } . } \\end{array}", + "type": "interline_equation", + "image_path": "203cbd0fb05cb2afb06a29343f45f7f23840efc5fae5b197db66f36fa3125896.jpg" + } + ] + } + ], + "index": 19, + "virtual_lines": [ + { + "bbox": [ + 163, + 344, + 447, + 360.3333333333333 + ], + "spans": [], + "index": 18 + }, + { + "bbox": [ + 163, + 360.3333333333333, + 447, + 376.66666666666663 + ], + "spans": [], + "index": 19 + }, + { + "bbox": [ + 163, + 376.66666666666663, + 447, + 392.99999999999994 + ], + "spans": [], + "index": 20 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 392, + 332, + 404 + ], + "lines": [ + { + "bbox": [ + 106, + 391, + 333, + 406 + ], + "spans": [ + { + "bbox": [ + 106, + 391, + 285, + 406 + ], + "score": 1.0, + "content": "By the above inequality, any critical point in", + "type": "text" + }, + { + "bbox": [ + 286, + 393, + 298, + 402 + ], + "score": 0.8, + "content": "\\mathcal { W }", + "type": "inline_equation" + }, + { + "bbox": [ + 298, + 391, + 333, + 406 + ], + "score": 1.0, + "content": "satisfies", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 21 + }, + { + "type": "interline_equation", + "bbox": [ + 196, + 405, + 413, + 430 + ], + "lines": [ + { + "bbox": [ + 196, + 405, + 413, + 430 + ], + "spans": [ + { + "bbox": [ + 196, + 405, + 413, + 430 + ], + "score": 0.88, + "content": "\\forall i , \\frac { \\partial L } { \\partial W _ { i } } = 0 \\Rightarrow ( W _ { H + 1 } W _ { H } \\cdot \\cdot \\cdot W _ { 1 } X - Y ) X ^ { T } = 0 ,", + "type": "interline_equation", + "image_path": "19b888bcd619523f06ef7c9ea0c0d8a5616a7d1822d9b24744b68284bd9e67bf.jpg" + } + ] + } + ], + "index": 22, + "virtual_lines": [ + { + "bbox": [ + 196, + 405, + 413, + 430 + ], + "spans": [], + "index": 22 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 430, + 504, + 454 + ], + "lines": [ + { + "bbox": [ + 105, + 429, + 505, + 444 + ], + "spans": [ + { + "bbox": [ + 105, + 429, + 178, + 444 + ], + "score": 1.0, + "content": "which means that", + "type": "text" + }, + { + "bbox": [ + 178, + 431, + 330, + 443 + ], + "score": 0.9, + "content": "W _ { H + 1 } W _ { H } \\cdot \\cdot \\cdot W _ { 1 } = Y X ^ { T } ( X X ^ { T } ) ^ { - 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 331, + 429, + 505, + 444 + ], + "score": 1.0, + "content": ". The product is the unique globally optimal", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 442, + 430, + 455 + ], + "spans": [ + { + "bbox": [ + 105, + 442, + 286, + 455 + ], + "score": 1.0, + "content": "solution (3) of the relaxed problem in (2), so", + "type": "text" + }, + { + "bbox": [ + 286, + 444, + 298, + 452 + ], + "score": 0.78, + "content": "W", + "type": "inline_equation" + }, + { + "bbox": [ + 298, + 442, + 418, + 455 + ], + "score": 1.0, + "content": "is a global minimum point of", + "type": "text" + }, + { + "bbox": [ + 418, + 443, + 426, + 452 + ], + "score": 0.82, + "content": "L", + "type": "inline_equation" + }, + { + "bbox": [ + 426, + 442, + 430, + 455 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 23.5 + }, + { + "type": "text", + "bbox": [ + 108, + 458, + 504, + 482 + ], + "lines": [ + { + "bbox": [ + 105, + 456, + 506, + 472 + ], + "spans": [ + { + "bbox": [ + 105, + 456, + 119, + 472 + ], + "score": 1.0, + "content": "(If", + "type": "text" + }, + { + "bbox": [ + 120, + 459, + 158, + 471 + ], + "score": 0.87, + "content": "d _ { x } \\leq d _ { y } ,", + "type": "inline_equation" + }, + { + "bbox": [ + 158, + 456, + 231, + 472 + ], + "score": 1.0, + "content": ") Consider (5) for", + "type": "text" + }, + { + "bbox": [ + 231, + 460, + 278, + 470 + ], + "score": 0.9, + "content": "i = H + 1", + "type": "inline_equation" + }, + { + "bbox": [ + 279, + 456, + 370, + 472 + ], + "score": 1.0, + "content": ". We can observe that", + "type": "text" + }, + { + "bbox": [ + 370, + 458, + 468, + 471 + ], + "score": 0.93, + "content": "W _ { 1 } ^ { T } \\cdot \\cdot \\cdot W _ { H } ^ { T } \\in \\mathbb { R } ^ { d _ { x } \\times d _ { H } }", + "type": "inline_equation" + }, + { + "bbox": [ + 468, + 456, + 506, + 472 + ], + "score": 1.0, + "content": "and that", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 107, + 470, + 242, + 483 + ], + "spans": [ + { + "bbox": [ + 107, + 471, + 144, + 482 + ], + "score": 0.91, + "content": "d _ { x } \\leq d _ { H }", + "type": "inline_equation" + }, + { + "bbox": [ + 144, + 470, + 242, + 483 + ], + "score": 1.0, + "content": ". Then by Lemma 3.3.2,", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 25.5 + }, + { + "type": "interline_equation", + "bbox": [ + 192, + 483, + 417, + 512 + ], + "lines": [ + { + "bbox": [ + 192, + 483, + 417, + 512 + ], + "spans": [ + { + "bbox": [ + 192, + 483, + 417, + 512 + ], + "score": 0.92, + "content": "\\left\\| \\frac { \\partial \\cal { L } } { \\partial W _ { H + 1 } } \\right\\| _ { \\mathrm { F } } ^ { 2 } \\geq \\epsilon ^ { 2 } \\left\\| ( W _ { H + 1 } W _ { H } \\cdot \\cdot \\cdot W _ { 1 } X - Y ) X ^ { T } \\right\\| _ { \\mathrm { F } } ^ { 2 } ,", + "type": "interline_equation", + "image_path": "d89b611db20a0dbe8e98cccf93c8394749acc2ac9cefc384594ca4193c5530ae.jpg" + } + ] + } + ], + "index": 27.5, + "virtual_lines": [ + { + "bbox": [ + 192, + 483, + 417, + 497.5 + ], + "spans": [], + "index": 27 + }, + { + "bbox": [ + 192, + 497.5, + 417, + 512.0 + ], + "spans": [], + "index": 28 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 513, + 377, + 524 + ], + "lines": [ + { + "bbox": [ + 105, + 511, + 379, + 526 + ], + "spans": [ + { + "bbox": [ + 105, + 511, + 379, + 526 + ], + "score": 1.0, + "content": "and the rest of the proof flows in a similar way as the previous case.", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 29 + }, + { + "type": "text", + "bbox": [ + 107, + 536, + 272, + 547 + ], + "lines": [ + { + "bbox": [ + 106, + 535, + 273, + 549 + ], + "spans": [ + { + "bbox": [ + 106, + 535, + 273, + 549 + ], + "score": 1.0, + "content": "The next proposition proves the theorem:", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 30 + }, + { + "type": "text", + "bbox": [ + 109, + 549, + 435, + 561 + ], + "lines": [ + { + "bbox": [ + 106, + 548, + 433, + 563 + ], + "spans": [ + { + "bbox": [ + 106, + 548, + 234, + 563 + ], + "score": 1.0, + "content": "Proposition 3.5. For any point", + "type": "text" + }, + { + "bbox": [ + 235, + 549, + 268, + 560 + ], + "score": 0.92, + "content": "W \\in \\nu _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 269, + 548, + 332, + 563 + ], + "score": 1.0, + "content": ", there exists an", + "type": "text" + }, + { + "bbox": [ + 332, + 550, + 356, + 559 + ], + "score": 0.88, + "content": "\\epsilon > 0", + "type": "inline_equation" + }, + { + "bbox": [ + 356, + 548, + 396, + 563 + ], + "score": 1.0, + "content": "such that", + "type": "text" + }, + { + "bbox": [ + 396, + 549, + 433, + 560 + ], + "score": 0.9, + "content": "W \\in \\mathcal { W } _ { \\epsilon }", + "type": "inline_equation" + } + ], + "index": 31 + } + ], + "index": 31 + }, + { + "type": "text", + "bbox": [ + 106, + 572, + 373, + 584 + ], + "lines": [ + { + "bbox": [ + 105, + 571, + 374, + 586 + ], + "spans": [ + { + "bbox": [ + 105, + 571, + 204, + 586 + ], + "score": 1.0, + "content": "Proof. Define a new set", + "type": "text" + }, + { + "bbox": [ + 205, + 573, + 216, + 583 + ], + "score": 0.83, + "content": "\\mathcal { W }", + "type": "inline_equation" + }, + { + "bbox": [ + 217, + 571, + 303, + 586 + ], + "score": 1.0, + "content": ", a “limit” version (as", + "type": "text" + }, + { + "bbox": [ + 303, + 573, + 329, + 583 + ], + "score": 0.89, + "content": "\\epsilon 0", + "type": "inline_equation" + }, + { + "bbox": [ + 330, + 571, + 344, + 586 + ], + "score": 1.0, + "content": ") of", + "type": "text" + }, + { + "bbox": [ + 344, + 573, + 359, + 583 + ], + "score": 0.89, + "content": "\\mathcal { W } _ { \\epsilon }", + "type": "inline_equation" + }, + { + "bbox": [ + 359, + 571, + 374, + 586 + ], + "score": 1.0, + "content": ", as", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 32 + }, + { + "type": "interline_equation", + "bbox": [ + 159, + 585, + 451, + 613 + ], + "lines": [ + { + "bbox": [ + 159, + 585, + 451, + 613 + ], + "spans": [ + { + "bbox": [ + 159, + 585, + 451, + 613 + ], + "score": 0.92, + "content": "\\begin{array} { r } { \\mathcal { W } : = \\{ \\{ ( W _ { 1 } , \\ldots , W _ { H + 1 } ) : \\mathrm { r a n k } ( W _ { H + 1 } \\cdot \\cdot \\cdot W _ { 2 } ) = d _ { y } \\} , \\mathrm { i f } \\ d _ { y } \\leq d _ { x } , } \\\\ { \\{ ( W _ { 1 } , \\ldots , W _ { H + 1 } ) : \\mathrm { r a n k } ( W _ { H } \\cdot \\cdot \\cdot W _ { 1 } ) = d _ { x } \\} , \\mathrm { i f } \\ d _ { x } \\leq d _ { y } . } \\end{array}", + "type": "interline_equation", + "image_path": "53917d40e08378e95d794b17cfba85e08f2d7177c755a23bbd12625a79647095.jpg" + } + ] + } + ], + "index": 33, + "virtual_lines": [ + { + "bbox": [ + 159, + 585, + 451, + 613 + ], + "spans": [], + "index": 33 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 614, + 349, + 626 + ], + "lines": [ + { + "bbox": [ + 106, + 613, + 349, + 627 + ], + "spans": [ + { + "bbox": [ + 106, + 613, + 163, + 627 + ], + "score": 1.0, + "content": "We show that", + "type": "text" + }, + { + "bbox": [ + 163, + 614, + 199, + 626 + ], + "score": 0.92, + "content": "\\nu _ { 1 } \\subset \\mathcal { W }", + "type": "inline_equation" + }, + { + "bbox": [ + 199, + 613, + 265, + 627 + ], + "score": 1.0, + "content": "by showing that", + "type": "text" + }, + { + "bbox": [ + 266, + 614, + 306, + 626 + ], + "score": 0.92, + "content": "\\mathcal { W } ^ { c } \\subset \\mathcal { V } _ { 1 } ^ { c }", + "type": "inline_equation" + }, + { + "bbox": [ + 306, + 613, + 349, + 627 + ], + "score": 1.0, + "content": ". Consider", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 34 + }, + { + "type": "interline_equation", + "bbox": [ + 158, + 628, + 452, + 655 + ], + "lines": [ + { + "bbox": [ + 158, + 628, + 452, + 655 + ], + "spans": [ + { + "bbox": [ + 158, + 628, + 452, + 655 + ], + "score": 0.89, + "content": "\\mathcal { W } ^ { c } = \\left\\{ \\begin{array} { l l } { \\{ ( W _ { 1 } , \\ldots , W _ { H + 1 } ) : \\operatorname { r a n k } ( W _ { H + 1 } \\cdot \\cdot \\cdot W _ { 2 } ) < d _ { y } \\} , } & { \\mathrm { i f ~ } d _ { y } \\leq d _ { x } , } \\\\ { \\{ ( W _ { 1 } , \\ldots , W _ { H + 1 } ) : \\operatorname { r a n k } ( W _ { H } \\cdot \\cdot \\cdot W _ { 1 } ) < d _ { x } \\} , } & { \\mathrm { i f ~ } d _ { x } \\leq d _ { y } . } \\end{array} \\right.", + "type": "interline_equation", + "image_path": "34aa988edfa931aa84a1b1ae46d068bec887b962474ac1136374521ee91f3f1a.jpg" + } + ] + } + ], + "index": 35, + "virtual_lines": [ + { + "bbox": [ + 158, + 628, + 452, + 655 + ], + "spans": [], + "index": 35 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 657, + 504, + 691 + ], + "lines": [ + { + "bbox": [ + 105, + 655, + 506, + 671 + ], + "spans": [ + { + "bbox": [ + 105, + 655, + 147, + 671 + ], + "score": 1.0, + "content": "Then any", + "type": "text" + }, + { + "bbox": [ + 147, + 658, + 188, + 668 + ], + "score": 0.91, + "content": "W \\in \\mathcal { W } ^ { c }", + "type": "inline_equation" + }, + { + "bbox": [ + 188, + 655, + 235, + 671 + ], + "score": 1.0, + "content": "must have", + "type": "text" + }, + { + "bbox": [ + 236, + 657, + 405, + 669 + ], + "score": 0.84, + "content": "\\mathrm { r a n k } ( W _ { H + 1 } \\cdot \\cdot \\cdot W _ { 1 } ) < \\mathrm { m i n } \\{ d _ { x } , d _ { y } \\} = k", + "type": "inline_equation" + }, + { + "bbox": [ + 405, + 655, + 420, + 671 + ], + "score": 1.0, + "content": ", so", + "type": "text" + }, + { + "bbox": [ + 420, + 657, + 457, + 669 + ], + "score": 0.87, + "content": "W \\in \\mathcal { V } _ { 1 } ^ { c }", + "type": "inline_equation" + }, + { + "bbox": [ + 457, + 655, + 506, + 671 + ], + "score": 1.0, + "content": ". Thus, any", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 106, + 666, + 505, + 682 + ], + "spans": [ + { + "bbox": [ + 106, + 668, + 142, + 679 + ], + "score": 0.91, + "content": "W \\in \\nu _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 143, + 666, + 182, + 682 + ], + "score": 1.0, + "content": "is also in", + "type": "text" + }, + { + "bbox": [ + 183, + 669, + 195, + 678 + ], + "score": 0.84, + "content": "\\mathcal { W }", + "type": "inline_equation" + }, + { + "bbox": [ + 195, + 666, + 236, + 682 + ], + "score": 1.0, + "content": ", so either", + "type": "text" + }, + { + "bbox": [ + 236, + 669, + 345, + 680 + ], + "score": 0.86, + "content": "\\operatorname { r a n k } ( W _ { H + 1 } \\cdot \\cdot \\cdot W _ { 2 } ) = d _ { y }", + "type": "inline_equation" + }, + { + "bbox": [ + 345, + 666, + 359, + 682 + ], + "score": 1.0, + "content": "or", + "type": "text" + }, + { + "bbox": [ + 360, + 669, + 457, + 680 + ], + "score": 0.72, + "content": "\\operatorname { r a n k } ( W _ { H } \\cdot \\cdot \\cdot W _ { 1 } ) = d _ { x }", + "type": "inline_equation" + }, + { + "bbox": [ + 457, + 666, + 505, + 682 + ], + "score": 1.0, + "content": ", depending", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 106, + 679, + 230, + 691 + ], + "spans": [ + { + "bbox": [ + 106, + 679, + 230, + 691 + ], + "score": 1.0, + "content": "on the cases. Then, we can set", + "type": "text" + } + ], + "index": 38 + } + ], + "index": 37 + }, + { + "type": "interline_equation", + "bbox": [ + 221, + 691, + 389, + 720 + ], + "lines": [ + { + "bbox": [ + 221, + 691, + 389, + 720 + ], + "spans": [ + { + "bbox": [ + 221, + 691, + 389, + 720 + ], + "score": 0.92, + "content": "\\epsilon = \\left\\{ \\begin{array} { l l } { \\sigma _ { \\operatorname* { m i n } } ( W _ { H + 1 } \\cdot \\cdot \\cdot W _ { 2 } ) , } & { \\mathrm { i f ~ } d _ { y } \\leq d _ { x } , } \\\\ { \\sigma _ { \\operatorname* { m i n } } ( W _ { H } \\cdot \\cdot \\cdot W _ { 1 } ) , } & { \\mathrm { i f ~ } d _ { x } \\leq d _ { y } . } \\end{array} \\right.", + "type": "interline_equation", + "image_path": "b2978b2b24104da391fed3da86e81035bc9d55da3da7f11a99049c105824839d.jpg" + } + ] + } + ], + "index": 39.5, + "virtual_lines": [ + { + "bbox": [ + 221, + 691, + 389, + 705.5 + ], + "spans": [], + "index": 39 + }, + { + "bbox": [ + 221, + 705.5, + 389, + 720.0 + ], + "spans": [], + "index": 40 + } + ] + }, + { + "type": "text", + "bbox": [ + 105, + 720, + 456, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 720, + 454, + 733 + ], + "spans": [ + { + "bbox": [ + 106, + 720, + 173, + 733 + ], + "score": 1.0, + "content": "We always have", + "type": "text" + }, + { + "bbox": [ + 173, + 721, + 196, + 731 + ], + "score": 0.89, + "content": "\\epsilon > 0", + "type": "inline_equation" + }, + { + "bbox": [ + 197, + 720, + 416, + 733 + ], + "score": 1.0, + "content": "because the matrices are full rank, and we can see that", + "type": "text" + }, + { + "bbox": [ + 416, + 721, + 454, + 732 + ], + "score": 0.91, + "content": "W \\in \\mathcal { W } _ { \\epsilon }", + "type": "inline_equation" + } + ], + "index": 41 + } + ], + "index": 41 + } + ], + "page_idx": 4, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 107, + 26, + 294, + 38 + ], + "lines": [ + { + "bbox": [ + 106, + 25, + 294, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 25, + 294, + 38 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2018", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 494, + 513, + 505, + 523 + ], + "lines": [ + { + "bbox": [ + 496, + 514, + 504, + 523 + ], + "spans": [ + { + "bbox": [ + 496, + 514, + 504, + 523 + ], + "score": 0.999, + "content": "□", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 494, + 721, + 505, + 731 + ], + "lines": [ + { + "bbox": [ + 496, + 722, + 505, + 731 + ], + "spans": [ + { + "bbox": [ + 496, + 722, + 505, + 731 + ], + "score": 0.999, + "content": "□", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 302, + 751, + 309, + 760 + ], + "lines": [ + { + "bbox": [ + 302, + 750, + 309, + 762 + ], + "spans": [ + { + "bbox": [ + 302, + 750, + 309, + 762 + ], + "score": 1.0, + "content": "5", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "title", + "bbox": [ + 107, + 82, + 236, + 93 + ], + "lines": [ + { + "bbox": [ + 106, + 81, + 237, + 95 + ], + "spans": [ + { + "bbox": [ + 106, + 81, + 237, + 95 + ], + "score": 1.0, + "content": "3.3 PROOF OF THEOREM 2.1", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "text", + "bbox": [ + 105, + 102, + 505, + 126 + ], + "lines": [ + { + "bbox": [ + 105, + 101, + 506, + 116 + ], + "spans": [ + { + "bbox": [ + 105, + 101, + 304, + 116 + ], + "score": 1.0, + "content": "We prove Theorem 2.1, which addresses the case", + "type": "text" + }, + { + "bbox": [ + 304, + 103, + 375, + 115 + ], + "score": 0.93, + "content": "k = \\operatorname* { m i n } \\{ d _ { x } , d _ { y } \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 376, + 101, + 506, + 116 + ], + "score": 1.0, + "content": ". First, recall that the set defined", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 114, + 180, + 126 + ], + "spans": [ + { + "bbox": [ + 105, + 114, + 180, + 126 + ], + "score": 1.0, + "content": "in Theorem 2.1 is", + "type": "text" + } + ], + "index": 2 + } + ], + "index": 1.5, + "bbox_fs": [ + 105, + 101, + 506, + 126 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 192, + 126, + 416, + 140 + ], + "lines": [ + { + "bbox": [ + 192, + 126, + 416, + 140 + ], + "spans": [ + { + "bbox": [ + 192, + 126, + 416, + 140 + ], + "score": 0.79, + "content": "\\mathcal { V } _ { 1 } : = \\left\\{ ( W _ { 1 } , \\ldots , W _ { H + 1 } ) : \\operatorname { r a n k } ( W _ { H + 1 } \\cdot \\cdot \\cdot W _ { 1 } ) = k \\right\\} .", + "type": "interline_equation", + "image_path": "22f37a06b5e57e58268a3726f1418e0c705e79519fba482e28d44631f77f280a.jpg" + } + ] + } + ], + "index": 3, + "virtual_lines": [ + { + "bbox": [ + 192, + 126, + 416, + 140 + ], + "spans": [], + "index": 3 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 141, + 504, + 174 + ], + "lines": [ + { + "bbox": [ + 105, + 140, + 505, + 153 + ], + "spans": [ + { + "bbox": [ + 105, + 140, + 289, + 153 + ], + "score": 1.0, + "content": "As seen in (3), the unique minimum point of", + "type": "text" + }, + { + "bbox": [ + 289, + 141, + 302, + 152 + ], + "score": 0.87, + "content": "L _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 302, + 140, + 339, + 153 + ], + "score": 1.0, + "content": "has rank", + "type": "text" + }, + { + "bbox": [ + 340, + 141, + 346, + 151 + ], + "score": 0.78, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 347, + 140, + 403, + 153 + ], + "score": 1.0, + "content": ". So, no point", + "type": "text" + }, + { + "bbox": [ + 404, + 141, + 440, + 153 + ], + "score": 0.91, + "content": "W \\in \\mathcal { V } _ { 1 } ^ { c }", + "type": "inline_equation" + }, + { + "bbox": [ + 441, + 140, + 505, + 153 + ], + "score": 1.0, + "content": "can be a global", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 151, + 505, + 164 + ], + "spans": [ + { + "bbox": [ + 105, + 151, + 160, + 164 + ], + "score": 1.0, + "content": "minimum of", + "type": "text" + }, + { + "bbox": [ + 160, + 153, + 168, + 162 + ], + "score": 0.82, + "content": "L", + "type": "inline_equation" + }, + { + "bbox": [ + 168, + 151, + 505, + 164 + ], + "score": 1.0, + "content": ". Therefore, by Kawaguchi (2016, Theorem 2.3.(iii)) and Lemma B.1, any critical", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 162, + 247, + 176 + ], + "spans": [ + { + "bbox": [ + 105, + 162, + 140, + 176 + ], + "score": 1.0, + "content": "point in", + "type": "text" + }, + { + "bbox": [ + 140, + 163, + 152, + 174 + ], + "score": 0.89, + "content": "\\mathcal { V } _ { 1 } ^ { c }", + "type": "inline_equation" + }, + { + "bbox": [ + 153, + 162, + 247, + 176 + ], + "score": 1.0, + "content": "must be a saddle point.", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 5, + "bbox_fs": [ + 105, + 140, + 505, + 176 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 179, + 505, + 235 + ], + "lines": [ + { + "bbox": [ + 105, + 179, + 505, + 192 + ], + "spans": [ + { + "bbox": [ + 105, + 179, + 338, + 192 + ], + "score": 1.0, + "content": "For the rest of our proof, we need to consider two cases:", + "type": "text" + }, + { + "bbox": [ + 338, + 180, + 374, + 192 + ], + "score": 0.93, + "content": "d _ { y } \\leq d _ { x }", + "type": "inline_equation" + }, + { + "bbox": [ + 375, + 179, + 393, + 192 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 393, + 180, + 429, + 192 + ], + "score": 0.91, + "content": "d _ { x } \\leq d _ { y }", + "type": "inline_equation" + }, + { + "bbox": [ + 430, + 179, + 444, + 192 + ], + "score": 1.0, + "content": ". If", + "type": "text" + }, + { + "bbox": [ + 444, + 180, + 480, + 192 + ], + "score": 0.92, + "content": "d _ { x } = d _ { y }", + "type": "inline_equation" + }, + { + "bbox": [ + 480, + 179, + 505, + 192 + ], + "score": 1.0, + "content": ", both", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 191, + 505, + 203 + ], + "spans": [ + { + "bbox": [ + 105, + 191, + 397, + 203 + ], + "score": 1.0, + "content": "cases work. The outline of the proof is as follows: we define a new set", + "type": "text" + }, + { + "bbox": [ + 397, + 191, + 412, + 201 + ], + "score": 0.89, + "content": "\\mathcal { W } _ { \\epsilon }", + "type": "inline_equation" + }, + { + "bbox": [ + 412, + 191, + 505, + 203 + ], + "score": 1.0, + "content": ", show that any critical", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 200, + 506, + 215 + ], + "spans": [ + { + "bbox": [ + 105, + 200, + 168, + 215 + ], + "score": 1.0, + "content": "point in the set", + "type": "text" + }, + { + "bbox": [ + 169, + 202, + 183, + 213 + ], + "score": 0.9, + "content": "\\mathcal { W } _ { \\epsilon }", + "type": "inline_equation" + }, + { + "bbox": [ + 184, + 200, + 375, + 215 + ], + "score": 1.0, + "content": "is a global minimum, and then show that every", + "type": "text" + }, + { + "bbox": [ + 376, + 202, + 411, + 213 + ], + "score": 0.92, + "content": "W \\in \\mathcal { V } _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 411, + 200, + 451, + 215 + ], + "score": 1.0, + "content": "is also in", + "type": "text" + }, + { + "bbox": [ + 451, + 202, + 465, + 213 + ], + "score": 0.9, + "content": "\\mathcal { W } _ { \\epsilon }", + "type": "inline_equation" + }, + { + "bbox": [ + 466, + 200, + 506, + 215 + ], + "score": 1.0, + "content": "for some", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 106, + 212, + 505, + 226 + ], + "spans": [ + { + "bbox": [ + 106, + 213, + 133, + 223 + ], + "score": 0.88, + "content": "\\epsilon > 0", + "type": "inline_equation" + }, + { + "bbox": [ + 133, + 212, + 293, + 226 + ], + "score": 1.0, + "content": ". This proves that any critical point of", + "type": "text" + }, + { + "bbox": [ + 293, + 212, + 319, + 225 + ], + "score": 0.93, + "content": "L ( W )", + "type": "inline_equation" + }, + { + "bbox": [ + 320, + 212, + 332, + 226 + ], + "score": 1.0, + "content": "in", + "type": "text" + }, + { + "bbox": [ + 333, + 213, + 344, + 223 + ], + "score": 0.89, + "content": "\\nu _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 345, + 212, + 449, + 226 + ], + "score": 1.0, + "content": "is also a critical point in", + "type": "text" + }, + { + "bbox": [ + 450, + 213, + 464, + 223 + ], + "score": 0.89, + "content": "\\mathcal { W } _ { \\epsilon }", + "type": "inline_equation" + }, + { + "bbox": [ + 465, + 212, + 505, + 226 + ], + "score": 1.0, + "content": "for some", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 224, + 237, + 236 + ], + "spans": [ + { + "bbox": [ + 106, + 224, + 130, + 234 + ], + "score": 0.86, + "content": "\\epsilon > 0", + "type": "inline_equation" + }, + { + "bbox": [ + 130, + 224, + 237, + 236 + ], + "score": 1.0, + "content": ", hence a global minimum.", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 9, + "bbox_fs": [ + 105, + 179, + 506, + 236 + ] + }, + { + "type": "text", + "bbox": [ + 108, + 240, + 294, + 252 + ], + "lines": [ + { + "bbox": [ + 105, + 238, + 296, + 254 + ], + "spans": [ + { + "bbox": [ + 105, + 238, + 296, + 254 + ], + "score": 1.0, + "content": "The following proposition proves the first step:", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 12, + "bbox_fs": [ + 105, + 238, + 296, + 254 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 253, + 462, + 266 + ], + "lines": [ + { + "bbox": [ + 105, + 252, + 463, + 268 + ], + "spans": [ + { + "bbox": [ + 105, + 252, + 229, + 268 + ], + "score": 1.0, + "content": "Proposition 3.4. Assume that", + "type": "text" + }, + { + "bbox": [ + 229, + 253, + 300, + 266 + ], + "score": 0.93, + "content": "k = \\operatorname* { m i n } \\{ d _ { x } , d _ { y } \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 300, + 252, + 338, + 268 + ], + "score": 1.0, + "content": ". For any", + "type": "text" + }, + { + "bbox": [ + 338, + 254, + 361, + 264 + ], + "score": 0.88, + "content": "\\epsilon > 0", + "type": "inline_equation" + }, + { + "bbox": [ + 362, + 252, + 463, + 268 + ], + "score": 1.0, + "content": ", define the following set:", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 13, + "bbox_fs": [ + 105, + 252, + 463, + 268 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 160, + 268, + 449, + 295 + ], + "lines": [ + { + "bbox": [ + 160, + 268, + 449, + 295 + ], + "spans": [ + { + "bbox": [ + 160, + 268, + 449, + 295 + ], + "score": 0.91, + "content": "\\mathcal { W } _ { \\epsilon } : = \\left\\{ \\begin{array} { l l } { \\{ ( W _ { 1 } , \\dots , W _ { H + 1 } ) : \\sigma _ { \\operatorname* { m i n } } ( W _ { H + 1 } \\cdot \\cdot \\cdot W _ { 2 } ) \\geq \\epsilon \\} , } & { \\mathrm { i f ~ } d _ { y } \\leq d _ { x } , } \\\\ { \\{ ( W _ { 1 } , \\dots , W _ { H + 1 } ) : \\sigma _ { \\operatorname* { m i n } } ( W _ { H } \\cdot \\cdot \\cdot W _ { 1 } ) \\geq \\epsilon \\} , } & { \\mathrm { i f ~ } d _ { x } \\leq d _ { y } . } \\end{array} \\right.", + "type": "interline_equation", + "image_path": "a5c01d59a18b54b64d8e3533acef12e3053e575a03c0a3b181ca09627ef55889.jpg" + } + ] + } + ], + "index": 14, + "virtual_lines": [ + { + "bbox": [ + 160, + 268, + 449, + 295 + ], + "spans": [], + "index": 14 + } + ] + }, + { + "type": "text", + "bbox": [ + 108, + 296, + 372, + 309 + ], + "lines": [ + { + "bbox": [ + 106, + 295, + 374, + 310 + ], + "spans": [ + { + "bbox": [ + 106, + 295, + 210, + 310 + ], + "score": 1.0, + "content": "Then any critical point of", + "type": "text" + }, + { + "bbox": [ + 210, + 297, + 236, + 309 + ], + "score": 0.9, + "content": "L ( W )", + "type": "inline_equation" + }, + { + "bbox": [ + 237, + 295, + 248, + 310 + ], + "score": 1.0, + "content": "in", + "type": "text" + }, + { + "bbox": [ + 248, + 297, + 263, + 308 + ], + "score": 0.87, + "content": "\\mathcal { W } _ { \\epsilon }", + "type": "inline_equation" + }, + { + "bbox": [ + 263, + 295, + 374, + 310 + ], + "score": 1.0, + "content": "is a global minimum point.", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 15, + "bbox_fs": [ + 106, + 295, + 374, + 310 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 319, + 503, + 343 + ], + "lines": [ + { + "bbox": [ + 102, + 316, + 503, + 336 + ], + "spans": [ + { + "bbox": [ + 102, + 316, + 148, + 336 + ], + "score": 1.0, + "content": "Proof. (If", + "type": "text" + }, + { + "bbox": [ + 149, + 320, + 184, + 332 + ], + "score": 0.89, + "content": "d _ { y } \\leq d _ { x } ,", + "type": "inline_equation" + }, + { + "bbox": [ + 184, + 316, + 293, + 336 + ], + "score": 1.0, + "content": ") Consider (5) in the case of", + "type": "text" + }, + { + "bbox": [ + 293, + 321, + 316, + 330 + ], + "score": 0.89, + "content": "i = 1", + "type": "inline_equation" + }, + { + "bbox": [ + 316, + 316, + 401, + 336 + ], + "score": 1.0, + "content": ". We can observe that", + "type": "text" + }, + { + "bbox": [ + 401, + 319, + 503, + 333 + ], + "score": 0.92, + "content": "W _ { 2 } ^ { T } \\cdot \\cdot \\cdot W _ { H + 1 } ^ { T } \\in \\mathbb { R } ^ { d _ { 1 } \\times d _ { y } }", + "type": "inline_equation" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 331, + 273, + 344 + ], + "spans": [ + { + "bbox": [ + 106, + 331, + 141, + 344 + ], + "score": 1.0, + "content": "and that", + "type": "text" + }, + { + "bbox": [ + 141, + 332, + 174, + 344 + ], + "score": 0.92, + "content": "d _ { 1 } \\geq d _ { y }", + "type": "inline_equation" + }, + { + "bbox": [ + 175, + 331, + 273, + 344 + ], + "score": 1.0, + "content": ". Then by Lemma 3.3.1,", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 16.5, + "bbox_fs": [ + 102, + 316, + 503, + 344 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 163, + 344, + 447, + 393 + ], + "lines": [ + { + "bbox": [ + 163, + 344, + 447, + 393 + ], + "spans": [ + { + "bbox": [ + 163, + 344, + 447, + 393 + ], + "score": 0.93, + "content": "\\begin{array} { r l } { { \\| \\frac { \\partial L } { \\partial W _ { 1 } } \\| _ { \\mathrm { F } } ^ { 2 } \\geq \\sigma _ { \\operatorname* { m i n } } ^ { 2 } ( W _ { H + 1 } \\cdot \\cdot \\cdot W _ { 2 } ) \\| ( W _ { H + 1 } W _ { H } \\cdot \\cdot \\cdot W _ { 1 } X - Y ) X ^ { T } \\| _ { \\mathrm { F } } ^ { 2 } } } \\\\ & { \\geq \\epsilon ^ { 2 } \\| ( W _ { H + 1 } W _ { H } \\cdot \\cdot \\cdot W _ { 1 } X - Y ) X ^ { T } \\| _ { \\mathrm { F } } ^ { 2 } . } \\end{array}", + "type": "interline_equation", + "image_path": "203cbd0fb05cb2afb06a29343f45f7f23840efc5fae5b197db66f36fa3125896.jpg" + } + ] + } + ], + "index": 19, + "virtual_lines": [ + { + "bbox": [ + 163, + 344, + 447, + 360.3333333333333 + ], + "spans": [], + "index": 18 + }, + { + "bbox": [ + 163, + 360.3333333333333, + 447, + 376.66666666666663 + ], + "spans": [], + "index": 19 + }, + { + "bbox": [ + 163, + 376.66666666666663, + 447, + 392.99999999999994 + ], + "spans": [], + "index": 20 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 392, + 332, + 404 + ], + "lines": [ + { + "bbox": [ + 106, + 391, + 333, + 406 + ], + "spans": [ + { + "bbox": [ + 106, + 391, + 285, + 406 + ], + "score": 1.0, + "content": "By the above inequality, any critical point in", + "type": "text" + }, + { + "bbox": [ + 286, + 393, + 298, + 402 + ], + "score": 0.8, + "content": "\\mathcal { W }", + "type": "inline_equation" + }, + { + "bbox": [ + 298, + 391, + 333, + 406 + ], + "score": 1.0, + "content": "satisfies", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 21, + "bbox_fs": [ + 106, + 391, + 333, + 406 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 196, + 405, + 413, + 430 + ], + "lines": [ + { + "bbox": [ + 196, + 405, + 413, + 430 + ], + "spans": [ + { + "bbox": [ + 196, + 405, + 413, + 430 + ], + "score": 0.88, + "content": "\\forall i , \\frac { \\partial L } { \\partial W _ { i } } = 0 \\Rightarrow ( W _ { H + 1 } W _ { H } \\cdot \\cdot \\cdot W _ { 1 } X - Y ) X ^ { T } = 0 ,", + "type": "interline_equation", + "image_path": "19b888bcd619523f06ef7c9ea0c0d8a5616a7d1822d9b24744b68284bd9e67bf.jpg" + } + ] + } + ], + "index": 22, + "virtual_lines": [ + { + "bbox": [ + 196, + 405, + 413, + 430 + ], + "spans": [], + "index": 22 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 430, + 504, + 454 + ], + "lines": [ + { + "bbox": [ + 105, + 429, + 505, + 444 + ], + "spans": [ + { + "bbox": [ + 105, + 429, + 178, + 444 + ], + "score": 1.0, + "content": "which means that", + "type": "text" + }, + { + "bbox": [ + 178, + 431, + 330, + 443 + ], + "score": 0.9, + "content": "W _ { H + 1 } W _ { H } \\cdot \\cdot \\cdot W _ { 1 } = Y X ^ { T } ( X X ^ { T } ) ^ { - 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 331, + 429, + 505, + 444 + ], + "score": 1.0, + "content": ". The product is the unique globally optimal", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 442, + 430, + 455 + ], + "spans": [ + { + "bbox": [ + 105, + 442, + 286, + 455 + ], + "score": 1.0, + "content": "solution (3) of the relaxed problem in (2), so", + "type": "text" + }, + { + "bbox": [ + 286, + 444, + 298, + 452 + ], + "score": 0.78, + "content": "W", + "type": "inline_equation" + }, + { + "bbox": [ + 298, + 442, + 418, + 455 + ], + "score": 1.0, + "content": "is a global minimum point of", + "type": "text" + }, + { + "bbox": [ + 418, + 443, + 426, + 452 + ], + "score": 0.82, + "content": "L", + "type": "inline_equation" + }, + { + "bbox": [ + 426, + 442, + 430, + 455 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 23.5, + "bbox_fs": [ + 105, + 429, + 505, + 455 + ] + }, + { + "type": "text", + "bbox": [ + 108, + 458, + 504, + 482 + ], + "lines": [ + { + "bbox": [ + 105, + 456, + 506, + 472 + ], + "spans": [ + { + "bbox": [ + 105, + 456, + 119, + 472 + ], + "score": 1.0, + "content": "(If", + "type": "text" + }, + { + "bbox": [ + 120, + 459, + 158, + 471 + ], + "score": 0.87, + "content": "d _ { x } \\leq d _ { y } ,", + "type": "inline_equation" + }, + { + "bbox": [ + 158, + 456, + 231, + 472 + ], + "score": 1.0, + "content": ") Consider (5) for", + "type": "text" + }, + { + "bbox": [ + 231, + 460, + 278, + 470 + ], + "score": 0.9, + "content": "i = H + 1", + "type": "inline_equation" + }, + { + "bbox": [ + 279, + 456, + 370, + 472 + ], + "score": 1.0, + "content": ". We can observe that", + "type": "text" + }, + { + "bbox": [ + 370, + 458, + 468, + 471 + ], + "score": 0.93, + "content": "W _ { 1 } ^ { T } \\cdot \\cdot \\cdot W _ { H } ^ { T } \\in \\mathbb { R } ^ { d _ { x } \\times d _ { H } }", + "type": "inline_equation" + }, + { + "bbox": [ + 468, + 456, + 506, + 472 + ], + "score": 1.0, + "content": "and that", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 107, + 470, + 242, + 483 + ], + "spans": [ + { + "bbox": [ + 107, + 471, + 144, + 482 + ], + "score": 0.91, + "content": "d _ { x } \\leq d _ { H }", + "type": "inline_equation" + }, + { + "bbox": [ + 144, + 470, + 242, + 483 + ], + "score": 1.0, + "content": ". Then by Lemma 3.3.2,", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 25.5, + "bbox_fs": [ + 105, + 456, + 506, + 483 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 192, + 483, + 417, + 512 + ], + "lines": [ + { + "bbox": [ + 192, + 483, + 417, + 512 + ], + "spans": [ + { + "bbox": [ + 192, + 483, + 417, + 512 + ], + "score": 0.92, + "content": "\\left\\| \\frac { \\partial \\cal { L } } { \\partial W _ { H + 1 } } \\right\\| _ { \\mathrm { F } } ^ { 2 } \\geq \\epsilon ^ { 2 } \\left\\| ( W _ { H + 1 } W _ { H } \\cdot \\cdot \\cdot W _ { 1 } X - Y ) X ^ { T } \\right\\| _ { \\mathrm { F } } ^ { 2 } ,", + "type": "interline_equation", + "image_path": "d89b611db20a0dbe8e98cccf93c8394749acc2ac9cefc384594ca4193c5530ae.jpg" + } + ] + } + ], + "index": 27.5, + "virtual_lines": [ + { + "bbox": [ + 192, + 483, + 417, + 497.5 + ], + "spans": [], + "index": 27 + }, + { + "bbox": [ + 192, + 497.5, + 417, + 512.0 + ], + "spans": [], + "index": 28 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 513, + 377, + 524 + ], + "lines": [ + { + "bbox": [ + 105, + 511, + 379, + 526 + ], + "spans": [ + { + "bbox": [ + 105, + 511, + 379, + 526 + ], + "score": 1.0, + "content": "and the rest of the proof flows in a similar way as the previous case.", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 29, + "bbox_fs": [ + 105, + 511, + 379, + 526 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 536, + 272, + 547 + ], + "lines": [ + { + "bbox": [ + 106, + 535, + 273, + 549 + ], + "spans": [ + { + "bbox": [ + 106, + 535, + 273, + 549 + ], + "score": 1.0, + "content": "The next proposition proves the theorem:", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 30, + "bbox_fs": [ + 106, + 535, + 273, + 549 + ] + }, + { + "type": "text", + "bbox": [ + 109, + 549, + 435, + 561 + ], + "lines": [ + { + "bbox": [ + 106, + 548, + 433, + 563 + ], + "spans": [ + { + "bbox": [ + 106, + 548, + 234, + 563 + ], + "score": 1.0, + "content": "Proposition 3.5. For any point", + "type": "text" + }, + { + "bbox": [ + 235, + 549, + 268, + 560 + ], + "score": 0.92, + "content": "W \\in \\nu _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 269, + 548, + 332, + 563 + ], + "score": 1.0, + "content": ", there exists an", + "type": "text" + }, + { + "bbox": [ + 332, + 550, + 356, + 559 + ], + "score": 0.88, + "content": "\\epsilon > 0", + "type": "inline_equation" + }, + { + "bbox": [ + 356, + 548, + 396, + 563 + ], + "score": 1.0, + "content": "such that", + "type": "text" + }, + { + "bbox": [ + 396, + 549, + 433, + 560 + ], + "score": 0.9, + "content": "W \\in \\mathcal { W } _ { \\epsilon }", + "type": "inline_equation" + } + ], + "index": 31 + } + ], + "index": 31, + "bbox_fs": [ + 106, + 548, + 433, + 563 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 572, + 373, + 584 + ], + "lines": [ + { + "bbox": [ + 105, + 571, + 374, + 586 + ], + "spans": [ + { + "bbox": [ + 105, + 571, + 204, + 586 + ], + "score": 1.0, + "content": "Proof. Define a new set", + "type": "text" + }, + { + "bbox": [ + 205, + 573, + 216, + 583 + ], + "score": 0.83, + "content": "\\mathcal { W }", + "type": "inline_equation" + }, + { + "bbox": [ + 217, + 571, + 303, + 586 + ], + "score": 1.0, + "content": ", a “limit” version (as", + "type": "text" + }, + { + "bbox": [ + 303, + 573, + 329, + 583 + ], + "score": 0.89, + "content": "\\epsilon 0", + "type": "inline_equation" + }, + { + "bbox": [ + 330, + 571, + 344, + 586 + ], + "score": 1.0, + "content": ") of", + "type": "text" + }, + { + "bbox": [ + 344, + 573, + 359, + 583 + ], + "score": 0.89, + "content": "\\mathcal { W } _ { \\epsilon }", + "type": "inline_equation" + }, + { + "bbox": [ + 359, + 571, + 374, + 586 + ], + "score": 1.0, + "content": ", as", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 32, + "bbox_fs": [ + 105, + 571, + 374, + 586 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 159, + 585, + 451, + 613 + ], + "lines": [ + { + "bbox": [ + 159, + 585, + 451, + 613 + ], + "spans": [ + { + "bbox": [ + 159, + 585, + 451, + 613 + ], + "score": 0.92, + "content": "\\begin{array} { r } { \\mathcal { W } : = \\{ \\{ ( W _ { 1 } , \\ldots , W _ { H + 1 } ) : \\mathrm { r a n k } ( W _ { H + 1 } \\cdot \\cdot \\cdot W _ { 2 } ) = d _ { y } \\} , \\mathrm { i f } \\ d _ { y } \\leq d _ { x } , } \\\\ { \\{ ( W _ { 1 } , \\ldots , W _ { H + 1 } ) : \\mathrm { r a n k } ( W _ { H } \\cdot \\cdot \\cdot W _ { 1 } ) = d _ { x } \\} , \\mathrm { i f } \\ d _ { x } \\leq d _ { y } . } \\end{array}", + "type": "interline_equation", + "image_path": "53917d40e08378e95d794b17cfba85e08f2d7177c755a23bbd12625a79647095.jpg" + } + ] + } + ], + "index": 33, + "virtual_lines": [ + { + "bbox": [ + 159, + 585, + 451, + 613 + ], + "spans": [], + "index": 33 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 614, + 349, + 626 + ], + "lines": [ + { + "bbox": [ + 106, + 613, + 349, + 627 + ], + "spans": [ + { + "bbox": [ + 106, + 613, + 163, + 627 + ], + "score": 1.0, + "content": "We show that", + "type": "text" + }, + { + "bbox": [ + 163, + 614, + 199, + 626 + ], + "score": 0.92, + "content": "\\nu _ { 1 } \\subset \\mathcal { W }", + "type": "inline_equation" + }, + { + "bbox": [ + 199, + 613, + 265, + 627 + ], + "score": 1.0, + "content": "by showing that", + "type": "text" + }, + { + "bbox": [ + 266, + 614, + 306, + 626 + ], + "score": 0.92, + "content": "\\mathcal { W } ^ { c } \\subset \\mathcal { V } _ { 1 } ^ { c }", + "type": "inline_equation" + }, + { + "bbox": [ + 306, + 613, + 349, + 627 + ], + "score": 1.0, + "content": ". Consider", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 34, + "bbox_fs": [ + 106, + 613, + 349, + 627 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 158, + 628, + 452, + 655 + ], + "lines": [ + { + "bbox": [ + 158, + 628, + 452, + 655 + ], + "spans": [ + { + "bbox": [ + 158, + 628, + 452, + 655 + ], + "score": 0.89, + "content": "\\mathcal { W } ^ { c } = \\left\\{ \\begin{array} { l l } { \\{ ( W _ { 1 } , \\ldots , W _ { H + 1 } ) : \\operatorname { r a n k } ( W _ { H + 1 } \\cdot \\cdot \\cdot W _ { 2 } ) < d _ { y } \\} , } & { \\mathrm { i f ~ } d _ { y } \\leq d _ { x } , } \\\\ { \\{ ( W _ { 1 } , \\ldots , W _ { H + 1 } ) : \\operatorname { r a n k } ( W _ { H } \\cdot \\cdot \\cdot W _ { 1 } ) < d _ { x } \\} , } & { \\mathrm { i f ~ } d _ { x } \\leq d _ { y } . } \\end{array} \\right.", + "type": "interline_equation", + "image_path": "34aa988edfa931aa84a1b1ae46d068bec887b962474ac1136374521ee91f3f1a.jpg" + } + ] + } + ], + "index": 35, + "virtual_lines": [ + { + "bbox": [ + 158, + 628, + 452, + 655 + ], + "spans": [], + "index": 35 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 657, + 504, + 691 + ], + "lines": [ + { + "bbox": [ + 105, + 655, + 506, + 671 + ], + "spans": [ + { + "bbox": [ + 105, + 655, + 147, + 671 + ], + "score": 1.0, + "content": "Then any", + "type": "text" + }, + { + "bbox": [ + 147, + 658, + 188, + 668 + ], + "score": 0.91, + "content": "W \\in \\mathcal { W } ^ { c }", + "type": "inline_equation" + }, + { + "bbox": [ + 188, + 655, + 235, + 671 + ], + "score": 1.0, + "content": "must have", + "type": "text" + }, + { + "bbox": [ + 236, + 657, + 405, + 669 + ], + "score": 0.84, + "content": "\\mathrm { r a n k } ( W _ { H + 1 } \\cdot \\cdot \\cdot W _ { 1 } ) < \\mathrm { m i n } \\{ d _ { x } , d _ { y } \\} = k", + "type": "inline_equation" + }, + { + "bbox": [ + 405, + 655, + 420, + 671 + ], + "score": 1.0, + "content": ", so", + "type": "text" + }, + { + "bbox": [ + 420, + 657, + 457, + 669 + ], + "score": 0.87, + "content": "W \\in \\mathcal { V } _ { 1 } ^ { c }", + "type": "inline_equation" + }, + { + "bbox": [ + 457, + 655, + 506, + 671 + ], + "score": 1.0, + "content": ". Thus, any", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 106, + 666, + 505, + 682 + ], + "spans": [ + { + "bbox": [ + 106, + 668, + 142, + 679 + ], + "score": 0.91, + "content": "W \\in \\nu _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 143, + 666, + 182, + 682 + ], + "score": 1.0, + "content": "is also in", + "type": "text" + }, + { + "bbox": [ + 183, + 669, + 195, + 678 + ], + "score": 0.84, + "content": "\\mathcal { W }", + "type": "inline_equation" + }, + { + "bbox": [ + 195, + 666, + 236, + 682 + ], + "score": 1.0, + "content": ", so either", + "type": "text" + }, + { + "bbox": [ + 236, + 669, + 345, + 680 + ], + "score": 0.86, + "content": "\\operatorname { r a n k } ( W _ { H + 1 } \\cdot \\cdot \\cdot W _ { 2 } ) = d _ { y }", + "type": "inline_equation" + }, + { + "bbox": [ + 345, + 666, + 359, + 682 + ], + "score": 1.0, + "content": "or", + "type": "text" + }, + { + "bbox": [ + 360, + 669, + 457, + 680 + ], + "score": 0.72, + "content": "\\operatorname { r a n k } ( W _ { H } \\cdot \\cdot \\cdot W _ { 1 } ) = d _ { x }", + "type": "inline_equation" + }, + { + "bbox": [ + 457, + 666, + 505, + 682 + ], + "score": 1.0, + "content": ", depending", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 106, + 679, + 230, + 691 + ], + "spans": [ + { + "bbox": [ + 106, + 679, + 230, + 691 + ], + "score": 1.0, + "content": "on the cases. Then, we can set", + "type": "text" + } + ], + "index": 38 + } + ], + "index": 37, + "bbox_fs": [ + 105, + 655, + 506, + 691 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 221, + 691, + 389, + 720 + ], + "lines": [ + { + "bbox": [ + 221, + 691, + 389, + 720 + ], + "spans": [ + { + "bbox": [ + 221, + 691, + 389, + 720 + ], + "score": 0.92, + "content": "\\epsilon = \\left\\{ \\begin{array} { l l } { \\sigma _ { \\operatorname* { m i n } } ( W _ { H + 1 } \\cdot \\cdot \\cdot W _ { 2 } ) , } & { \\mathrm { i f ~ } d _ { y } \\leq d _ { x } , } \\\\ { \\sigma _ { \\operatorname* { m i n } } ( W _ { H } \\cdot \\cdot \\cdot W _ { 1 } ) , } & { \\mathrm { i f ~ } d _ { x } \\leq d _ { y } . } \\end{array} \\right.", + "type": "interline_equation", + "image_path": "b2978b2b24104da391fed3da86e81035bc9d55da3da7f11a99049c105824839d.jpg" + } + ] + } + ], + "index": 39.5, + "virtual_lines": [ + { + "bbox": [ + 221, + 691, + 389, + 705.5 + ], + "spans": [], + "index": 39 + }, + { + "bbox": [ + 221, + 705.5, + 389, + 720.0 + ], + "spans": [], + "index": 40 + } + ] + }, + { + "type": "text", + "bbox": [ + 105, + 720, + 456, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 720, + 454, + 733 + ], + "spans": [ + { + "bbox": [ + 106, + 720, + 173, + 733 + ], + "score": 1.0, + "content": "We always have", + "type": "text" + }, + { + "bbox": [ + 173, + 721, + 196, + 731 + ], + "score": 0.89, + "content": "\\epsilon > 0", + "type": "inline_equation" + }, + { + "bbox": [ + 197, + 720, + 416, + 733 + ], + "score": 1.0, + "content": "because the matrices are full rank, and we can see that", + "type": "text" + }, + { + "bbox": [ + 416, + 721, + 454, + 732 + ], + "score": 0.91, + "content": "W \\in \\mathcal { W } _ { \\epsilon }", + "type": "inline_equation" + } + ], + "index": 41 + } + ], + "index": 41, + "bbox_fs": [ + 106, + 720, + 454, + 733 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "title", + "bbox": [ + 107, + 82, + 237, + 94 + ], + "lines": [ + { + "bbox": [ + 106, + 82, + 237, + 94 + ], + "spans": [ + { + "bbox": [ + 106, + 82, + 237, + 94 + ], + "score": 1.0, + "content": "3.4 PROOF OF THEOREM 2.2", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "text", + "bbox": [ + 105, + 102, + 504, + 126 + ], + "lines": [ + { + "bbox": [ + 105, + 102, + 505, + 115 + ], + "spans": [ + { + "bbox": [ + 105, + 102, + 363, + 115 + ], + "score": 1.0, + "content": "In this section we prove Theorem 2.2, which tackles the case", + "type": "text" + }, + { + "bbox": [ + 363, + 103, + 439, + 115 + ], + "score": 0.92, + "content": "k < \\operatorname* { m i n } \\{ d _ { x } , d _ { y } \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 439, + 102, + 505, + 115 + ], + "score": 1.0, + "content": ". Note that this", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 114, + 273, + 126 + ], + "spans": [ + { + "bbox": [ + 105, + 114, + 222, + 126 + ], + "score": 1.0, + "content": "assumption also implies that", + "type": "text" + }, + { + "bbox": [ + 222, + 114, + 269, + 126 + ], + "score": 0.92, + "content": "1 \\leq p \\leq H", + "type": "inline_equation" + }, + { + "bbox": [ + 269, + 114, + 273, + 126 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 2 + } + ], + "index": 1.5 + }, + { + "type": "text", + "bbox": [ + 108, + 131, + 266, + 142 + ], + "lines": [ + { + "bbox": [ + 106, + 131, + 266, + 144 + ], + "spans": [ + { + "bbox": [ + 106, + 131, + 266, + 144 + ], + "score": 1.0, + "content": "As for the proof of Theorem 2.1, define", + "type": "text" + } + ], + "index": 3 + } + ], + "index": 3 + }, + { + "type": "interline_equation", + "bbox": [ + 193, + 147, + 417, + 161 + ], + "lines": [ + { + "bbox": [ + 193, + 147, + 417, + 161 + ], + "spans": [ + { + "bbox": [ + 193, + 147, + 417, + 161 + ], + "score": 0.88, + "content": "\\mathcal { V } _ { 1 } : = \\left\\{ ( W _ { 1 } , \\ldots , W _ { H + 1 } ) : \\operatorname { r a n k } ( W _ { H + 1 } \\cdot \\cdot \\cdot W _ { 1 } ) = k \\right\\} .", + "type": "interline_equation", + "image_path": "c679bef42b4da682812002d2606aa607c6d16cb36d2f7f11f1657f207b7208f6.jpg" + } + ] + } + ], + "index": 4, + "virtual_lines": [ + { + "bbox": [ + 193, + 147, + 417, + 161 + ], + "spans": [], + "index": 4 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 165, + 506, + 210 + ], + "lines": [ + { + "bbox": [ + 106, + 164, + 505, + 178 + ], + "spans": [ + { + "bbox": [ + 106, + 164, + 365, + 178 + ], + "score": 1.0, + "content": "The globally optimal point of the relaxed problem (2) has rank", + "type": "text" + }, + { + "bbox": [ + 365, + 166, + 372, + 176 + ], + "score": 0.82, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 372, + 164, + 505, + 178 + ], + "score": 1.0, + "content": ", as seen in (4). Thus, any point", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 176, + 505, + 188 + ], + "spans": [ + { + "bbox": [ + 106, + 176, + 151, + 188 + ], + "score": 1.0, + "content": "outside of", + "type": "text" + }, + { + "bbox": [ + 151, + 177, + 163, + 187 + ], + "score": 0.88, + "content": "\\nu _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 163, + 176, + 505, + 188 + ], + "score": 1.0, + "content": "cannot be a global minimum. Then, by Kawaguchi (2016, Theorem 2.3.(iii)) and", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 186, + 506, + 200 + ], + "spans": [ + { + "bbox": [ + 105, + 186, + 303, + 200 + ], + "score": 1.0, + "content": "Lemma B.1, it follows that any critical point in", + "type": "text" + }, + { + "bbox": [ + 304, + 187, + 316, + 199 + ], + "score": 0.89, + "content": "\\mathcal { V } _ { 1 } ^ { c }", + "type": "inline_equation" + }, + { + "bbox": [ + 317, + 186, + 506, + 200 + ], + "score": 1.0, + "content": "must be a saddle point. The remaining proof", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 198, + 200, + 210 + ], + "spans": [ + { + "bbox": [ + 106, + 198, + 183, + 210 + ], + "score": 1.0, + "content": "considers points in", + "type": "text" + }, + { + "bbox": [ + 184, + 198, + 195, + 209 + ], + "score": 0.89, + "content": "\\mathcal { V } _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 196, + 198, + 200, + 210 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 6.5 + }, + { + "type": "text", + "bbox": [ + 106, + 214, + 454, + 227 + ], + "lines": [ + { + "bbox": [ + 105, + 214, + 455, + 227 + ], + "spans": [ + { + "bbox": [ + 105, + 214, + 455, + 227 + ], + "score": 1.0, + "content": "For this section, let us introduce some additional notations to ease presentation. Define", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 9 + }, + { + "type": "interline_equation", + "bbox": [ + 120, + 230, + 489, + 264 + ], + "lines": [ + { + "bbox": [ + 120, + 230, + 489, + 264 + ], + "spans": [ + { + "bbox": [ + 120, + 230, + 489, + 264 + ], + "score": 0.92, + "content": "\\begin{array} { r l } & { E : = ( W _ { H + 1 } \\cdot \\cdot \\cdot W _ { 1 } X - Y ) X ^ { T } \\in \\mathbb { R } ^ { d _ { y } \\times d _ { x } } , } \\\\ & { A _ { i } : = W _ { i + 1 } ^ { T } \\cdot \\cdot \\cdot W _ { H + 1 } ^ { T } \\in \\mathbb { R } ^ { d _ { i } \\times d _ { y } } , \\ B _ { i } : = W _ { 1 } ^ { T } \\cdot \\cdot \\cdot W _ { i - 1 } ^ { T } \\in \\mathbb { R } ^ { d _ { x } \\times d _ { i - 1 } } , \\ i = 1 , \\dots , H + 1 , } \\end{array}", + "type": "interline_equation", + "image_path": "457a0ca2581a2e0aea55b134454c35b398708916810fcb378d1b119dc2bf6fbc.jpg" + } + ] + } + ], + "index": 11, + "virtual_lines": [ + { + "bbox": [ + 120, + 230, + 489, + 241.33333333333334 + ], + "spans": [], + "index": 10 + }, + { + "bbox": [ + 120, + 241.33333333333334, + 489, + 252.66666666666669 + ], + "spans": [], + "index": 11 + }, + { + "bbox": [ + 120, + 252.66666666666669, + 489, + 264.0 + ], + "spans": [], + "index": 12 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 267, + 392, + 281 + ], + "lines": [ + { + "bbox": [ + 105, + 266, + 393, + 283 + ], + "spans": [ + { + "bbox": [ + 105, + 266, + 136, + 283 + ], + "score": 1.0, + "content": "so that", + "type": "text" + }, + { + "bbox": [ + 136, + 267, + 197, + 282 + ], + "score": 0.94, + "content": "\\begin{array} { r } { \\frac { \\partial L } { \\partial W _ { i } } = A _ { i } E B _ { i } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 197, + 266, + 248, + 283 + ], + "score": 1.0, + "content": ". Notice that", + "type": "text" + }, + { + "bbox": [ + 248, + 269, + 274, + 280 + ], + "score": 0.92, + "content": "A _ { H + 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 275, + 266, + 292, + 283 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 293, + 269, + 306, + 280 + ], + "score": 0.89, + "content": "B _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 306, + 266, + 393, + 283 + ], + "score": 1.0, + "content": "are identity matrices.", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 13 + }, + { + "type": "text", + "bbox": [ + 107, + 286, + 505, + 334 + ], + "lines": [ + { + "bbox": [ + 105, + 286, + 505, + 299 + ], + "spans": [ + { + "bbox": [ + 105, + 286, + 208, + 299 + ], + "score": 1.0, + "content": "Now consider any tuple", + "type": "text" + }, + { + "bbox": [ + 208, + 287, + 247, + 298 + ], + "score": 0.91, + "content": "W \\in \\mathcal { V } _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 248, + 286, + 349, + 299 + ], + "score": 1.0, + "content": ". Since the full product", + "type": "text" + }, + { + "bbox": [ + 349, + 287, + 406, + 298 + ], + "score": 0.91, + "content": "W _ { H + 1 } \\cdots W _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 406, + 286, + 447, + 299 + ], + "score": 1.0, + "content": "has rank", + "type": "text" + }, + { + "bbox": [ + 447, + 288, + 454, + 297 + ], + "score": 0.8, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 454, + 286, + 505, + 299 + ], + "score": 1.0, + "content": ", any partial", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 297, + 505, + 311 + ], + "spans": [ + { + "bbox": [ + 105, + 297, + 144, + 311 + ], + "score": 1.0, + "content": "products", + "type": "text" + }, + { + "bbox": [ + 144, + 298, + 156, + 309 + ], + "score": 0.89, + "content": "A _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 156, + 297, + 175, + 311 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 175, + 298, + 187, + 309 + ], + "score": 0.88, + "content": "B _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 187, + 297, + 234, + 311 + ], + "score": 1.0, + "content": "must have", + "type": "text" + }, + { + "bbox": [ + 234, + 298, + 294, + 309 + ], + "score": 0.9, + "content": "\\operatorname { r a n k } ( A _ { i } ) \\geq k", + "type": "inline_equation" + }, + { + "bbox": [ + 294, + 297, + 314, + 311 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 314, + 298, + 374, + 310 + ], + "score": 0.89, + "content": "\\operatorname { r a n k } ( B _ { i } ) \\geq k", + "type": "inline_equation" + }, + { + "bbox": [ + 374, + 297, + 406, + 311 + ], + "score": 1.0, + "content": ", for all", + "type": "text" + }, + { + "bbox": [ + 406, + 299, + 411, + 308 + ], + "score": 0.73, + "content": "i", + "type": "inline_equation" + }, + { + "bbox": [ + 411, + 297, + 480, + 311 + ], + "score": 1.0, + "content": ". Then, consider", + "type": "text" + }, + { + "bbox": [ + 480, + 298, + 505, + 310 + ], + "score": 0.9, + "content": "A _ { p } \\in", + "type": "inline_equation" + } + ], + "index": 15 + }, + { + "bbox": [ + 107, + 306, + 506, + 325 + ], + "spans": [ + { + "bbox": [ + 107, + 309, + 134, + 320 + ], + "score": 0.91, + "content": "\\mathbb { R } ^ { k \\times d _ { y } }", + "type": "inline_equation" + }, + { + "bbox": [ + 134, + 306, + 151, + 325 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 152, + 309, + 214, + 322 + ], + "score": 0.92, + "content": "B _ { p + 1 } \\in \\mathbb { R } ^ { d _ { x } \\times k }", + "type": "inline_equation" + }, + { + "bbox": [ + 214, + 306, + 243, + 325 + ], + "score": 1.0, + "content": ". Since", + "type": "text" + }, + { + "bbox": [ + 243, + 310, + 302, + 322 + ], + "score": 0.86, + "content": "\\operatorname { r a n k } ( A _ { p } ) \\leq k", + "type": "inline_equation" + }, + { + "bbox": [ + 302, + 306, + 319, + 325 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 319, + 310, + 388, + 322 + ], + "score": 0.89, + "content": "\\operatorname { r a n k } ( B _ { p + 1 } ) \\leq k", + "type": "inline_equation" + }, + { + "bbox": [ + 389, + 306, + 465, + 325 + ], + "score": 1.0, + "content": ", we can see that ra", + "type": "text" + }, + { + "bbox": [ + 466, + 311, + 506, + 323 + ], + "score": 0.7, + "content": "\\operatorname { l k } ( A _ { p } ) =", + "type": "inline_equation" + } + ], + "index": 16 + }, + { + "bbox": [ + 107, + 321, + 429, + 334 + ], + "spans": [ + { + "bbox": [ + 107, + 321, + 176, + 334 + ], + "score": 0.9, + "content": "\\mathrm { r a n k } ( B _ { p + 1 } ) \\dot { = } k", + "type": "inline_equation" + }, + { + "bbox": [ + 177, + 321, + 249, + 334 + ], + "score": 1.0, + "content": ". Also, notice that", + "type": "text" + }, + { + "bbox": [ + 249, + 322, + 317, + 333 + ], + "score": 0.92, + "content": "A _ { i } = W _ { i + 1 } A _ { i + 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 318, + 321, + 335, + 334 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 336, + 322, + 394, + 333 + ], + "score": 0.92, + "content": "B _ { i + 1 } = B _ { i } W _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 395, + 321, + 429, + 334 + ], + "score": 1.0, + "content": ", so that", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 15.5 + }, + { + "type": "interline_equation", + "bbox": [ + 112, + 337, + 500, + 351 + ], + "lines": [ + { + "bbox": [ + 112, + 337, + 500, + 351 + ], + "spans": [ + { + "bbox": [ + 112, + 337, + 500, + 351 + ], + "score": 0.85, + "content": "\\operatorname { r a n k } ( A _ { 1 } ) \\leq \\operatorname { r a n k } ( A _ { 2 } ) \\leq \\cdots \\leq \\operatorname { r a n k } ( A _ { p } ) { \\mathrm { ~ a n d ~ } } \\operatorname { r a n k } ( B _ { H + 1 } ) \\leq \\operatorname { r a n k } ( B _ { H } ) \\leq \\cdots \\leq \\operatorname { r a n k } ( B _ { p + 1 } ) .", + "type": "interline_equation", + "image_path": "78fca07875c35b77f1ef8b61935e19ed1918a92b34cddd97c725afe3afde15fa.jpg" + } + ] + } + ], + "index": 18, + "virtual_lines": [ + { + "bbox": [ + 112, + 337, + 500, + 351 + ], + "spans": [], + "index": 18 + } + ] + }, + { + "type": "text", + "bbox": [ + 109, + 354, + 495, + 368 + ], + "lines": [ + { + "bbox": [ + 106, + 353, + 494, + 369 + ], + "spans": [ + { + "bbox": [ + 106, + 353, + 182, + 369 + ], + "score": 1.0, + "content": "However, we have", + "type": "text" + }, + { + "bbox": [ + 182, + 355, + 241, + 367 + ], + "score": 0.91, + "content": "k \\leq \\mathrm { r a n k } ( A _ { 1 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 241, + 353, + 259, + 369 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 260, + 355, + 331, + 367 + ], + "score": 0.91, + "content": "k \\leq \\mathrm { r a n k } ( B _ { H + 1 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 332, + 353, + 457, + 369 + ], + "score": 1.0, + "content": ", so the ranks are all identically", + "type": "text" + }, + { + "bbox": [ + 458, + 356, + 464, + 365 + ], + "score": 0.8, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 465, + 353, + 494, + 369 + ], + "score": 1.0, + "content": ". Also,", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 19 + }, + { + "type": "interline_equation", + "bbox": [ + 124, + 371, + 484, + 385 + ], + "lines": [ + { + "bbox": [ + 124, + 371, + 484, + 385 + ], + "spans": [ + { + "bbox": [ + 124, + 371, + 484, + 385 + ], + "score": 0.83, + "content": "\\operatorname { r o w } ( A _ { 1 } ) \\subset \\operatorname { r o w } ( A _ { 2 } ) \\subset \\cdots \\subset \\operatorname { r o w } ( A _ { p } ) { \\mathrm { ~ a n d ~ } } \\operatorname { c o l } ( B _ { H + 1 } ) \\subset \\operatorname { c o l } ( B _ { H } ) \\subset \\cdots \\subset \\operatorname { c o l } ( B _ { p + 1 } ) ,", + "type": "interline_equation", + "image_path": "0a448f4a37400c1b93651acd842dd80b9a9c5f15ec945d705774d5cc8d02e96e.jpg" + } + ] + } + ], + "index": 20, + "virtual_lines": [ + { + "bbox": [ + 124, + 371, + 484, + 385 + ], + "spans": [], + "index": 20 + } + ] + }, + { + "type": "text", + "bbox": [ + 114, + 389, + 486, + 401 + ], + "lines": [ + { + "bbox": [ + 111, + 388, + 488, + 404 + ], + "spans": [ + { + "bbox": [ + 111, + 388, + 440, + 404 + ], + "score": 1.0, + "content": "ut it was just shown that the these spaces have the same dimensions, which equals", + "type": "text" + }, + { + "bbox": [ + 440, + 390, + 446, + 399 + ], + "score": 0.82, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 447, + 388, + 488, + 404 + ], + "score": 1.0, + "content": ", meaning", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 21 + }, + { + "type": "interline_equation", + "bbox": [ + 124, + 406, + 484, + 419 + ], + "lines": [ + { + "bbox": [ + 124, + 406, + 484, + 419 + ], + "spans": [ + { + "bbox": [ + 124, + 406, + 484, + 419 + ], + "score": 0.37, + "content": "\\operatorname { r o w } ( A _ { 1 } ) = \\operatorname { r o w } ( A _ { 2 } ) = \\cdots = \\operatorname { r o w } ( A _ { p } ) { \\mathrm { ~ a n d ~ } } \\operatorname { c o l } ( B _ { H + 1 } ) = \\operatorname { c o l } ( B _ { H } ) = \\cdots = \\operatorname { c o l } ( B _ { p + 1 } ) .", + "type": "interline_equation", + "image_path": "ac83487c40ffb9f00f68793f0a49ee68473c88a8a0dfe1fc025822a98b405cf4.jpg" + } + ] + } + ], + "index": 22, + "virtual_lines": [ + { + "bbox": [ + 124, + 406, + 484, + 419 + ], + "spans": [], + "index": 22 + } + ] + }, + { + "type": "text", + "bbox": [ + 108, + 424, + 501, + 446 + ], + "lines": [ + { + "bbox": [ + 106, + 424, + 504, + 436 + ], + "spans": [ + { + "bbox": [ + 106, + 424, + 504, + 436 + ], + "score": 1.0, + "content": "Using this observation, we can now state a proposition showing necessary and sufficient conditions", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 434, + 309, + 447 + ], + "spans": [ + { + "bbox": [ + 106, + 434, + 150, + 447 + ], + "score": 1.0, + "content": "for a tuple", + "type": "text" + }, + { + "bbox": [ + 150, + 435, + 185, + 446 + ], + "score": 0.92, + "content": "W \\in \\mathcal { V } _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 185, + 434, + 279, + 447 + ], + "score": 1.0, + "content": "to be a critical point of", + "type": "text" + }, + { + "bbox": [ + 279, + 435, + 305, + 447 + ], + "score": 0.92, + "content": "L ( W )", + "type": "inline_equation" + }, + { + "bbox": [ + 306, + 434, + 309, + 447 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 23.5 + }, + { + "type": "text", + "bbox": [ + 106, + 448, + 500, + 462 + ], + "lines": [ + { + "bbox": [ + 105, + 447, + 500, + 463 + ], + "spans": [ + { + "bbox": [ + 105, + 447, + 209, + 463 + ], + "score": 1.0, + "content": "Proposition 3.6. A tuple", + "type": "text" + }, + { + "bbox": [ + 209, + 449, + 244, + 460 + ], + "score": 0.91, + "content": "W \\in \\nu _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 244, + 447, + 326, + 463 + ], + "score": 1.0, + "content": "is a critical point of", + "type": "text" + }, + { + "bbox": [ + 327, + 450, + 334, + 459 + ], + "score": 0.71, + "content": "L", + "type": "inline_equation" + }, + { + "bbox": [ + 335, + 447, + 389, + 463 + ], + "score": 1.0, + "content": "if and only if", + "type": "text" + }, + { + "bbox": [ + 389, + 450, + 428, + 461 + ], + "score": 0.91, + "content": "A _ { p } E = 0", + "type": "inline_equation" + }, + { + "bbox": [ + 429, + 447, + 447, + 463 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 447, + 450, + 496, + 462 + ], + "score": 0.91, + "content": "E B _ { p + 1 } = 0", + "type": "inline_equation" + }, + { + "bbox": [ + 496, + 447, + 500, + 463 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 25 + }, + { + "type": "text", + "bbox": [ + 106, + 473, + 505, + 515 + ], + "lines": [ + { + "bbox": [ + 105, + 472, + 504, + 490 + ], + "spans": [ + { + "bbox": [ + 105, + 472, + 172, + 490 + ], + "score": 1.0, + "content": "Proof. (If part)", + "type": "text" + }, + { + "bbox": [ + 173, + 475, + 218, + 487 + ], + "score": 0.92, + "content": "A _ { p } E = 0", + "type": "inline_equation" + }, + { + "bbox": [ + 218, + 472, + 272, + 490 + ], + "score": 1.0, + "content": "implies that", + "type": "text" + }, + { + "bbox": [ + 272, + 474, + 453, + 487 + ], + "score": 0.91, + "content": "\\operatorname { c o l } ( E ) \\subset \\operatorname { r o w } ( A _ { p } ) ^ { \\perp } = \\cdot \\cdot \\cdot = \\operatorname { r o w } ( A _ { 1 } ) ^ { \\perp }", + "type": "inline_equation" + }, + { + "bbox": [ + 453, + 472, + 472, + 490 + ], + "score": 1.0, + "content": ", so", + "type": "text" + }, + { + "bbox": [ + 472, + 473, + 504, + 488 + ], + "score": 0.85, + "content": "\\begin{array} { r } { \\frac { \\partial L } { \\partial W _ { i } } \\ = } \\end{array}", + "type": "inline_equation" + } + ], + "index": 26 + }, + { + "bbox": [ + 107, + 487, + 505, + 501 + ], + "spans": [ + { + "bbox": [ + 107, + 488, + 195, + 500 + ], + "score": 0.92, + "content": "A _ { i } E B _ { i } = 0 \\cdot B _ { i } = 0 .", + "type": "inline_equation" + }, + { + "bbox": [ + 196, + 487, + 213, + 501 + ], + "score": 1.0, + "content": ", for", + "type": "text" + }, + { + "bbox": [ + 214, + 489, + 265, + 500 + ], + "score": 0.89, + "content": "i = 1 , \\ldots , p", + "type": "inline_equation" + }, + { + "bbox": [ + 265, + 487, + 312, + 501 + ], + "score": 1.0, + "content": ". Similarly,", + "type": "text" + }, + { + "bbox": [ + 312, + 489, + 363, + 501 + ], + "score": 0.92, + "content": "E B _ { p + 1 } = 0", + "type": "inline_equation" + }, + { + "bbox": [ + 363, + 487, + 397, + 501 + ], + "score": 1.0, + "content": "implies", + "type": "text" + }, + { + "bbox": [ + 397, + 488, + 505, + 501 + ], + "score": 0.83, + "content": "\\operatorname { r o w } ( E ) \\subset \\operatorname { c o l } ( B _ { p + 1 } ) ^ { \\perp } =", + "type": "inline_equation" + } + ], + "index": 27 + }, + { + "bbox": [ + 107, + 497, + 424, + 517 + ], + "spans": [ + { + "bbox": [ + 107, + 501, + 184, + 514 + ], + "score": 0.88, + "content": "\\cdot \\cdot \\cdot = \\mathrm { c o l } ( B _ { H + 1 } ) ^ { \\perp }", + "type": "inline_equation" + }, + { + "bbox": [ + 184, + 497, + 200, + 517 + ], + "score": 1.0, + "content": ", so", + "type": "text" + }, + { + "bbox": [ + 200, + 500, + 316, + 515 + ], + "score": 0.9, + "content": "\\begin{array} { r } { \\frac { \\partial L } { \\partial W _ { i } } = A _ { i } E B _ { i } = A _ { i } \\cdot 0 = 0 } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 316, + 497, + 331, + 517 + ], + "score": 1.0, + "content": "for", + "type": "text" + }, + { + "bbox": [ + 332, + 502, + 420, + 513 + ], + "score": 0.91, + "content": "i = p + 1 , \\ldots , H + 1", + "type": "inline_equation" + }, + { + "bbox": [ + 420, + 497, + 424, + 517 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 27 + }, + { + "type": "text", + "bbox": [ + 106, + 519, + 381, + 535 + ], + "lines": [ + { + "bbox": [ + 106, + 519, + 381, + 537 + ], + "spans": [ + { + "bbox": [ + 106, + 519, + 199, + 537 + ], + "score": 1.0, + "content": "(Only if part) We have", + "type": "text" + }, + { + "bbox": [ + 199, + 520, + 279, + 536 + ], + "score": 0.95, + "content": "\\begin{array} { r } { \\frac { \\partial L } { \\partial W _ { i } } = A _ { i } E B _ { i } = 0 } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 279, + 519, + 306, + 537 + ], + "score": 1.0, + "content": "for all", + "type": "text" + }, + { + "bbox": [ + 307, + 522, + 311, + 531 + ], + "score": 0.7, + "content": "i", + "type": "inline_equation" + }, + { + "bbox": [ + 312, + 519, + 381, + 537 + ], + "score": 1.0, + "content": ". This means that", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 29 + }, + { + "type": "interline_equation", + "bbox": [ + 173, + 539, + 437, + 572 + ], + "lines": [ + { + "bbox": [ + 173, + 539, + 437, + 572 + ], + "spans": [ + { + "bbox": [ + 173, + 539, + 437, + 572 + ], + "score": 0.88, + "content": "\\begin{array} { r l } & { \\quad \\mathrm { c o l } ( E B _ { i } ) \\subset \\mathrm { r o w } ( A _ { i } ) ^ { \\perp } = \\mathrm { r o w } ( A _ { p } ) ^ { \\perp } \\mathrm { f o r } i = 1 , \\ldots , p } \\\\ & { \\quad \\mathrm { r o w } ( A _ { i } E ) \\subset \\mathrm { c o l } ( B _ { i } ) ^ { \\perp } = \\mathrm { c o l } ( B _ { p + 1 } ) ^ { \\perp } \\mathrm { f o r } i = p + 1 , \\ldots , H + 1 . } \\end{array}", + "type": "interline_equation", + "image_path": "d5940b8d098dcccc7c6173cead353f7832657fbb9d17335d5fc15b7cdcf2527d.jpg" + } + ] + } + ], + "index": 31, + "virtual_lines": [ + { + "bbox": [ + 173, + 539, + 437, + 550.0 + ], + "spans": [], + "index": 30 + }, + { + "bbox": [ + 173, + 550.0, + 437, + 561.0 + ], + "spans": [], + "index": 31 + }, + { + "bbox": [ + 173, + 561.0, + 437, + 572.0 + ], + "spans": [], + "index": 32 + } + ] + }, + { + "type": "text", + "bbox": [ + 103, + 577, + 504, + 602 + ], + "lines": [ + { + "bbox": [ + 105, + 575, + 505, + 591 + ], + "spans": [ + { + "bbox": [ + 105, + 575, + 175, + 591 + ], + "score": 1.0, + "content": "Now recall that", + "type": "text" + }, + { + "bbox": [ + 175, + 578, + 189, + 588 + ], + "score": 0.89, + "content": "B _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 189, + 575, + 210, + 591 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 210, + 577, + 236, + 589 + ], + "score": 0.92, + "content": "A _ { H + 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 237, + 575, + 344, + 591 + ], + "score": 1.0, + "content": "are identity matrices, so", + "type": "text" + }, + { + "bbox": [ + 344, + 576, + 435, + 589 + ], + "score": 0.88, + "content": "\\mathrm { c o l } ( E ) \\subset \\mathrm { r o w } ( A _ { p } ) ^ { \\perp }", + "type": "inline_equation" + }, + { + "bbox": [ + 436, + 575, + 457, + 591 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 457, + 577, + 505, + 589 + ], + "score": 0.79, + "content": "\\operatorname { r o w } ( E ) ~ \\subset", + "type": "inline_equation" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 588, + 328, + 603 + ], + "spans": [ + { + "bbox": [ + 106, + 588, + 156, + 602 + ], + "score": 0.92, + "content": "\\mathrm { c o l } ( B _ { p + 1 } ) ^ { \\perp }", + "type": "inline_equation" + }, + { + "bbox": [ + 156, + 588, + 216, + 603 + ], + "score": 1.0, + "content": ", which proves", + "type": "text" + }, + { + "bbox": [ + 216, + 590, + 256, + 601 + ], + "score": 0.92, + "content": "A _ { p } E = 0", + "type": "inline_equation" + }, + { + "bbox": [ + 256, + 588, + 273, + 603 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 274, + 590, + 323, + 601 + ], + "score": 0.92, + "content": "E B _ { p + 1 } = 0", + "type": "inline_equation" + }, + { + "bbox": [ + 324, + 588, + 328, + 603 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 33.5 + }, + { + "type": "text", + "bbox": [ + 106, + 613, + 506, + 649 + ], + "lines": [ + { + "bbox": [ + 105, + 613, + 505, + 625 + ], + "spans": [ + { + "bbox": [ + 105, + 613, + 505, + 625 + ], + "score": 1.0, + "content": "Now we present a proposition that specifies the necessary and sufficient condition in which a critical", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 623, + 506, + 637 + ], + "spans": [ + { + "bbox": [ + 105, + 623, + 140, + 637 + ], + "score": 1.0, + "content": "point of", + "type": "text" + }, + { + "bbox": [ + 140, + 624, + 166, + 636 + ], + "score": 0.91, + "content": "L ( W )", + "type": "inline_equation" + }, + { + "bbox": [ + 167, + 623, + 177, + 637 + ], + "score": 1.0, + "content": "in", + "type": "text" + }, + { + "bbox": [ + 178, + 625, + 190, + 635 + ], + "score": 0.88, + "content": "\\nu _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 190, + 623, + 417, + 637 + ], + "score": 1.0, + "content": "is a global minimum. Recall that when we take the SVD", + "type": "text" + }, + { + "bbox": [ + 417, + 623, + 506, + 636 + ], + "score": 0.91, + "content": "{ \\cal Y } X ^ { T } ( X X ^ { T } ) ^ { - 1 } X =", + "type": "inline_equation" + } + ], + "index": 36 + }, + { + "bbox": [ + 106, + 635, + 444, + 650 + ], + "spans": [ + { + "bbox": [ + 106, + 636, + 137, + 648 + ], + "score": 0.85, + "content": "U { \\boldsymbol { \\Sigma } } V ^ { T }", + "type": "inline_equation" + }, + { + "bbox": [ + 137, + 635, + 141, + 650 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 141, + 636, + 189, + 648 + ], + "score": 0.88, + "content": "\\hat { U } \\in \\mathbb { R } ^ { d _ { y } \\times k }", + "type": "inline_equation" + }, + { + "bbox": [ + 189, + 635, + 375, + 650 + ], + "score": 1.0, + "content": "is defined to be a matrix consisting of the first", + "type": "text" + }, + { + "bbox": [ + 375, + 637, + 382, + 647 + ], + "score": 0.78, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 382, + 635, + 430, + 650 + ], + "score": 1.0, + "content": "columns of", + "type": "text" + }, + { + "bbox": [ + 430, + 638, + 439, + 647 + ], + "score": 0.84, + "content": "U", + "type": "inline_equation" + }, + { + "bbox": [ + 439, + 635, + 444, + 650 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 37 + } + ], + "index": 36 + }, + { + "type": "text", + "bbox": [ + 105, + 650, + 506, + 677 + ], + "lines": [ + { + "bbox": [ + 105, + 649, + 506, + 666 + ], + "spans": [ + { + "bbox": [ + 105, + 649, + 248, + 666 + ], + "score": 1.0, + "content": "Proposition 3.7. A critical point", + "type": "text" + }, + { + "bbox": [ + 248, + 651, + 289, + 663 + ], + "score": 0.9, + "content": "W \\in \\mathsf { V } _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 289, + 649, + 303, + 666 + ], + "score": 1.0, + "content": "of", + "type": "text" + }, + { + "bbox": [ + 303, + 651, + 329, + 664 + ], + "score": 0.91, + "content": "L ( W )", + "type": "inline_equation" + }, + { + "bbox": [ + 330, + 649, + 496, + 666 + ], + "score": 1.0, + "content": "is a global minimum point if and only", + "type": "text" + }, + { + "bbox": [ + 497, + 652, + 506, + 663 + ], + "score": 0.4, + "content": "i f", + "type": "inline_equation" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 662, + 287, + 678 + ], + "spans": [ + { + "bbox": [ + 105, + 663, + 284, + 677 + ], + "score": 0.87, + "content": "\\operatorname { c o l } ( W _ { H + 1 } \\cdot \\cdot \\cdot W _ { p + 1 } ) = \\operatorname { r o w } ( A _ { p } ) = \\operatorname { c o l } ( { \\hat { U } } )", + "type": "inline_equation" + }, + { + "bbox": [ + 284, + 662, + 287, + 678 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 39 + } + ], + "index": 38.5 + }, + { + "type": "text", + "bbox": [ + 106, + 687, + 505, + 714 + ], + "lines": [ + { + "bbox": [ + 106, + 687, + 504, + 700 + ], + "spans": [ + { + "bbox": [ + 106, + 687, + 162, + 700 + ], + "score": 1.0, + "content": "Proof. Since", + "type": "text" + }, + { + "bbox": [ + 162, + 689, + 174, + 698 + ], + "score": 0.75, + "content": "W", + "type": "inline_equation" + }, + { + "bbox": [ + 175, + 687, + 371, + 700 + ], + "score": 1.0, + "content": "is a critical point, by Proposition 3.6 we have", + "type": "text" + }, + { + "bbox": [ + 371, + 689, + 415, + 700 + ], + "score": 0.92, + "content": "A _ { p } E = 0", + "type": "inline_equation" + }, + { + "bbox": [ + 416, + 687, + 504, + 700 + ], + "score": 1.0, + "content": ". Also note from the", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 104, + 698, + 360, + 716 + ], + "spans": [ + { + "bbox": [ + 104, + 698, + 162, + 716 + ], + "score": 1.0, + "content": "definitions of", + "type": "text" + }, + { + "bbox": [ + 162, + 701, + 174, + 712 + ], + "score": 0.88, + "content": "A _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 174, + 698, + 199, + 716 + ], + "score": 1.0, + "content": "’s and", + "type": "text" + }, + { + "bbox": [ + 199, + 701, + 210, + 712 + ], + "score": 0.89, + "content": "B _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 211, + 698, + 236, + 716 + ], + "score": 1.0, + "content": "’s that", + "type": "text" + }, + { + "bbox": [ + 236, + 700, + 342, + 714 + ], + "score": 0.93, + "content": "W _ { H + 1 } \\cdot \\cdot \\cdot W _ { 1 } = A _ { p } ^ { T } B _ { p + 1 } ^ { T }", + "type": "inline_equation" + }, + { + "bbox": [ + 343, + 698, + 360, + 716 + ], + "score": 1.0, + "content": ", so", + "type": "text" + } + ], + "index": 41 + } + ], + "index": 40.5 + }, + { + "type": "interline_equation", + "bbox": [ + 159, + 718, + 451, + 734 + ], + "lines": [ + { + "bbox": [ + 159, + 718, + 451, + 734 + ], + "spans": [ + { + "bbox": [ + 159, + 718, + 451, + 734 + ], + "score": 0.9, + "content": "A _ { p } E = A _ { p } ( A _ { p } ^ { T } B _ { p + 1 } ^ { T } X - Y ) X ^ { T } = A _ { p } A _ { p } ^ { T } B _ { p + 1 } ^ { T } X X ^ { T } - A _ { p } Y X ^ { T } = 0 .", + "type": "interline_equation", + "image_path": "133b976ba55711afec0c475d6297affe61800958d98c72ff831b8a6b9f66f5a1.jpg" + } + ] + } + ], + "index": 42, + "virtual_lines": [ + { + "bbox": [ + 159, + 718, + 451, + 734 + ], + "spans": [], + "index": 42 + } + ] + } + ], + "page_idx": 5, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 107, + 26, + 294, + 38 + ], + "lines": [ + { + "bbox": [ + 106, + 25, + 294, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 25, + 294, + 38 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2018", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 302, + 751, + 309, + 760 + ], + "lines": [ + { + "bbox": [ + 302, + 751, + 309, + 762 + ], + "spans": [ + { + "bbox": [ + 302, + 751, + 309, + 762 + ], + "score": 1.0, + "content": "6", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 495, + 590, + 504, + 600 + ], + "lines": [ + { + "bbox": [ + 496, + 591, + 504, + 600 + ], + "spans": [ + { + "bbox": [ + 496, + 591, + 504, + 600 + ], + "score": 0.999, + "content": "□", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "title", + "bbox": [ + 107, + 82, + 237, + 94 + ], + "lines": [ + { + "bbox": [ + 106, + 82, + 237, + 94 + ], + "spans": [ + { + "bbox": [ + 106, + 82, + 237, + 94 + ], + "score": 1.0, + "content": "3.4 PROOF OF THEOREM 2.2", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "text", + "bbox": [ + 105, + 102, + 504, + 126 + ], + "lines": [ + { + "bbox": [ + 105, + 102, + 505, + 115 + ], + "spans": [ + { + "bbox": [ + 105, + 102, + 363, + 115 + ], + "score": 1.0, + "content": "In this section we prove Theorem 2.2, which tackles the case", + "type": "text" + }, + { + "bbox": [ + 363, + 103, + 439, + 115 + ], + "score": 0.92, + "content": "k < \\operatorname* { m i n } \\{ d _ { x } , d _ { y } \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 439, + 102, + 505, + 115 + ], + "score": 1.0, + "content": ". Note that this", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 114, + 273, + 126 + ], + "spans": [ + { + "bbox": [ + 105, + 114, + 222, + 126 + ], + "score": 1.0, + "content": "assumption also implies that", + "type": "text" + }, + { + "bbox": [ + 222, + 114, + 269, + 126 + ], + "score": 0.92, + "content": "1 \\leq p \\leq H", + "type": "inline_equation" + }, + { + "bbox": [ + 269, + 114, + 273, + 126 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 2 + } + ], + "index": 1.5, + "bbox_fs": [ + 105, + 102, + 505, + 126 + ] + }, + { + "type": "text", + "bbox": [ + 108, + 131, + 266, + 142 + ], + "lines": [ + { + "bbox": [ + 106, + 131, + 266, + 144 + ], + "spans": [ + { + "bbox": [ + 106, + 131, + 266, + 144 + ], + "score": 1.0, + "content": "As for the proof of Theorem 2.1, define", + "type": "text" + } + ], + "index": 3 + } + ], + "index": 3, + "bbox_fs": [ + 106, + 131, + 266, + 144 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 193, + 147, + 417, + 161 + ], + "lines": [ + { + "bbox": [ + 193, + 147, + 417, + 161 + ], + "spans": [ + { + "bbox": [ + 193, + 147, + 417, + 161 + ], + "score": 0.88, + "content": "\\mathcal { V } _ { 1 } : = \\left\\{ ( W _ { 1 } , \\ldots , W _ { H + 1 } ) : \\operatorname { r a n k } ( W _ { H + 1 } \\cdot \\cdot \\cdot W _ { 1 } ) = k \\right\\} .", + "type": "interline_equation", + "image_path": "c679bef42b4da682812002d2606aa607c6d16cb36d2f7f11f1657f207b7208f6.jpg" + } + ] + } + ], + "index": 4, + "virtual_lines": [ + { + "bbox": [ + 193, + 147, + 417, + 161 + ], + "spans": [], + "index": 4 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 165, + 506, + 210 + ], + "lines": [ + { + "bbox": [ + 106, + 164, + 505, + 178 + ], + "spans": [ + { + "bbox": [ + 106, + 164, + 365, + 178 + ], + "score": 1.0, + "content": "The globally optimal point of the relaxed problem (2) has rank", + "type": "text" + }, + { + "bbox": [ + 365, + 166, + 372, + 176 + ], + "score": 0.82, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 372, + 164, + 505, + 178 + ], + "score": 1.0, + "content": ", as seen in (4). Thus, any point", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 176, + 505, + 188 + ], + "spans": [ + { + "bbox": [ + 106, + 176, + 151, + 188 + ], + "score": 1.0, + "content": "outside of", + "type": "text" + }, + { + "bbox": [ + 151, + 177, + 163, + 187 + ], + "score": 0.88, + "content": "\\nu _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 163, + 176, + 505, + 188 + ], + "score": 1.0, + "content": "cannot be a global minimum. Then, by Kawaguchi (2016, Theorem 2.3.(iii)) and", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 186, + 506, + 200 + ], + "spans": [ + { + "bbox": [ + 105, + 186, + 303, + 200 + ], + "score": 1.0, + "content": "Lemma B.1, it follows that any critical point in", + "type": "text" + }, + { + "bbox": [ + 304, + 187, + 316, + 199 + ], + "score": 0.89, + "content": "\\mathcal { V } _ { 1 } ^ { c }", + "type": "inline_equation" + }, + { + "bbox": [ + 317, + 186, + 506, + 200 + ], + "score": 1.0, + "content": "must be a saddle point. The remaining proof", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 198, + 200, + 210 + ], + "spans": [ + { + "bbox": [ + 106, + 198, + 183, + 210 + ], + "score": 1.0, + "content": "considers points in", + "type": "text" + }, + { + "bbox": [ + 184, + 198, + 195, + 209 + ], + "score": 0.89, + "content": "\\mathcal { V } _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 196, + 198, + 200, + 210 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 6.5, + "bbox_fs": [ + 105, + 164, + 506, + 210 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 214, + 454, + 227 + ], + "lines": [ + { + "bbox": [ + 105, + 214, + 455, + 227 + ], + "spans": [ + { + "bbox": [ + 105, + 214, + 455, + 227 + ], + "score": 1.0, + "content": "For this section, let us introduce some additional notations to ease presentation. Define", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 9, + "bbox_fs": [ + 105, + 214, + 455, + 227 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 120, + 230, + 489, + 264 + ], + "lines": [ + { + "bbox": [ + 120, + 230, + 489, + 264 + ], + "spans": [ + { + "bbox": [ + 120, + 230, + 489, + 264 + ], + "score": 0.92, + "content": "\\begin{array} { r l } & { E : = ( W _ { H + 1 } \\cdot \\cdot \\cdot W _ { 1 } X - Y ) X ^ { T } \\in \\mathbb { R } ^ { d _ { y } \\times d _ { x } } , } \\\\ & { A _ { i } : = W _ { i + 1 } ^ { T } \\cdot \\cdot \\cdot W _ { H + 1 } ^ { T } \\in \\mathbb { R } ^ { d _ { i } \\times d _ { y } } , \\ B _ { i } : = W _ { 1 } ^ { T } \\cdot \\cdot \\cdot W _ { i - 1 } ^ { T } \\in \\mathbb { R } ^ { d _ { x } \\times d _ { i - 1 } } , \\ i = 1 , \\dots , H + 1 , } \\end{array}", + "type": "interline_equation", + "image_path": "457a0ca2581a2e0aea55b134454c35b398708916810fcb378d1b119dc2bf6fbc.jpg" + } + ] + } + ], + "index": 11, + "virtual_lines": [ + { + "bbox": [ + 120, + 230, + 489, + 241.33333333333334 + ], + "spans": [], + "index": 10 + }, + { + "bbox": [ + 120, + 241.33333333333334, + 489, + 252.66666666666669 + ], + "spans": [], + "index": 11 + }, + { + "bbox": [ + 120, + 252.66666666666669, + 489, + 264.0 + ], + "spans": [], + "index": 12 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 267, + 392, + 281 + ], + "lines": [ + { + "bbox": [ + 105, + 266, + 393, + 283 + ], + "spans": [ + { + "bbox": [ + 105, + 266, + 136, + 283 + ], + "score": 1.0, + "content": "so that", + "type": "text" + }, + { + "bbox": [ + 136, + 267, + 197, + 282 + ], + "score": 0.94, + "content": "\\begin{array} { r } { \\frac { \\partial L } { \\partial W _ { i } } = A _ { i } E B _ { i } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 197, + 266, + 248, + 283 + ], + "score": 1.0, + "content": ". Notice that", + "type": "text" + }, + { + "bbox": [ + 248, + 269, + 274, + 280 + ], + "score": 0.92, + "content": "A _ { H + 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 275, + 266, + 292, + 283 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 293, + 269, + 306, + 280 + ], + "score": 0.89, + "content": "B _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 306, + 266, + 393, + 283 + ], + "score": 1.0, + "content": "are identity matrices.", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 13, + "bbox_fs": [ + 105, + 266, + 393, + 283 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 286, + 505, + 334 + ], + "lines": [ + { + "bbox": [ + 105, + 286, + 505, + 299 + ], + "spans": [ + { + "bbox": [ + 105, + 286, + 208, + 299 + ], + "score": 1.0, + "content": "Now consider any tuple", + "type": "text" + }, + { + "bbox": [ + 208, + 287, + 247, + 298 + ], + "score": 0.91, + "content": "W \\in \\mathcal { V } _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 248, + 286, + 349, + 299 + ], + "score": 1.0, + "content": ". Since the full product", + "type": "text" + }, + { + "bbox": [ + 349, + 287, + 406, + 298 + ], + "score": 0.91, + "content": "W _ { H + 1 } \\cdots W _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 406, + 286, + 447, + 299 + ], + "score": 1.0, + "content": "has rank", + "type": "text" + }, + { + "bbox": [ + 447, + 288, + 454, + 297 + ], + "score": 0.8, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 454, + 286, + 505, + 299 + ], + "score": 1.0, + "content": ", any partial", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 297, + 505, + 311 + ], + "spans": [ + { + "bbox": [ + 105, + 297, + 144, + 311 + ], + "score": 1.0, + "content": "products", + "type": "text" + }, + { + "bbox": [ + 144, + 298, + 156, + 309 + ], + "score": 0.89, + "content": "A _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 156, + 297, + 175, + 311 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 175, + 298, + 187, + 309 + ], + "score": 0.88, + "content": "B _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 187, + 297, + 234, + 311 + ], + "score": 1.0, + "content": "must have", + "type": "text" + }, + { + "bbox": [ + 234, + 298, + 294, + 309 + ], + "score": 0.9, + "content": "\\operatorname { r a n k } ( A _ { i } ) \\geq k", + "type": "inline_equation" + }, + { + "bbox": [ + 294, + 297, + 314, + 311 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 314, + 298, + 374, + 310 + ], + "score": 0.89, + "content": "\\operatorname { r a n k } ( B _ { i } ) \\geq k", + "type": "inline_equation" + }, + { + "bbox": [ + 374, + 297, + 406, + 311 + ], + "score": 1.0, + "content": ", for all", + "type": "text" + }, + { + "bbox": [ + 406, + 299, + 411, + 308 + ], + "score": 0.73, + "content": "i", + "type": "inline_equation" + }, + { + "bbox": [ + 411, + 297, + 480, + 311 + ], + "score": 1.0, + "content": ". Then, consider", + "type": "text" + }, + { + "bbox": [ + 480, + 298, + 505, + 310 + ], + "score": 0.9, + "content": "A _ { p } \\in", + "type": "inline_equation" + } + ], + "index": 15 + }, + { + "bbox": [ + 107, + 306, + 506, + 325 + ], + "spans": [ + { + "bbox": [ + 107, + 309, + 134, + 320 + ], + "score": 0.91, + "content": "\\mathbb { R } ^ { k \\times d _ { y } }", + "type": "inline_equation" + }, + { + "bbox": [ + 134, + 306, + 151, + 325 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 152, + 309, + 214, + 322 + ], + "score": 0.92, + "content": "B _ { p + 1 } \\in \\mathbb { R } ^ { d _ { x } \\times k }", + "type": "inline_equation" + }, + { + "bbox": [ + 214, + 306, + 243, + 325 + ], + "score": 1.0, + "content": ". Since", + "type": "text" + }, + { + "bbox": [ + 243, + 310, + 302, + 322 + ], + "score": 0.86, + "content": "\\operatorname { r a n k } ( A _ { p } ) \\leq k", + "type": "inline_equation" + }, + { + "bbox": [ + 302, + 306, + 319, + 325 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 319, + 310, + 388, + 322 + ], + "score": 0.89, + "content": "\\operatorname { r a n k } ( B _ { p + 1 } ) \\leq k", + "type": "inline_equation" + }, + { + "bbox": [ + 389, + 306, + 465, + 325 + ], + "score": 1.0, + "content": ", we can see that ra", + "type": "text" + }, + { + "bbox": [ + 466, + 311, + 506, + 323 + ], + "score": 0.7, + "content": "\\operatorname { l k } ( A _ { p } ) =", + "type": "inline_equation" + } + ], + "index": 16 + }, + { + "bbox": [ + 107, + 321, + 429, + 334 + ], + "spans": [ + { + "bbox": [ + 107, + 321, + 176, + 334 + ], + "score": 0.9, + "content": "\\mathrm { r a n k } ( B _ { p + 1 } ) \\dot { = } k", + "type": "inline_equation" + }, + { + "bbox": [ + 177, + 321, + 249, + 334 + ], + "score": 1.0, + "content": ". Also, notice that", + "type": "text" + }, + { + "bbox": [ + 249, + 322, + 317, + 333 + ], + "score": 0.92, + "content": "A _ { i } = W _ { i + 1 } A _ { i + 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 318, + 321, + 335, + 334 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 336, + 322, + 394, + 333 + ], + "score": 0.92, + "content": "B _ { i + 1 } = B _ { i } W _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 395, + 321, + 429, + 334 + ], + "score": 1.0, + "content": ", so that", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 15.5, + "bbox_fs": [ + 105, + 286, + 506, + 334 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 112, + 337, + 500, + 351 + ], + "lines": [ + { + "bbox": [ + 112, + 337, + 500, + 351 + ], + "spans": [ + { + "bbox": [ + 112, + 337, + 500, + 351 + ], + "score": 0.85, + "content": "\\operatorname { r a n k } ( A _ { 1 } ) \\leq \\operatorname { r a n k } ( A _ { 2 } ) \\leq \\cdots \\leq \\operatorname { r a n k } ( A _ { p } ) { \\mathrm { ~ a n d ~ } } \\operatorname { r a n k } ( B _ { H + 1 } ) \\leq \\operatorname { r a n k } ( B _ { H } ) \\leq \\cdots \\leq \\operatorname { r a n k } ( B _ { p + 1 } ) .", + "type": "interline_equation", + "image_path": "78fca07875c35b77f1ef8b61935e19ed1918a92b34cddd97c725afe3afde15fa.jpg" + } + ] + } + ], + "index": 18, + "virtual_lines": [ + { + "bbox": [ + 112, + 337, + 500, + 351 + ], + "spans": [], + "index": 18 + } + ] + }, + { + "type": "text", + "bbox": [ + 109, + 354, + 495, + 368 + ], + "lines": [ + { + "bbox": [ + 106, + 353, + 494, + 369 + ], + "spans": [ + { + "bbox": [ + 106, + 353, + 182, + 369 + ], + "score": 1.0, + "content": "However, we have", + "type": "text" + }, + { + "bbox": [ + 182, + 355, + 241, + 367 + ], + "score": 0.91, + "content": "k \\leq \\mathrm { r a n k } ( A _ { 1 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 241, + 353, + 259, + 369 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 260, + 355, + 331, + 367 + ], + "score": 0.91, + "content": "k \\leq \\mathrm { r a n k } ( B _ { H + 1 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 332, + 353, + 457, + 369 + ], + "score": 1.0, + "content": ", so the ranks are all identically", + "type": "text" + }, + { + "bbox": [ + 458, + 356, + 464, + 365 + ], + "score": 0.8, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 465, + 353, + 494, + 369 + ], + "score": 1.0, + "content": ". Also,", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 19, + "bbox_fs": [ + 106, + 353, + 494, + 369 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 124, + 371, + 484, + 385 + ], + "lines": [ + { + "bbox": [ + 124, + 371, + 484, + 385 + ], + "spans": [ + { + "bbox": [ + 124, + 371, + 484, + 385 + ], + "score": 0.83, + "content": "\\operatorname { r o w } ( A _ { 1 } ) \\subset \\operatorname { r o w } ( A _ { 2 } ) \\subset \\cdots \\subset \\operatorname { r o w } ( A _ { p } ) { \\mathrm { ~ a n d ~ } } \\operatorname { c o l } ( B _ { H + 1 } ) \\subset \\operatorname { c o l } ( B _ { H } ) \\subset \\cdots \\subset \\operatorname { c o l } ( B _ { p + 1 } ) ,", + "type": "interline_equation", + "image_path": "0a448f4a37400c1b93651acd842dd80b9a9c5f15ec945d705774d5cc8d02e96e.jpg" + } + ] + } + ], + "index": 20, + "virtual_lines": [ + { + "bbox": [ + 124, + 371, + 484, + 385 + ], + "spans": [], + "index": 20 + } + ] + }, + { + "type": "text", + "bbox": [ + 114, + 389, + 486, + 401 + ], + "lines": [ + { + "bbox": [ + 111, + 388, + 488, + 404 + ], + "spans": [ + { + "bbox": [ + 111, + 388, + 440, + 404 + ], + "score": 1.0, + "content": "ut it was just shown that the these spaces have the same dimensions, which equals", + "type": "text" + }, + { + "bbox": [ + 440, + 390, + 446, + 399 + ], + "score": 0.82, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 447, + 388, + 488, + 404 + ], + "score": 1.0, + "content": ", meaning", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 21, + "bbox_fs": [ + 111, + 388, + 488, + 404 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 124, + 406, + 484, + 419 + ], + "lines": [ + { + "bbox": [ + 124, + 406, + 484, + 419 + ], + "spans": [ + { + "bbox": [ + 124, + 406, + 484, + 419 + ], + "score": 0.37, + "content": "\\operatorname { r o w } ( A _ { 1 } ) = \\operatorname { r o w } ( A _ { 2 } ) = \\cdots = \\operatorname { r o w } ( A _ { p } ) { \\mathrm { ~ a n d ~ } } \\operatorname { c o l } ( B _ { H + 1 } ) = \\operatorname { c o l } ( B _ { H } ) = \\cdots = \\operatorname { c o l } ( B _ { p + 1 } ) .", + "type": "interline_equation", + "image_path": "ac83487c40ffb9f00f68793f0a49ee68473c88a8a0dfe1fc025822a98b405cf4.jpg" + } + ] + } + ], + "index": 22, + "virtual_lines": [ + { + "bbox": [ + 124, + 406, + 484, + 419 + ], + "spans": [], + "index": 22 + } + ] + }, + { + "type": "text", + "bbox": [ + 108, + 424, + 501, + 446 + ], + "lines": [ + { + "bbox": [ + 106, + 424, + 504, + 436 + ], + "spans": [ + { + "bbox": [ + 106, + 424, + 504, + 436 + ], + "score": 1.0, + "content": "Using this observation, we can now state a proposition showing necessary and sufficient conditions", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 434, + 309, + 447 + ], + "spans": [ + { + "bbox": [ + 106, + 434, + 150, + 447 + ], + "score": 1.0, + "content": "for a tuple", + "type": "text" + }, + { + "bbox": [ + 150, + 435, + 185, + 446 + ], + "score": 0.92, + "content": "W \\in \\mathcal { V } _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 185, + 434, + 279, + 447 + ], + "score": 1.0, + "content": "to be a critical point of", + "type": "text" + }, + { + "bbox": [ + 279, + 435, + 305, + 447 + ], + "score": 0.92, + "content": "L ( W )", + "type": "inline_equation" + }, + { + "bbox": [ + 306, + 434, + 309, + 447 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 23.5, + "bbox_fs": [ + 106, + 424, + 504, + 447 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 448, + 500, + 462 + ], + "lines": [ + { + "bbox": [ + 105, + 447, + 500, + 463 + ], + "spans": [ + { + "bbox": [ + 105, + 447, + 209, + 463 + ], + "score": 1.0, + "content": "Proposition 3.6. A tuple", + "type": "text" + }, + { + "bbox": [ + 209, + 449, + 244, + 460 + ], + "score": 0.91, + "content": "W \\in \\nu _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 244, + 447, + 326, + 463 + ], + "score": 1.0, + "content": "is a critical point of", + "type": "text" + }, + { + "bbox": [ + 327, + 450, + 334, + 459 + ], + "score": 0.71, + "content": "L", + "type": "inline_equation" + }, + { + "bbox": [ + 335, + 447, + 389, + 463 + ], + "score": 1.0, + "content": "if and only if", + "type": "text" + }, + { + "bbox": [ + 389, + 450, + 428, + 461 + ], + "score": 0.91, + "content": "A _ { p } E = 0", + "type": "inline_equation" + }, + { + "bbox": [ + 429, + 447, + 447, + 463 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 447, + 450, + 496, + 462 + ], + "score": 0.91, + "content": "E B _ { p + 1 } = 0", + "type": "inline_equation" + }, + { + "bbox": [ + 496, + 447, + 500, + 463 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 25, + "bbox_fs": [ + 105, + 447, + 500, + 463 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 473, + 505, + 515 + ], + "lines": [ + { + "bbox": [ + 105, + 472, + 504, + 490 + ], + "spans": [ + { + "bbox": [ + 105, + 472, + 172, + 490 + ], + "score": 1.0, + "content": "Proof. (If part)", + "type": "text" + }, + { + "bbox": [ + 173, + 475, + 218, + 487 + ], + "score": 0.92, + "content": "A _ { p } E = 0", + "type": "inline_equation" + }, + { + "bbox": [ + 218, + 472, + 272, + 490 + ], + "score": 1.0, + "content": "implies that", + "type": "text" + }, + { + "bbox": [ + 272, + 474, + 453, + 487 + ], + "score": 0.91, + "content": "\\operatorname { c o l } ( E ) \\subset \\operatorname { r o w } ( A _ { p } ) ^ { \\perp } = \\cdot \\cdot \\cdot = \\operatorname { r o w } ( A _ { 1 } ) ^ { \\perp }", + "type": "inline_equation" + }, + { + "bbox": [ + 453, + 472, + 472, + 490 + ], + "score": 1.0, + "content": ", so", + "type": "text" + }, + { + "bbox": [ + 472, + 473, + 504, + 488 + ], + "score": 0.85, + "content": "\\begin{array} { r } { \\frac { \\partial L } { \\partial W _ { i } } \\ = } \\end{array}", + "type": "inline_equation" + } + ], + "index": 26 + }, + { + "bbox": [ + 107, + 487, + 505, + 501 + ], + "spans": [ + { + "bbox": [ + 107, + 488, + 195, + 500 + ], + "score": 0.92, + "content": "A _ { i } E B _ { i } = 0 \\cdot B _ { i } = 0 .", + "type": "inline_equation" + }, + { + "bbox": [ + 196, + 487, + 213, + 501 + ], + "score": 1.0, + "content": ", for", + "type": "text" + }, + { + "bbox": [ + 214, + 489, + 265, + 500 + ], + "score": 0.89, + "content": "i = 1 , \\ldots , p", + "type": "inline_equation" + }, + { + "bbox": [ + 265, + 487, + 312, + 501 + ], + "score": 1.0, + "content": ". Similarly,", + "type": "text" + }, + { + "bbox": [ + 312, + 489, + 363, + 501 + ], + "score": 0.92, + "content": "E B _ { p + 1 } = 0", + "type": "inline_equation" + }, + { + "bbox": [ + 363, + 487, + 397, + 501 + ], + "score": 1.0, + "content": "implies", + "type": "text" + }, + { + "bbox": [ + 397, + 488, + 505, + 501 + ], + "score": 0.83, + "content": "\\operatorname { r o w } ( E ) \\subset \\operatorname { c o l } ( B _ { p + 1 } ) ^ { \\perp } =", + "type": "inline_equation" + } + ], + "index": 27 + }, + { + "bbox": [ + 107, + 497, + 424, + 517 + ], + "spans": [ + { + "bbox": [ + 107, + 501, + 184, + 514 + ], + "score": 0.88, + "content": "\\cdot \\cdot \\cdot = \\mathrm { c o l } ( B _ { H + 1 } ) ^ { \\perp }", + "type": "inline_equation" + }, + { + "bbox": [ + 184, + 497, + 200, + 517 + ], + "score": 1.0, + "content": ", so", + "type": "text" + }, + { + "bbox": [ + 200, + 500, + 316, + 515 + ], + "score": 0.9, + "content": "\\begin{array} { r } { \\frac { \\partial L } { \\partial W _ { i } } = A _ { i } E B _ { i } = A _ { i } \\cdot 0 = 0 } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 316, + 497, + 331, + 517 + ], + "score": 1.0, + "content": "for", + "type": "text" + }, + { + "bbox": [ + 332, + 502, + 420, + 513 + ], + "score": 0.91, + "content": "i = p + 1 , \\ldots , H + 1", + "type": "inline_equation" + }, + { + "bbox": [ + 420, + 497, + 424, + 517 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 27, + "bbox_fs": [ + 105, + 472, + 505, + 517 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 519, + 381, + 535 + ], + "lines": [ + { + "bbox": [ + 106, + 519, + 381, + 537 + ], + "spans": [ + { + "bbox": [ + 106, + 519, + 199, + 537 + ], + "score": 1.0, + "content": "(Only if part) We have", + "type": "text" + }, + { + "bbox": [ + 199, + 520, + 279, + 536 + ], + "score": 0.95, + "content": "\\begin{array} { r } { \\frac { \\partial L } { \\partial W _ { i } } = A _ { i } E B _ { i } = 0 } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 279, + 519, + 306, + 537 + ], + "score": 1.0, + "content": "for all", + "type": "text" + }, + { + "bbox": [ + 307, + 522, + 311, + 531 + ], + "score": 0.7, + "content": "i", + "type": "inline_equation" + }, + { + "bbox": [ + 312, + 519, + 381, + 537 + ], + "score": 1.0, + "content": ". This means that", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 29, + "bbox_fs": [ + 106, + 519, + 381, + 537 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 173, + 539, + 437, + 572 + ], + "lines": [ + { + "bbox": [ + 173, + 539, + 437, + 572 + ], + "spans": [ + { + "bbox": [ + 173, + 539, + 437, + 572 + ], + "score": 0.88, + "content": "\\begin{array} { r l } & { \\quad \\mathrm { c o l } ( E B _ { i } ) \\subset \\mathrm { r o w } ( A _ { i } ) ^ { \\perp } = \\mathrm { r o w } ( A _ { p } ) ^ { \\perp } \\mathrm { f o r } i = 1 , \\ldots , p } \\\\ & { \\quad \\mathrm { r o w } ( A _ { i } E ) \\subset \\mathrm { c o l } ( B _ { i } ) ^ { \\perp } = \\mathrm { c o l } ( B _ { p + 1 } ) ^ { \\perp } \\mathrm { f o r } i = p + 1 , \\ldots , H + 1 . } \\end{array}", + "type": "interline_equation", + "image_path": "d5940b8d098dcccc7c6173cead353f7832657fbb9d17335d5fc15b7cdcf2527d.jpg" + } + ] + } + ], + "index": 31, + "virtual_lines": [ + { + "bbox": [ + 173, + 539, + 437, + 550.0 + ], + "spans": [], + "index": 30 + }, + { + "bbox": [ + 173, + 550.0, + 437, + 561.0 + ], + "spans": [], + "index": 31 + }, + { + "bbox": [ + 173, + 561.0, + 437, + 572.0 + ], + "spans": [], + "index": 32 + } + ] + }, + { + "type": "text", + "bbox": [ + 103, + 577, + 504, + 602 + ], + "lines": [ + { + "bbox": [ + 105, + 575, + 505, + 591 + ], + "spans": [ + { + "bbox": [ + 105, + 575, + 175, + 591 + ], + "score": 1.0, + "content": "Now recall that", + "type": "text" + }, + { + "bbox": [ + 175, + 578, + 189, + 588 + ], + "score": 0.89, + "content": "B _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 189, + 575, + 210, + 591 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 210, + 577, + 236, + 589 + ], + "score": 0.92, + "content": "A _ { H + 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 237, + 575, + 344, + 591 + ], + "score": 1.0, + "content": "are identity matrices, so", + "type": "text" + }, + { + "bbox": [ + 344, + 576, + 435, + 589 + ], + "score": 0.88, + "content": "\\mathrm { c o l } ( E ) \\subset \\mathrm { r o w } ( A _ { p } ) ^ { \\perp }", + "type": "inline_equation" + }, + { + "bbox": [ + 436, + 575, + 457, + 591 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 457, + 577, + 505, + 589 + ], + "score": 0.79, + "content": "\\operatorname { r o w } ( E ) ~ \\subset", + "type": "inline_equation" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 588, + 328, + 603 + ], + "spans": [ + { + "bbox": [ + 106, + 588, + 156, + 602 + ], + "score": 0.92, + "content": "\\mathrm { c o l } ( B _ { p + 1 } ) ^ { \\perp }", + "type": "inline_equation" + }, + { + "bbox": [ + 156, + 588, + 216, + 603 + ], + "score": 1.0, + "content": ", which proves", + "type": "text" + }, + { + "bbox": [ + 216, + 590, + 256, + 601 + ], + "score": 0.92, + "content": "A _ { p } E = 0", + "type": "inline_equation" + }, + { + "bbox": [ + 256, + 588, + 273, + 603 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 274, + 590, + 323, + 601 + ], + "score": 0.92, + "content": "E B _ { p + 1 } = 0", + "type": "inline_equation" + }, + { + "bbox": [ + 324, + 588, + 328, + 603 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 33.5, + "bbox_fs": [ + 105, + 575, + 505, + 603 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 613, + 506, + 649 + ], + "lines": [ + { + "bbox": [ + 105, + 613, + 505, + 625 + ], + "spans": [ + { + "bbox": [ + 105, + 613, + 505, + 625 + ], + "score": 1.0, + "content": "Now we present a proposition that specifies the necessary and sufficient condition in which a critical", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 623, + 506, + 637 + ], + "spans": [ + { + "bbox": [ + 105, + 623, + 140, + 637 + ], + "score": 1.0, + "content": "point of", + "type": "text" + }, + { + "bbox": [ + 140, + 624, + 166, + 636 + ], + "score": 0.91, + "content": "L ( W )", + "type": "inline_equation" + }, + { + "bbox": [ + 167, + 623, + 177, + 637 + ], + "score": 1.0, + "content": "in", + "type": "text" + }, + { + "bbox": [ + 178, + 625, + 190, + 635 + ], + "score": 0.88, + "content": "\\nu _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 190, + 623, + 417, + 637 + ], + "score": 1.0, + "content": "is a global minimum. Recall that when we take the SVD", + "type": "text" + }, + { + "bbox": [ + 417, + 623, + 506, + 636 + ], + "score": 0.91, + "content": "{ \\cal Y } X ^ { T } ( X X ^ { T } ) ^ { - 1 } X =", + "type": "inline_equation" + } + ], + "index": 36 + }, + { + "bbox": [ + 106, + 635, + 444, + 650 + ], + "spans": [ + { + "bbox": [ + 106, + 636, + 137, + 648 + ], + "score": 0.85, + "content": "U { \\boldsymbol { \\Sigma } } V ^ { T }", + "type": "inline_equation" + }, + { + "bbox": [ + 137, + 635, + 141, + 650 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 141, + 636, + 189, + 648 + ], + "score": 0.88, + "content": "\\hat { U } \\in \\mathbb { R } ^ { d _ { y } \\times k }", + "type": "inline_equation" + }, + { + "bbox": [ + 189, + 635, + 375, + 650 + ], + "score": 1.0, + "content": "is defined to be a matrix consisting of the first", + "type": "text" + }, + { + "bbox": [ + 375, + 637, + 382, + 647 + ], + "score": 0.78, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 382, + 635, + 430, + 650 + ], + "score": 1.0, + "content": "columns of", + "type": "text" + }, + { + "bbox": [ + 430, + 638, + 439, + 647 + ], + "score": 0.84, + "content": "U", + "type": "inline_equation" + }, + { + "bbox": [ + 439, + 635, + 444, + 650 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 37 + } + ], + "index": 36, + "bbox_fs": [ + 105, + 613, + 506, + 650 + ] + }, + { + "type": "text", + "bbox": [ + 105, + 650, + 506, + 677 + ], + "lines": [ + { + "bbox": [ + 105, + 649, + 506, + 666 + ], + "spans": [ + { + "bbox": [ + 105, + 649, + 248, + 666 + ], + "score": 1.0, + "content": "Proposition 3.7. A critical point", + "type": "text" + }, + { + "bbox": [ + 248, + 651, + 289, + 663 + ], + "score": 0.9, + "content": "W \\in \\mathsf { V } _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 289, + 649, + 303, + 666 + ], + "score": 1.0, + "content": "of", + "type": "text" + }, + { + "bbox": [ + 303, + 651, + 329, + 664 + ], + "score": 0.91, + "content": "L ( W )", + "type": "inline_equation" + }, + { + "bbox": [ + 330, + 649, + 496, + 666 + ], + "score": 1.0, + "content": "is a global minimum point if and only", + "type": "text" + }, + { + "bbox": [ + 497, + 652, + 506, + 663 + ], + "score": 0.4, + "content": "i f", + "type": "inline_equation" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 662, + 287, + 678 + ], + "spans": [ + { + "bbox": [ + 105, + 663, + 284, + 677 + ], + "score": 0.87, + "content": "\\operatorname { c o l } ( W _ { H + 1 } \\cdot \\cdot \\cdot W _ { p + 1 } ) = \\operatorname { r o w } ( A _ { p } ) = \\operatorname { c o l } ( { \\hat { U } } )", + "type": "inline_equation" + }, + { + "bbox": [ + 284, + 662, + 287, + 678 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 39 + } + ], + "index": 38.5, + "bbox_fs": [ + 105, + 649, + 506, + 678 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 687, + 505, + 714 + ], + "lines": [ + { + "bbox": [ + 106, + 687, + 504, + 700 + ], + "spans": [ + { + "bbox": [ + 106, + 687, + 162, + 700 + ], + "score": 1.0, + "content": "Proof. Since", + "type": "text" + }, + { + "bbox": [ + 162, + 689, + 174, + 698 + ], + "score": 0.75, + "content": "W", + "type": "inline_equation" + }, + { + "bbox": [ + 175, + 687, + 371, + 700 + ], + "score": 1.0, + "content": "is a critical point, by Proposition 3.6 we have", + "type": "text" + }, + { + "bbox": [ + 371, + 689, + 415, + 700 + ], + "score": 0.92, + "content": "A _ { p } E = 0", + "type": "inline_equation" + }, + { + "bbox": [ + 416, + 687, + 504, + 700 + ], + "score": 1.0, + "content": ". Also note from the", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 104, + 698, + 360, + 716 + ], + "spans": [ + { + "bbox": [ + 104, + 698, + 162, + 716 + ], + "score": 1.0, + "content": "definitions of", + "type": "text" + }, + { + "bbox": [ + 162, + 701, + 174, + 712 + ], + "score": 0.88, + "content": "A _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 174, + 698, + 199, + 716 + ], + "score": 1.0, + "content": "’s and", + "type": "text" + }, + { + "bbox": [ + 199, + 701, + 210, + 712 + ], + "score": 0.89, + "content": "B _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 211, + 698, + 236, + 716 + ], + "score": 1.0, + "content": "’s that", + "type": "text" + }, + { + "bbox": [ + 236, + 700, + 342, + 714 + ], + "score": 0.93, + "content": "W _ { H + 1 } \\cdot \\cdot \\cdot W _ { 1 } = A _ { p } ^ { T } B _ { p + 1 } ^ { T }", + "type": "inline_equation" + }, + { + "bbox": [ + 343, + 698, + 360, + 716 + ], + "score": 1.0, + "content": ", so", + "type": "text" + } + ], + "index": 41 + } + ], + "index": 40.5, + "bbox_fs": [ + 104, + 687, + 504, + 716 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 159, + 718, + 451, + 734 + ], + "lines": [ + { + "bbox": [ + 159, + 718, + 451, + 734 + ], + "spans": [ + { + "bbox": [ + 159, + 718, + 451, + 734 + ], + "score": 0.9, + "content": "A _ { p } E = A _ { p } ( A _ { p } ^ { T } B _ { p + 1 } ^ { T } X - Y ) X ^ { T } = A _ { p } A _ { p } ^ { T } B _ { p + 1 } ^ { T } X X ^ { T } - A _ { p } Y X ^ { T } = 0 .", + "type": "interline_equation", + "image_path": "133b976ba55711afec0c475d6297affe61800958d98c72ff831b8a6b9f66f5a1.jpg" + } + ] + } + ], + "index": 42, + "virtual_lines": [ + { + "bbox": [ + 159, + 718, + 451, + 734 + ], + "spans": [], + "index": 42 + } + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 104, + 80, + 477, + 95 + ], + "lines": [ + { + "bbox": [ + 104, + 79, + 480, + 98 + ], + "spans": [ + { + "bbox": [ + 104, + 79, + 143, + 98 + ], + "score": 1.0, + "content": "Because", + "type": "text" + }, + { + "bbox": [ + 143, + 82, + 201, + 95 + ], + "score": 0.91, + "content": "\\operatorname { r a n k } ( A _ { p } ) = k", + "type": "inline_equation" + }, + { + "bbox": [ + 202, + 79, + 222, + 98 + ], + "score": 1.0, + "content": ", and", + "type": "text" + }, + { + "bbox": [ + 222, + 81, + 284, + 96 + ], + "score": 0.93, + "content": "A _ { p } A _ { p } ^ { T } \\in \\mathbb { R } ^ { k \\times k }", + "type": "inline_equation" + }, + { + "bbox": [ + 285, + 79, + 348, + 98 + ], + "score": 1.0, + "content": "is invertible, so", + "type": "text" + }, + { + "bbox": [ + 349, + 83, + 372, + 95 + ], + "score": 0.92, + "content": "B _ { p + 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 372, + 79, + 480, + 98 + ], + "score": 1.0, + "content": "is determined uniquely as", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "interline_equation", + "bbox": [ + 223, + 99, + 386, + 115 + ], + "lines": [ + { + "bbox": [ + 223, + 99, + 386, + 115 + ], + "spans": [ + { + "bbox": [ + 223, + 99, + 386, + 115 + ], + "score": 0.9, + "content": "B _ { p + 1 } ^ { T } = ( A _ { p } A _ { p } ^ { T } ) ^ { - 1 } A _ { p } Y X ^ { T } ( X X ^ { T } ) ^ { - 1 } ,", + "type": "interline_equation", + "image_path": "3f382d3e4ff4a1b35d769099e63fd05d0fa5b93e67a767b203c9be0a5066976d.jpg" + } + ] + } + ], + "index": 1, + "virtual_lines": [ + { + "bbox": [ + 223, + 99, + 386, + 115 + ], + "spans": [], + "index": 1 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 118, + 126, + 129 + ], + "lines": [ + { + "bbox": [ + 105, + 118, + 126, + 130 + ], + "spans": [ + { + "bbox": [ + 105, + 118, + 126, + 130 + ], + "score": 1.0, + "content": "thus", + "type": "text" + } + ], + "index": 2 + } + ], + "index": 2 + }, + { + "type": "interline_equation", + "bbox": [ + 175, + 127, + 434, + 143 + ], + "lines": [ + { + "bbox": [ + 175, + 127, + 434, + 143 + ], + "spans": [ + { + "bbox": [ + 175, + 127, + 434, + 143 + ], + "score": 0.89, + "content": "W _ { H + 1 } \\cdot \\cdot \\cdot W _ { 1 } = A _ { p } ^ { T } B _ { p + 1 } ^ { T } = A _ { p } ^ { T } ( A _ { p } A _ { p } ^ { T } ) ^ { - 1 } A _ { p } Y X ^ { T } ( X X ^ { T } ) ^ { - 1 } .", + "type": "interline_equation", + "image_path": "50024e11e17c8f44f44de7917ec1f340e6228deb04d0464fb5b18e278656552d.jpg" + } + ] + } + ], + "index": 3, + "virtual_lines": [ + { + "bbox": [ + 175, + 127, + 434, + 143 + ], + "spans": [], + "index": 3 + } + ] + }, + { + "type": "text", + "bbox": [ + 105, + 144, + 396, + 155 + ], + "lines": [ + { + "bbox": [ + 106, + 141, + 396, + 157 + ], + "spans": [ + { + "bbox": [ + 106, + 141, + 207, + 157 + ], + "score": 1.0, + "content": "Comparing this with (4),", + "type": "text" + }, + { + "bbox": [ + 208, + 144, + 219, + 154 + ], + "score": 0.74, + "content": "W", + "type": "inline_equation" + }, + { + "bbox": [ + 220, + 141, + 396, + 157 + ], + "score": 1.0, + "content": "is a global minimum solution if and only if", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 4 + }, + { + "type": "interline_equation", + "bbox": [ + 149, + 159, + 461, + 175 + ], + "lines": [ + { + "bbox": [ + 149, + 159, + 461, + 175 + ], + "spans": [ + { + "bbox": [ + 149, + 159, + 461, + 175 + ], + "score": 0.9, + "content": "\\hat { U } \\hat { U } ^ { T } Y X ^ { T } ( X X ^ { T } ) ^ { - 1 } = W _ { H + 1 } \\cdot \\cdot \\cdot W _ { 1 } = A _ { p } ^ { T } ( A _ { p } A _ { p } ^ { T } ) ^ { - 1 } A _ { p } Y X ^ { T } ( X X ^ { T } ) ^ { - 1 } .", + "type": "interline_equation", + "image_path": "80172eca2523982a8d9e7e2f5f364646b3e49446dc8e4b179867abda9a089964.jpg" + } + ] + } + ], + "index": 5, + "virtual_lines": [ + { + "bbox": [ + 149, + 159, + 461, + 175 + ], + "spans": [], + "index": 5 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 179, + 506, + 234 + ], + "lines": [ + { + "bbox": [ + 104, + 177, + 507, + 196 + ], + "spans": [ + { + "bbox": [ + 104, + 177, + 251, + 196 + ], + "score": 1.0, + "content": "This equation holds if and only if", + "type": "text" + }, + { + "bbox": [ + 252, + 179, + 363, + 195 + ], + "score": 0.93, + "content": "A _ { p } ^ { T } ( A _ { p } A _ { p } ^ { T } ) ^ { - 1 } A _ { p } \\ = \\ \\hat { U } \\hat { U } ^ { T }", + "type": "inline_equation" + }, + { + "bbox": [ + 363, + 177, + 507, + 196 + ], + "score": 1.0, + "content": ", meaning that they are projecting", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 192, + 506, + 209 + ], + "spans": [ + { + "bbox": [ + 106, + 193, + 173, + 207 + ], + "score": 0.92, + "content": "Y X ^ { T } ( X X ^ { T } ) ^ { - 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 174, + 192, + 363, + 209 + ], + "score": 1.0, + "content": "onto the same subspace. The projection matrix", + "type": "text" + }, + { + "bbox": [ + 363, + 193, + 434, + 208 + ], + "score": 0.93, + "content": "A _ { p } ^ { T } ( A _ { p } A _ { p } ^ { T } ) ^ { - 1 } A _ { p }", + "type": "inline_equation" + }, + { + "bbox": [ + 434, + 192, + 464, + 209 + ], + "score": 1.0, + "content": "is onto", + "type": "text" + }, + { + "bbox": [ + 465, + 194, + 501, + 207 + ], + "score": 0.61, + "content": "\\operatorname { r o w } ( A _ { p } )", + "type": "inline_equation" + }, + { + "bbox": [ + 501, + 192, + 506, + 209 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 206, + 506, + 222 + ], + "spans": [ + { + "bbox": [ + 105, + 206, + 131, + 222 + ], + "score": 1.0, + "content": "while", + "type": "text" + }, + { + "bbox": [ + 131, + 207, + 154, + 219 + ], + "score": 0.87, + "content": "\\hat { U } \\hat { U } ^ { T }", + "type": "inline_equation" + }, + { + "bbox": [ + 154, + 206, + 185, + 222 + ], + "score": 1.0, + "content": "is onto", + "type": "text" + }, + { + "bbox": [ + 185, + 207, + 213, + 221 + ], + "score": 0.9, + "content": "\\mathrm { c o l } ( { \\hat { U } } )", + "type": "inline_equation" + }, + { + "bbox": [ + 213, + 206, + 331, + 222 + ], + "score": 1.0, + "content": ". From this, we conclude that", + "type": "text" + }, + { + "bbox": [ + 331, + 209, + 343, + 219 + ], + "score": 0.81, + "content": "W", + "type": "inline_equation" + }, + { + "bbox": [ + 343, + 206, + 506, + 222 + ], + "score": 1.0, + "content": "is a global minimum point if and only if", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 107, + 219, + 505, + 235 + ], + "spans": [ + { + "bbox": [ + 107, + 220, + 185, + 234 + ], + "score": 0.9, + "content": "\\operatorname { r o w } ( A _ { p } ) = \\operatorname { c o l } ( \\hat { U } )", + "type": "inline_equation" + }, + { + "bbox": [ + 185, + 219, + 189, + 235 + ], + "score": 1.0, + "content": ".", + "type": "text" + }, + { + "bbox": [ + 494, + 222, + 505, + 233 + ], + "score": 0.999, + "content": "□", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 7.5 + }, + { + "type": "text", + "bbox": [ + 107, + 245, + 505, + 279 + ], + "lines": [ + { + "bbox": [ + 105, + 244, + 506, + 258 + ], + "spans": [ + { + "bbox": [ + 105, + 244, + 289, + 258 + ], + "score": 1.0, + "content": "From Proposition 3.7, we can define the set", + "type": "text" + }, + { + "bbox": [ + 290, + 246, + 302, + 257 + ], + "score": 0.91, + "content": "\\nu _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 302, + 244, + 506, + 258 + ], + "score": 1.0, + "content": "that appeared in Theorem 2.2, and conclude that", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 256, + 505, + 269 + ], + "spans": [ + { + "bbox": [ + 105, + 256, + 198, + 269 + ], + "score": 1.0, + "content": "every critical point of", + "type": "text" + }, + { + "bbox": [ + 199, + 257, + 225, + 268 + ], + "score": 0.92, + "content": "L ( W )", + "type": "inline_equation" + }, + { + "bbox": [ + 225, + 256, + 238, + 269 + ], + "score": 1.0, + "content": "in", + "type": "text" + }, + { + "bbox": [ + 238, + 257, + 250, + 267 + ], + "score": 0.88, + "content": "\\nu _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 250, + 256, + 505, + 269 + ], + "score": 1.0, + "content": "is a global minimum, and any other critical points are saddle", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 104, + 267, + 137, + 281 + ], + "spans": [ + { + "bbox": [ + 104, + 267, + 137, + 281 + ], + "score": 1.0, + "content": "points.", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 11 + }, + { + "type": "title", + "bbox": [ + 108, + 295, + 403, + 307 + ], + "lines": [ + { + "bbox": [ + 105, + 294, + 404, + 309 + ], + "spans": [ + { + "bbox": [ + 105, + 294, + 404, + 309 + ], + "score": 1.0, + "content": "4 EXTENSION TO DEEP NONLINEAR NEURAL NETWORKS", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 13 + }, + { + "type": "text", + "bbox": [ + 106, + 318, + 505, + 430 + ], + "lines": [ + { + "bbox": [ + 105, + 318, + 505, + 332 + ], + "spans": [ + { + "bbox": [ + 105, + 318, + 505, + 332 + ], + "score": 1.0, + "content": "In this section, we present some sufficient conditions for global optimality for deep nonlinear neu-", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 329, + 506, + 343 + ], + "spans": [ + { + "bbox": [ + 105, + 329, + 415, + 343 + ], + "score": 1.0, + "content": "ral networks via a function space view. Given a smooth nonlinear function", + "type": "text" + }, + { + "bbox": [ + 415, + 331, + 427, + 341 + ], + "score": 0.85, + "content": "h ^ { * }", + "type": "inline_equation" + }, + { + "bbox": [ + 427, + 329, + 506, + 343 + ], + "score": 1.0, + "content": "that maps input to", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 341, + 505, + 354 + ], + "spans": [ + { + "bbox": [ + 105, + 341, + 505, + 354 + ], + "score": 1.0, + "content": "output, Bartlett et al. (2017) described a method to decompose it into a number of smooth nonlinear", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 352, + 506, + 366 + ], + "spans": [ + { + "bbox": [ + 105, + 352, + 146, + 366 + ], + "score": 1.0, + "content": "functions", + "type": "text" + }, + { + "bbox": [ + 147, + 352, + 239, + 364 + ], + "score": 0.91, + "content": "h ^ { * } = h _ { H + 1 } \\circ \\cdot \\cdot \\cdot \\circ h _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 240, + 352, + 268, + 366 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 268, + 352, + 279, + 363 + ], + "score": 0.88, + "content": "h _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 279, + 352, + 506, + 366 + ], + "score": 1.0, + "content": "’s are close to identity. Using Frechet derivatives of the ´", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 363, + 505, + 376 + ], + "spans": [ + { + "bbox": [ + 105, + 363, + 285, + 376 + ], + "score": 1.0, + "content": "population risk with respect to each function", + "type": "text" + }, + { + "bbox": [ + 285, + 364, + 295, + 374 + ], + "score": 0.88, + "content": "h _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 296, + 363, + 404, + 376 + ], + "score": 1.0, + "content": ", they showed that when all", + "type": "text" + }, + { + "bbox": [ + 405, + 363, + 415, + 374 + ], + "score": 0.86, + "content": "h _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 415, + 363, + 505, + 376 + ], + "score": 1.0, + "content": "’s are close to identity,", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 374, + 506, + 387 + ], + "spans": [ + { + "bbox": [ + 105, + 374, + 506, + 387 + ], + "score": 1.0, + "content": "any critical point of the population risk is a global minimum. One can see that these results are direct", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 104, + 385, + 506, + 398 + ], + "spans": [ + { + "bbox": [ + 104, + 385, + 506, + 398 + ], + "score": 1.0, + "content": "generalization of Theorems 2.1 and 2.2 of Hardt & Ma (2017) to nonlinear networks and utilize the", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 396, + 506, + 408 + ], + "spans": [ + { + "bbox": [ + 105, + 396, + 506, + 408 + ], + "score": 1.0, + "content": "classical “small gain” arguments often used in nonlinear analysis and control (Khalil, 1996; Zames,", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 406, + 506, + 420 + ], + "spans": [ + { + "bbox": [ + 105, + 406, + 506, + 420 + ], + "score": 1.0, + "content": "1966). Motivated by this result, we extended Theorem 2.1 to deep nonlinear neural networks and", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 417, + 385, + 431 + ], + "spans": [ + { + "bbox": [ + 105, + 417, + 385, + 431 + ], + "score": 1.0, + "content": "obtained sufficient conditions for global optimality in function space.", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 18.5 + }, + { + "type": "title", + "bbox": [ + 106, + 443, + 306, + 454 + ], + "lines": [ + { + "bbox": [ + 105, + 442, + 307, + 455 + ], + "spans": [ + { + "bbox": [ + 105, + 442, + 307, + 455 + ], + "score": 1.0, + "content": "4.1 PROBLEM FORMULATION AND NOTATION", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 24 + }, + { + "type": "text", + "bbox": [ + 106, + 462, + 505, + 530 + ], + "lines": [ + { + "bbox": [ + 105, + 462, + 505, + 476 + ], + "spans": [ + { + "bbox": [ + 105, + 462, + 177, + 476 + ], + "score": 1.0, + "content": "Suppose the data", + "type": "text" + }, + { + "bbox": [ + 177, + 462, + 215, + 474 + ], + "score": 0.93, + "content": "X \\in \\mathbb { R } ^ { d _ { x } }", + "type": "inline_equation" + }, + { + "bbox": [ + 215, + 462, + 327, + 476 + ], + "score": 1.0, + "content": "and its corresponding label", + "type": "text" + }, + { + "bbox": [ + 327, + 462, + 365, + 474 + ], + "score": 0.92, + "content": "Y \\in \\mathbb { R } ^ { d _ { y } }", + "type": "inline_equation" + }, + { + "bbox": [ + 365, + 462, + 505, + 476 + ], + "score": 1.0, + "content": "are drawn from some distribution.", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 473, + 504, + 486 + ], + "spans": [ + { + "bbox": [ + 105, + 473, + 214, + 486 + ], + "score": 1.0, + "content": "Notice that in this section,", + "type": "text" + }, + { + "bbox": [ + 215, + 475, + 225, + 484 + ], + "score": 0.84, + "content": "X", + "type": "inline_equation" + }, + { + "bbox": [ + 225, + 473, + 243, + 486 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 244, + 475, + 253, + 484 + ], + "score": 0.82, + "content": "Y", + "type": "inline_equation" + }, + { + "bbox": [ + 253, + 473, + 494, + 486 + ], + "score": 1.0, + "content": "are random vectors instead of matrices. We want to predict", + "type": "text" + }, + { + "bbox": [ + 495, + 475, + 504, + 484 + ], + "score": 0.78, + "content": "Y", + "type": "inline_equation" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 484, + 506, + 498 + ], + "spans": [ + { + "bbox": [ + 105, + 484, + 131, + 498 + ], + "score": 1.0, + "content": "given", + "type": "text" + }, + { + "bbox": [ + 131, + 486, + 141, + 495 + ], + "score": 0.8, + "content": "X", + "type": "inline_equation" + }, + { + "bbox": [ + 142, + 484, + 330, + 498 + ], + "score": 1.0, + "content": "with a deep nonlinear neural network that has", + "type": "text" + }, + { + "bbox": [ + 330, + 486, + 340, + 495 + ], + "score": 0.8, + "content": "H", + "type": "inline_equation" + }, + { + "bbox": [ + 340, + 484, + 506, + 498 + ], + "score": 1.0, + "content": "hidden layers. We express each layer of", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 495, + 505, + 509 + ], + "spans": [ + { + "bbox": [ + 105, + 495, + 205, + 509 + ], + "score": 1.0, + "content": "the network as functions", + "type": "text" + }, + { + "bbox": [ + 206, + 496, + 278, + 507 + ], + "score": 0.91, + "content": "h _ { i } : \\mathbb { R } ^ { d _ { i - 1 } } \\mathbb { R } ^ { d _ { i } }", + "type": "inline_equation" + }, + { + "bbox": [ + 278, + 495, + 505, + 509 + ], + "score": 1.0, + "content": ", so the entire network can be expressed as a composition", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 507, + 505, + 520 + ], + "spans": [ + { + "bbox": [ + 106, + 507, + 160, + 520 + ], + "score": 1.0, + "content": "of functions:", + "type": "text" + }, + { + "bbox": [ + 160, + 507, + 247, + 519 + ], + "score": 0.91, + "content": "h _ { H + 1 } \\circ h _ { H } \\circ \\cdot \\cdot \\cdot \\circ h _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 247, + 507, + 375, + 520 + ], + "score": 1.0, + "content": ". Our goal is to obtain functions", + "type": "text" + }, + { + "bbox": [ + 375, + 507, + 432, + 519 + ], + "score": 0.92, + "content": "h _ { 1 } , \\ldots , h _ { H + 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 432, + 507, + 505, + 520 + ], + "score": 1.0, + "content": "that minimize the", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 518, + 215, + 530 + ], + "spans": [ + { + "bbox": [ + 105, + 518, + 215, + 530 + ], + "score": 1.0, + "content": "population risk functional:", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 27.5 + }, + { + "type": "interline_equation", + "bbox": [ + 171, + 533, + 438, + 557 + ], + "lines": [ + { + "bbox": [ + 171, + 533, + 438, + 557 + ], + "spans": [ + { + "bbox": [ + 171, + 533, + 438, + 557 + ], + "score": 0.93, + "content": "L ( h ) = L ( h _ { 1 } , \\dots , h _ { H + 1 } ) : = { \\frac { 1 } { 2 } } \\mathbb { E } \\left[ \\left\\| h _ { H + 1 } \\circ \\cdot \\cdot \\cdot \\circ h _ { 1 } ( X ) - Y \\right\\| _ { 2 } ^ { 2 } \\right] ,", + "type": "interline_equation", + "image_path": "afe0acc547105b8021c065714cc2c759a5acc643f44b9fe9c8f1ed31a123d9f3.jpg" + } + ] + } + ], + "index": 31, + "virtual_lines": [ + { + "bbox": [ + 171, + 533, + 438, + 557 + ], + "spans": [], + "index": 31 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 560, + 505, + 594 + ], + "lines": [ + { + "bbox": [ + 105, + 560, + 506, + 574 + ], + "spans": [ + { + "bbox": [ + 105, + 560, + 133, + 574 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 133, + 561, + 140, + 571 + ], + "score": 0.83, + "content": "h", + "type": "inline_equation" + }, + { + "bbox": [ + 141, + 560, + 249, + 574 + ], + "score": 1.0, + "content": "is a shorthand notation for", + "type": "text" + }, + { + "bbox": [ + 249, + 561, + 313, + 573 + ], + "score": 0.9, + "content": "\\left( h _ { 1 } , \\ldots , h _ { H + 1 } \\right)", + "type": "inline_equation" + }, + { + "bbox": [ + 314, + 560, + 506, + 574 + ], + "score": 1.0, + "content": ". 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1 } = W _ { H + 1 } \\cdot \\cdot \\cdot W _ { 1 } = A _ { p } ^ { T } ( A _ { p } A _ { p } ^ { T } ) ^ { - 1 } A _ { p } Y X ^ { T } ( X X ^ { T } ) ^ { - 1 } .", + "type": "interline_equation", + "image_path": "80172eca2523982a8d9e7e2f5f364646b3e49446dc8e4b179867abda9a089964.jpg" + } + ] + } + ], + "index": 5, + "virtual_lines": [ + { + "bbox": [ + 149, + 159, + 461, + 175 + ], + "spans": [], + "index": 5 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 179, + 506, + 234 + ], + "lines": [ + { + "bbox": [ + 104, + 177, + 507, + 196 + ], + "spans": [ + { + "bbox": [ + 104, + 177, + 251, + 196 + ], + "score": 1.0, + "content": "This equation holds if and only if", + "type": "text" + }, + { + "bbox": [ + 252, + 179, + 363, + 195 + ], + "score": 0.93, + "content": "A _ { p } ^ { T } ( A _ { p } A _ { p } ^ { T } ) ^ { - 1 } A _ { p } \\ = \\ \\hat { U } \\hat { U } ^ { T }", + "type": "inline_equation" + }, + { + "bbox": [ + 363, + 177, + 507, + 196 + ], + "score": 1.0, + "content": ", meaning that they are projecting", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 192, + 506, + 209 + ], + "spans": [ + { + "bbox": [ + 106, + 193, + 173, + 207 + ], + "score": 0.92, + "content": "Y X ^ { T } ( X X ^ { T } ) ^ { - 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The projection matrix", + "type": "text" + }, + { + "bbox": [ + 363, + 193, + 434, + 208 + ], + "score": 0.93, + "content": "A _ { p } ^ { T } ( A _ { p } A _ { p } ^ { T } ) ^ { - 1 } A _ { p }", + "type": "inline_equation" + }, + { + "bbox": [ + 434, + 192, + 464, + 209 + ], + "score": 1.0, + "content": "is onto", + "type": "text" + }, + { + "bbox": [ + 465, + 194, + 501, + 207 + ], + "score": 0.61, + "content": "\\operatorname { r o w } ( A _ { p } )", + "type": "inline_equation" + }, + { + "bbox": [ + 501, + 192, + 506, + 209 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 206, + 506, + 222 + ], + "spans": [ + { + "bbox": [ + 105, + 206, + 131, + 222 + ], + "score": 1.0, + "content": "while", + "type": "text" + }, + { + "bbox": [ + 131, + 207, + 154, + 219 + ], + "score": 0.87, + "content": "\\hat { U } \\hat { U } ^ { T }", + "type": "inline_equation" + }, + { + "bbox": [ + 154, + 206, + 185, + 222 + ], + "score": 1.0, + "content": "is onto", + "type": "text" + }, + { + "bbox": [ + 185, + 207, + 213, + 221 + ], + "score": 0.9, + "content": "\\mathrm { c o l } ( { \\hat { U } } )", + "type": "inline_equation" + }, + { + "bbox": [ + 213, + 206, + 331, + 222 + ], + "score": 1.0, + "content": ". From this, we conclude that", + "type": "text" + }, + { + "bbox": [ + 331, + 209, + 343, + 219 + ], + "score": 0.81, + "content": "W", + "type": "inline_equation" + }, + { + "bbox": [ + 343, + 206, + 506, + 222 + ], + "score": 1.0, + "content": "is a global minimum point if and only if", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 107, + 219, + 505, + 235 + ], + "spans": [ + { + "bbox": [ + 107, + 220, + 185, + 234 + ], + "score": 0.9, + "content": "\\operatorname { r o w } ( A _ { p } ) = \\operatorname { c o l } ( \\hat { U } )", + "type": "inline_equation" + }, + { + "bbox": [ + 185, + 219, + 189, + 235 + ], + "score": 1.0, + "content": ".", + "type": "text" + }, + { + "bbox": [ + 494, + 222, + 505, + 233 + ], + "score": 0.999, + "content": "□", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 7.5, + "bbox_fs": [ + 104, + 177, + 507, + 235 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 245, + 505, + 279 + ], + "lines": [ + { + "bbox": [ + 105, + 244, + 506, + 258 + ], + "spans": [ + { + "bbox": [ + 105, + 244, + 289, + 258 + ], + "score": 1.0, + "content": "From Proposition 3.7, we can define the set", + "type": "text" + }, + { + "bbox": [ + 290, + 246, + 302, + 257 + ], + "score": 0.91, + "content": "\\nu _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 302, + 244, + 506, + 258 + ], + "score": 1.0, + "content": "that appeared in Theorem 2.2, and conclude that", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 256, + 505, + 269 + ], + "spans": [ + { + "bbox": [ + 105, + 256, + 198, + 269 + ], + "score": 1.0, + "content": "every critical point of", + "type": "text" + }, + { + "bbox": [ + 199, + 257, + 225, + 268 + ], + "score": 0.92, + "content": "L ( W )", + "type": "inline_equation" + }, + { + "bbox": [ + 225, + 256, + 238, + 269 + ], + "score": 1.0, + "content": "in", + "type": "text" + }, + { + "bbox": [ + 238, + 257, + 250, + 267 + ], + "score": 0.88, + "content": "\\nu _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 250, + 256, + 505, + 269 + ], + "score": 1.0, + "content": "is a global minimum, and any other critical points are saddle", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 104, + 267, + 137, + 281 + ], + "spans": [ + { + "bbox": [ + 104, + 267, + 137, + 281 + ], + "score": 1.0, + "content": "points.", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 11, + "bbox_fs": [ + 104, + 244, + 506, + 281 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 295, + 403, + 307 + ], + "lines": [ + { + "bbox": [ + 105, + 294, + 404, + 309 + ], + "spans": [ + { + "bbox": [ + 105, + 294, + 404, + 309 + ], + "score": 1.0, + "content": "4 EXTENSION TO DEEP NONLINEAR NEURAL NETWORKS", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 13 + }, + { + "type": "text", + "bbox": [ + 106, + 318, + 505, + 430 + ], + "lines": [ + { + "bbox": [ + 105, + 318, + 505, + 332 + ], + "spans": [ + { + "bbox": [ + 105, + 318, + 505, + 332 + ], + "score": 1.0, + "content": "In this section, we present some sufficient conditions for global optimality for deep nonlinear neu-", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 329, + 506, + 343 + ], + "spans": [ + { + "bbox": [ + 105, + 329, + 415, + 343 + ], + "score": 1.0, + "content": "ral networks via a function space view. Given a smooth nonlinear function", + "type": "text" + }, + { + "bbox": [ + 415, + 331, + 427, + 341 + ], + "score": 0.85, + "content": "h ^ { * }", + "type": "inline_equation" + }, + { + "bbox": [ + 427, + 329, + 506, + 343 + ], + "score": 1.0, + "content": "that maps input to", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 341, + 505, + 354 + ], + "spans": [ + { + "bbox": [ + 105, + 341, + 505, + 354 + ], + "score": 1.0, + "content": "output, Bartlett et al. (2017) described a method to decompose it into a number of smooth nonlinear", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 352, + 506, + 366 + ], + "spans": [ + { + "bbox": [ + 105, + 352, + 146, + 366 + ], + "score": 1.0, + "content": "functions", + "type": "text" + }, + { + "bbox": [ + 147, + 352, + 239, + 364 + ], + "score": 0.91, + "content": "h ^ { * } = h _ { H + 1 } \\circ \\cdot \\cdot \\cdot \\circ h _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 240, + 352, + 268, + 366 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 268, + 352, + 279, + 363 + ], + "score": 0.88, + "content": "h _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 279, + 352, + 506, + 366 + ], + "score": 1.0, + "content": "’s are close to identity. Using Frechet derivatives of the ´", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 363, + 505, + 376 + ], + "spans": [ + { + "bbox": [ + 105, + 363, + 285, + 376 + ], + "score": 1.0, + "content": "population risk with respect to each function", + "type": "text" + }, + { + "bbox": [ + 285, + 364, + 295, + 374 + ], + "score": 0.88, + "content": "h _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 296, + 363, + 404, + 376 + ], + "score": 1.0, + "content": ", they showed that when all", + "type": "text" + }, + { + "bbox": [ + 405, + 363, + 415, + 374 + ], + "score": 0.86, + "content": "h _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 415, + 363, + 505, + 376 + ], + "score": 1.0, + "content": "’s are close to identity,", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 374, + 506, + 387 + ], + "spans": [ + { + "bbox": [ + 105, + 374, + 506, + 387 + ], + "score": 1.0, + "content": "any critical point of the population risk is a global minimum. One can see that these results are direct", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 104, + 385, + 506, + 398 + ], + "spans": [ + { + "bbox": [ + 104, + 385, + 506, + 398 + ], + "score": 1.0, + "content": "generalization of Theorems 2.1 and 2.2 of Hardt & Ma (2017) to nonlinear networks and utilize the", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 396, + 506, + 408 + ], + "spans": [ + { + "bbox": [ + 105, + 396, + 506, + 408 + ], + "score": 1.0, + "content": "classical “small gain” arguments often used in nonlinear analysis and control (Khalil, 1996; Zames,", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 406, + 506, + 420 + ], + "spans": [ + { + "bbox": [ + 105, + 406, + 506, + 420 + ], + "score": 1.0, + "content": "1966). Motivated by this result, we extended Theorem 2.1 to deep nonlinear neural networks and", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 417, + 385, + 431 + ], + "spans": [ + { + "bbox": [ + 105, + 417, + 385, + 431 + ], + "score": 1.0, + "content": "obtained sufficient conditions for global optimality in function space.", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 18.5, + "bbox_fs": [ + 104, + 318, + 506, + 431 + ] + }, + { + "type": "title", + "bbox": [ + 106, + 443, + 306, + 454 + ], + "lines": [ + { + "bbox": [ + 105, + 442, + 307, + 455 + ], + "spans": [ + { + "bbox": [ + 105, + 442, + 307, + 455 + ], + "score": 1.0, + "content": "4.1 PROBLEM FORMULATION AND NOTATION", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 24 + }, + { + "type": "text", + "bbox": [ + 106, + 462, + 505, + 530 + ], + "lines": [ + { + "bbox": [ + 105, + 462, + 505, + 476 + ], + "spans": [ + { + "bbox": [ + 105, + 462, + 177, + 476 + ], + "score": 1.0, + "content": "Suppose the data", + "type": "text" + }, + { + "bbox": [ + 177, + 462, + 215, + 474 + ], + "score": 0.93, + "content": "X \\in \\mathbb { R } ^ { d _ { x } }", + "type": "inline_equation" + }, + { + "bbox": [ + 215, + 462, + 327, + 476 + ], + "score": 1.0, + "content": "and its corresponding label", + "type": "text" + }, + { + "bbox": [ + 327, + 462, + 365, + 474 + ], + "score": 0.92, + "content": "Y \\in \\mathbb { R } ^ { d _ { y } }", + "type": "inline_equation" + }, + { + "bbox": [ + 365, + 462, + 505, + 476 + ], + "score": 1.0, + "content": "are drawn from some distribution.", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 473, + 504, + 486 + ], + "spans": [ + { + "bbox": [ + 105, + 473, + 214, + 486 + ], + "score": 1.0, + "content": "Notice that in this section,", + "type": "text" + }, + { + "bbox": [ + 215, + 475, + 225, + 484 + ], + "score": 0.84, + "content": "X", + "type": "inline_equation" + }, + { + "bbox": [ + 225, + 473, + 243, + 486 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 244, + 475, + 253, + 484 + ], + "score": 0.82, + "content": "Y", + "type": "inline_equation" + }, + { + "bbox": [ + 253, + 473, + 494, + 486 + ], + "score": 1.0, + "content": "are random vectors instead of matrices. We want to predict", + "type": "text" + }, + { + "bbox": [ + 495, + 475, + 504, + 484 + ], + "score": 0.78, + "content": "Y", + "type": "inline_equation" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 484, + 506, + 498 + ], + "spans": [ + { + "bbox": [ + 105, + 484, + 131, + 498 + ], + "score": 1.0, + "content": "given", + "type": "text" + }, + { + "bbox": [ + 131, + 486, + 141, + 495 + ], + "score": 0.8, + "content": "X", + "type": "inline_equation" + }, + { + "bbox": [ + 142, + 484, + 330, + 498 + ], + "score": 1.0, + "content": "with a deep nonlinear neural network that has", + "type": "text" + }, + { + "bbox": [ + 330, + 486, + 340, + 495 + ], + "score": 0.8, + "content": "H", + "type": "inline_equation" + }, + { + "bbox": [ + 340, + 484, + 506, + 498 + ], + "score": 1.0, + "content": "hidden layers. We express each layer of", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 495, + 505, + 509 + ], + "spans": [ + { + "bbox": [ + 105, + 495, + 205, + 509 + ], + "score": 1.0, + "content": "the network as functions", + "type": "text" + }, + { + "bbox": [ + 206, + 496, + 278, + 507 + ], + "score": 0.91, + "content": "h _ { i } : \\mathbb { R } ^ { d _ { i - 1 } } \\mathbb { R } ^ { d _ { i } }", + "type": "inline_equation" + }, + { + "bbox": [ + 278, + 495, + 505, + 509 + ], + "score": 1.0, + "content": ", so the entire network can be expressed as a composition", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 507, + 505, + 520 + ], + "spans": [ + { + "bbox": [ + 106, + 507, + 160, + 520 + ], + "score": 1.0, + "content": "of functions:", + "type": "text" + }, + { + "bbox": [ + 160, + 507, + 247, + 519 + ], + "score": 0.91, + "content": "h _ { H + 1 } \\circ h _ { H } \\circ \\cdot \\cdot \\cdot \\circ h _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 247, + 507, + 375, + 520 + ], + "score": 1.0, + "content": ". Our goal is to obtain functions", + "type": "text" + }, + { + "bbox": [ + 375, + 507, + 432, + 519 + ], + "score": 0.92, + "content": "h _ { 1 } , \\ldots , h _ { H + 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 432, + 507, + 505, + 520 + ], + "score": 1.0, + "content": "that minimize the", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 518, + 215, + 530 + ], + "spans": [ + { + "bbox": [ + 105, + 518, + 215, + 530 + ], + "score": 1.0, + "content": "population risk functional:", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 27.5, + "bbox_fs": [ + 105, + 462, + 506, + 530 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 171, + 533, + 438, + 557 + ], + "lines": [ + { + "bbox": [ + 171, + 533, + 438, + 557 + ], + "spans": [ + { + "bbox": [ + 171, + 533, + 438, + 557 + ], + "score": 0.93, + "content": "L ( h ) = L ( h _ { 1 } , \\dots , h _ { H + 1 } ) : = { \\frac { 1 } { 2 } } \\mathbb { E } \\left[ \\left\\| h _ { H + 1 } \\circ \\cdot \\cdot \\cdot \\circ h _ { 1 } ( X ) - Y \\right\\| _ { 2 } ^ { 2 } \\right] ,", + "type": "interline_equation", + "image_path": "afe0acc547105b8021c065714cc2c759a5acc643f44b9fe9c8f1ed31a123d9f3.jpg" + } + ] + } + ], + "index": 31, + "virtual_lines": [ + { + "bbox": [ + 171, + 533, + 438, + 557 + ], + "spans": [], + "index": 31 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 560, + 505, + 594 + ], + "lines": [ + { + "bbox": [ + 105, + 560, + 506, + 574 + ], + "spans": [ + { + "bbox": [ + 105, + 560, + 133, + 574 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 133, + 561, + 140, + 571 + ], + "score": 0.83, + "content": "h", + "type": "inline_equation" + }, + { + "bbox": [ + 141, + 560, + 249, + 574 + ], + "score": 1.0, + "content": "is a shorthand notation for", + "type": "text" + }, + { + "bbox": [ + 249, + 561, + 313, + 573 + ], + "score": 0.9, + "content": "\\left( h _ { 1 } , \\ldots , h _ { H + 1 } \\right)", + "type": "inline_equation" + }, + { + "bbox": [ + 314, + 560, + 506, + 574 + ], + "score": 1.0, + "content": ". It is well-known that the minimizer of squared", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 570, + 505, + 585 + ], + "spans": [ + { + "bbox": [ + 105, + 570, + 273, + 585 + ], + "score": 1.0, + "content": "error risk is the conditional expectation of", + "type": "text" + }, + { + "bbox": [ + 273, + 573, + 282, + 582 + ], + "score": 0.8, + "content": "Y", + "type": "inline_equation" + }, + { + "bbox": [ + 282, + 570, + 307, + 585 + ], + "score": 1.0, + "content": "given", + "type": "text" + }, + { + "bbox": [ + 307, + 574, + 316, + 582 + ], + "score": 0.85, + "content": "X", + "type": "inline_equation" + }, + { + "bbox": [ + 317, + 570, + 406, + 585 + ], + "score": 1.0, + "content": ", which we will denote", + "type": "text" + }, + { + "bbox": [ + 407, + 572, + 501, + 584 + ], + "score": 0.92, + "content": "h ^ { * } ( x ) = \\mathbb { E } [ Y \\mid X = x ]", + "type": "inline_equation" + }, + { + "bbox": [ + 501, + 570, + 505, + 585 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 582, + 349, + 595 + ], + "spans": [ + { + "bbox": [ + 106, + 582, + 349, + 595 + ], + "score": 1.0, + "content": "With this, we can separate the risk functional into two terms", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 33, + "bbox_fs": [ + 105, + 560, + 506, + 595 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 196, + 598, + 414, + 622 + ], + "lines": [ + { + "bbox": [ + 196, + 598, + 414, + 622 + ], + "spans": [ + { + "bbox": [ + 196, + 598, + 414, + 622 + ], + "score": 0.93, + "content": "L ( h ) = \\frac { 1 } { 2 } \\mathbb { E } \\left[ \\| h _ { H + 1 } \\circ \\cdot \\cdot \\cdot \\circ h _ { 1 } ( X ) - h ^ { * } ( X ) \\| _ { 2 } ^ { 2 } \\right] + C ,", + "type": "interline_equation", + "image_path": "8c415fa4fcd13bf7a42910aebc33db6f4e718ca6b7dbaa68cd608ee8825aeea2.jpg" + } + ] + } + ], + "index": 35, + "virtual_lines": [ + { + "bbox": [ + 196, + 598, + 414, + 622 + ], + "spans": [], + "index": 35 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 624, + 504, + 648 + ], + "lines": [ + { + "bbox": [ + 105, + 623, + 504, + 639 + ], + "spans": [ + { + "bbox": [ + 105, + 623, + 183, + 639 + ], + "score": 1.0, + "content": "where the constant", + "type": "text" + }, + { + "bbox": [ + 183, + 626, + 192, + 635 + ], + "score": 0.83, + "content": "C", + "type": "inline_equation" + }, + { + "bbox": [ + 192, + 623, + 364, + 639 + ], + "score": 1.0, + "content": "denotes the variance that is independent of", + "type": "text" + }, + { + "bbox": [ + 365, + 626, + 421, + 637 + ], + "score": 0.91, + "content": "h _ { 1 } , \\ldots , h _ { H + 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 421, + 623, + 473, + 639 + ], + "score": 1.0, + "content": ". Note that if", + "type": "text" + }, + { + "bbox": [ + 473, + 626, + 504, + 636 + ], + "score": 0.9, + "content": "h _ { H + 1 } \\circ", + "type": "inline_equation" + } + ], + "index": 36 + }, + { + "bbox": [ + 107, + 635, + 502, + 648 + ], + "spans": [ + { + "bbox": [ + 107, + 636, + 162, + 647 + ], + "score": 0.9, + "content": "\\cdots \\circ h _ { 1 } = h ^ { * }", + "type": "inline_equation" + }, + { + "bbox": [ + 163, + 635, + 285, + 648 + ], + "score": 1.0, + "content": "almost surely, the first term in", + "type": "text" + }, + { + "bbox": [ + 285, + 636, + 307, + 648 + ], + "score": 0.92, + "content": "L ( h )", + "type": "inline_equation" + }, + { + "bbox": [ + 307, + 635, + 433, + 648 + ], + "score": 1.0, + "content": "vanishes and the optimal value", + "type": "text" + }, + { + "bbox": [ + 433, + 636, + 445, + 646 + ], + "score": 0.88, + "content": "L ^ { * }", + "type": "inline_equation" + }, + { + "bbox": [ + 446, + 635, + 457, + 648 + ], + "score": 1.0, + "content": "of", + "type": "text" + }, + { + "bbox": [ + 458, + 636, + 479, + 648 + ], + "score": 0.92, + "content": "L ( h )", + "type": "inline_equation" + }, + { + "bbox": [ + 479, + 635, + 489, + 648 + ], + "score": 1.0, + "content": "is", + "type": "text" + }, + { + "bbox": [ + 490, + 637, + 498, + 646 + ], + "score": 0.83, + "content": "C", + "type": "inline_equation" + }, + { + "bbox": [ + 498, + 635, + 502, + 648 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 37 + } + ], + "index": 36.5, + "bbox_fs": [ + 105, + 623, + 504, + 648 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 659, + 351, + 671 + ], + "lines": [ + { + "bbox": [ + 105, + 657, + 352, + 674 + ], + "spans": [ + { + "bbox": [ + 105, + 657, + 352, + 674 + ], + "score": 1.0, + "content": "Assumptions. Define the function spaces as the following:", + "type": "text" + } + ], + "index": 38 + } + ], + "index": 38, + "bbox_fs": [ + 105, + 657, + 352, + 674 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 136, + 674, + 478, + 732 + ], + "lines": [ + { + "bbox": [ + 136, + 674, + 478, + 732 + ], + "spans": [ + { + "bbox": [ + 136, + 674, + 478, + 732 + ], + "score": 0.9, + "content": "\\begin{array} { r l } & { \\mathcal { F } : = \\{ h : \\mathbb { R } ^ { d _ { x } } \\mathbb { R } ^ { d _ { y } } \\ | \\ h \\mathrm { ~ i s ~ d i f f e r e n t i a b l e } , h ( 0 ) = 0 , \\ \\mathrm { a n d ~ } \\underset { x } { \\operatorname* { s u p } } \\ \\frac { \\| h ( x ) \\| _ { 2 } } { \\| x \\| _ { 2 } } < \\infty \\} , } \\\\ & { \\mathcal { F } _ { i } : = \\{ h : \\mathbb { R } ^ { d _ { i } - 1 } \\mathbb { R } ^ { d _ { i } } \\ | \\ h \\mathrm { ~ i s ~ d i f f e r e n t i a b l e } , h ( 0 ) = 0 , \\ \\mathrm { a n d ~ } \\underset { x } { \\operatorname* { s u p } } \\ \\frac { \\| h ( x ) \\| _ { 2 } } { \\| x \\| _ { 2 } } < \\infty \\} , } \\end{array}", + "type": "interline_equation", + "image_path": "410c833d12584394e2aee75ace129b8403a9c22a994e906ccd3f92ff033264fc.jpg" + } + ] + } + ], + "index": 40, + "virtual_lines": [ + { + "bbox": [ + 136, + 674, + 478, + 693.3333333333334 + ], + "spans": [], + "index": 39 + }, + { + "bbox": [ + 136, + 693.3333333333334, + 478, + 712.6666666666667 + ], + "spans": [], + "index": 40 + }, + { + "bbox": [ + 136, + 712.6666666666667, + 478, + 732.0000000000001 + ], + "spans": [], + "index": 41 + } + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 82, + 505, + 149 + ], + "lines": [ + { + "bbox": [ + 105, + 80, + 506, + 96 + ], + "spans": [ + { + "bbox": [ + 105, + 80, + 133, + 96 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 134, + 83, + 145, + 93 + ], + "score": 0.88, + "content": "{ \\mathcal { F } } _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 146, + 80, + 222, + 96 + ], + "score": 1.0, + "content": "are defined for all", + "type": "text" + }, + { + "bbox": [ + 223, + 83, + 297, + 94 + ], + "score": 0.91, + "content": "i = 1 , \\ldots , H + 1", + "type": "inline_equation" + }, + { + "bbox": [ + 297, + 80, + 355, + 96 + ], + "score": 1.0, + "content": ". Assume that", + "type": "text" + }, + { + "bbox": [ + 356, + 83, + 390, + 93 + ], + "score": 0.91, + "content": "h ^ { \\ast } \\in \\mathcal { F }", + "type": "inline_equation" + }, + { + "bbox": [ + 390, + 80, + 506, + 96 + ], + "score": 1.0, + "content": ", and that we are optimizing", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 107, + 92, + 506, + 107 + ], + "spans": [ + { + "bbox": [ + 107, + 93, + 128, + 105 + ], + "score": 0.9, + "content": "L ( h )", + "type": "inline_equation" + }, + { + "bbox": [ + 128, + 92, + 151, + 107 + ], + "score": 1.0, + "content": "with", + "type": "text" + }, + { + "bbox": [ + 151, + 94, + 275, + 106 + ], + "score": 0.91, + "content": "h _ { 1 } \\in \\mathcal { F } _ { 1 } , \\ldots , h _ { H + 1 } \\in \\mathcal { F } _ { H + 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 276, + 92, + 416, + 107 + ], + "score": 1.0, + "content": ". 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Notice that", + "type": "text" + }, + { + "bbox": [ + 383, + 105, + 467, + 116 + ], + "score": 0.91, + "content": "h _ { H + 1 } \\circ \\cdot \\cdot \\cdot \\circ h _ { 1 } \\in \\mathcal { F }", + "type": "inline_equation" + }, + { + "bbox": [ + 467, + 104, + 505, + 117 + ], + "score": 1.0, + "content": ", because", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 115, + 504, + 127 + ], + "spans": [ + { + "bbox": [ + 105, + 115, + 504, + 127 + ], + "score": 1.0, + "content": "a composition of differentiable functions is also differentiable, and a composition of sublinear func-", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 126, + 506, + 139 + ], + "spans": [ + { + "bbox": [ + 105, + 126, + 286, + 139 + ], + "score": 1.0, + "content": "tions is also sublinear. We also assume that", + "type": "text" + }, + { + "bbox": [ + 286, + 126, + 361, + 138 + ], + "score": 0.92, + "content": "d _ { i } \\geq \\operatorname* { m i n } \\{ d _ { x } , d _ { y } \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 362, + 126, + 390, + 139 + ], + "score": 1.0, + "content": "for all", + "type": "text" + }, + { + "bbox": [ + 391, + 127, + 464, + 138 + ], + "score": 0.92, + "content": "i = 1 , \\ldots , H + 1", + "type": "inline_equation" + }, + { + "bbox": [ + 464, + 126, + 506, + 139 + ], + "score": 1.0, + "content": ", which is", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 137, + 355, + 150 + ], + "spans": [ + { + "bbox": [ + 105, + 137, + 216, + 150 + ], + "score": 1.0, + "content": "identical to the assumption", + "type": "text" + }, + { + "bbox": [ + 217, + 137, + 287, + 149 + ], + "score": 0.94, + "content": "k = \\operatorname* { m i n } \\{ d _ { x } , d _ { y } \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 288, + 137, + 355, + 150 + ], + "score": 1.0, + "content": "in Theorem 2.1.", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 2.5 + }, + { + "type": "text", + "bbox": [ + 106, + 160, + 505, + 218 + ], + "lines": [ + { + "bbox": [ + 106, + 160, + 505, + 174 + ], + "spans": [ + { + "bbox": [ + 106, + 160, + 386, + 174 + ], + "score": 1.0, + "content": "Notation. To simplify multiple composition of functions, we denote", + "type": "text" + }, + { + "bbox": [ + 386, + 162, + 505, + 173 + ], + "score": 0.9, + "content": "h _ { i : j } = h _ { i } \\circ h _ { i - 1 } \\circ \\cdot \\cdot \\cdot \\circ h _ { j + 1 } \\circ", + "type": "inline_equation" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 172, + 506, + 186 + ], + "spans": [ + { + "bbox": [ + 106, + 173, + 117, + 185 + ], + "score": 0.86, + "content": "h _ { j }", + "type": "inline_equation" + }, + { + "bbox": [ + 118, + 172, + 217, + 186 + ], + "score": 1.0, + "content": ". 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Let", + "type": "text" + }, + { + "bbox": [ + 460, + 184, + 504, + 196 + ], + "score": 0.89, + "content": "D _ { h _ { i } } [ L ( h ) ]", + "type": "inline_equation" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 194, + 504, + 208 + ], + "spans": [ + { + "bbox": [ + 105, + 194, + 216, + 208 + ], + "score": 1.0, + "content": "be the Frechet derivative of ´", + "type": "text" + }, + { + "bbox": [ + 216, + 196, + 237, + 207 + ], + "score": 0.94, + "content": "L ( h )", + "type": "inline_equation" + }, + { + "bbox": [ + 237, + 194, + 298, + 208 + ], + "score": 1.0, + "content": "with respect to", + "type": "text" + }, + { + "bbox": [ + 298, + 196, + 308, + 206 + ], + "score": 0.88, + "content": "h _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 308, + 194, + 358, + 208 + ], + "score": 1.0, + "content": "evaluated at", + "type": "text" + }, + { + "bbox": [ + 358, + 196, + 365, + 205 + ], + "score": 0.81, + "content": "h", + "type": "inline_equation" + }, + { + "bbox": [ + 366, + 194, + 460, + 208 + ], + "score": 1.0, + "content": ". The Frechet derivative ´", + "type": "text" + }, + { + "bbox": [ + 460, + 196, + 504, + 207 + ], + "score": 0.89, + "content": "D _ { h _ { i } } [ L ( h ) ]", + "type": "inline_equation" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 205, + 505, + 219 + ], + "spans": [ + { + "bbox": [ + 105, + 205, + 314, + 219 + ], + "score": 1.0, + "content": "is a linear functional that maps a function (direction)", + "type": "text" + }, + { + "bbox": [ + 314, + 207, + 343, + 218 + ], + "score": 0.92, + "content": "\\eta \\in \\mathcal { F } _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 344, + 205, + 505, + 219 + ], + "score": 1.0, + "content": "to a real number (directional derivative).", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 8 + }, + { + "type": "title", + "bbox": [ + 106, + 231, + 349, + 243 + ], + "lines": [ + { + "bbox": [ + 106, + 231, + 352, + 244 + ], + "spans": [ + { + "bbox": [ + 106, + 231, + 352, + 244 + ], + "score": 1.0, + "content": "4.2 SUFFICIENT CONDITIONS FOR GLOBAL OPTIMALITY", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 11 + }, + { + "type": "text", + "bbox": [ + 105, + 251, + 505, + 274 + ], + "lines": [ + { + "bbox": [ + 105, + 250, + 504, + 265 + ], + "spans": [ + { + "bbox": [ + 105, + 250, + 438, + 265 + ], + "score": 1.0, + "content": "Here, we present two theorems which give sufficient conditions for a critical point", + "type": "text" + }, + { + "bbox": [ + 439, + 252, + 504, + 264 + ], + "score": 0.91, + "content": "( D _ { h _ { i } } [ L ( h ) ] = 0", + "type": "inline_equation" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 262, + 486, + 275 + ], + "spans": [ + { + "bbox": [ + 106, + 262, + 133, + 275 + ], + "score": 1.0, + "content": "for all", + "type": "text" + }, + { + "bbox": [ + 133, + 264, + 138, + 272 + ], + "score": 0.67, + "content": "i", + "type": "inline_equation" + }, + { + "bbox": [ + 138, + 262, + 486, + 275 + ], + "score": 1.0, + "content": ") in the function space to be a global optimum. The proofs are deferred to Appendix A.", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 12.5 + }, + { + "type": "text", + "bbox": [ + 107, + 277, + 400, + 290 + ], + "lines": [ + { + "bbox": [ + 106, + 277, + 400, + 291 + ], + "spans": [ + { + "bbox": [ + 106, + 277, + 241, + 291 + ], + "score": 1.0, + "content": "Theorem 4.1. Consider the case", + "type": "text" + }, + { + "bbox": [ + 241, + 277, + 275, + 290 + ], + "score": 0.92, + "content": "d _ { x } \\geq d _ { y }", + "type": "inline_equation" + }, + { + "bbox": [ + 276, + 277, + 336, + 291 + ], + "score": 1.0, + "content": ". If there exists", + "type": "text" + }, + { + "bbox": [ + 336, + 278, + 360, + 288 + ], + "score": 0.88, + "content": "\\epsilon > 0", + "type": "inline_equation" + }, + { + "bbox": [ + 360, + 277, + 400, + 291 + ], + "score": 1.0, + "content": "such that", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 14 + }, + { + "type": "text", + "bbox": [ + 128, + 298, + 424, + 331 + ], + "lines": [ + { + "bbox": [ + 129, + 297, + 423, + 313 + ], + "spans": [ + { + "bbox": [ + 129, + 297, + 142, + 313 + ], + "score": 1.0, + "content": "1.", + "type": "text" + }, + { + "bbox": [ + 143, + 298, + 240, + 312 + ], + "score": 0.89, + "content": "J [ h _ { H + 1 : 2 } ] ( z ) \\in \\mathbb { R } ^ { d _ { y } \\times d _ { 1 } }", + "type": "inline_equation" + }, + { + "bbox": [ + 241, + 297, + 257, + 313 + ], + "score": 1.0, + "content": "has", + "type": "text" + }, + { + "bbox": [ + 257, + 298, + 423, + 312 + ], + "score": 0.59, + "content": "\\sigma _ { \\operatorname* { m i n } } ( J [ h _ { H + 1 : 2 } ] ( z ) ) \\geq \\epsilon f o r \\ : a l l \\ : z \\in \\mathbb { R } ^ { d _ { 1 } } ,", + "type": "inline_equation", + "image_path": "aec8497d8822eb241bfff78f4099996eef6556b290f68cbe1ee0e96b4347c0bc.jpg" + } + ], + "index": 15 + }, + { + "bbox": [ + 129, + 318, + 278, + 332 + ], + "spans": [ + { + "bbox": [ + 129, + 318, + 142, + 332 + ], + "score": 1.0, + "content": "2.", + "type": "text" + }, + { + "bbox": [ + 142, + 318, + 186, + 331 + ], + "score": 0.92, + "content": "h _ { H + 1 : 2 } ( z )", + "type": "inline_equation" + }, + { + "bbox": [ + 186, + 318, + 278, + 332 + ], + "score": 1.0, + "content": "is twice-differentiable,", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 15.5 + }, + { + "type": "text", + "bbox": [ + 107, + 338, + 487, + 352 + ], + "lines": [ + { + "bbox": [ + 106, + 339, + 488, + 353 + ], + "spans": [ + { + "bbox": [ + 106, + 339, + 207, + 353 + ], + "score": 1.0, + "content": "then any critical point of", + "type": "text" + }, + { + "bbox": [ + 208, + 339, + 228, + 352 + ], + "score": 0.91, + "content": "L ( h )", + "type": "inline_equation" + }, + { + "bbox": [ + 229, + 339, + 278, + 353 + ], + "score": 1.0, + "content": ", in terms of", + "type": "text" + }, + { + "bbox": [ + 279, + 339, + 399, + 353 + ], + "score": 0.93, + "content": "D _ { h _ { 1 } } [ L ( h ) ] , \\ldots , D _ { h _ { H + 1 } } [ L ( h ) ] ,", + "type": "inline_equation" + }, + { + "bbox": [ + 399, + 339, + 488, + 353 + ], + "score": 1.0, + "content": ", is a global minimum.", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 17 + }, + { + "type": "text", + "bbox": [ + 107, + 354, + 504, + 378 + ], + "lines": [ + { + "bbox": [ + 105, + 353, + 506, + 367 + ], + "spans": [ + { + "bbox": [ + 105, + 353, + 241, + 367 + ], + "score": 1.0, + "content": "Theorem 4.2. Consider the case", + "type": "text" + }, + { + "bbox": [ + 241, + 355, + 276, + 367 + ], + "score": 0.92, + "content": "d _ { x } \\leq d _ { y }", + "type": "inline_equation" + }, + { + "bbox": [ + 276, + 353, + 402, + 367 + ], + "score": 1.0, + "content": ". Assume that there exists some", + "type": "text" + }, + { + "bbox": [ + 402, + 354, + 483, + 367 + ], + "score": 0.93, + "content": "j \\in \\{ 1 , \\dots , H + 1 \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 483, + 353, + 506, + 367 + ], + "score": 1.0, + "content": "such", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 365, + 360, + 379 + ], + "spans": [ + { + "bbox": [ + 105, + 365, + 124, + 379 + ], + "score": 1.0, + "content": "that", + "type": "text" + }, + { + "bbox": [ + 125, + 366, + 169, + 378 + ], + "score": 0.93, + "content": "d _ { x } = d _ { j - 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 169, + 365, + 187, + 379 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 187, + 366, + 221, + 378 + ], + "score": 0.92, + "content": "d _ { y } \\leq d _ { j }", + "type": "inline_equation" + }, + { + "bbox": [ + 221, + 365, + 277, + 379 + ], + "score": 1.0, + "content": ". If there exist", + "type": "text" + }, + { + "bbox": [ + 278, + 366, + 318, + 377 + ], + "score": 0.91, + "content": "\\epsilon _ { 1 } , \\epsilon _ { 2 } > 0", + "type": "inline_equation" + }, + { + "bbox": [ + 318, + 365, + 360, + 379 + ], + "score": 1.0, + "content": "such that", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 18.5 + }, + { + "type": "text", + "bbox": [ + 131, + 386, + 316, + 400 + ], + "lines": [ + { + "bbox": [ + 128, + 385, + 317, + 401 + ], + "spans": [ + { + "bbox": [ + 128, + 385, + 142, + 401 + ], + "score": 1.0, + "content": "1.", + "type": "text" + }, + { + "bbox": [ + 142, + 387, + 263, + 400 + ], + "score": 0.9, + "content": "h _ { j - 1 : 1 } : \\mathbb { R } ^ { d _ { x } } \\mathbb { R } ^ { d _ { j - 1 } } = \\mathbb { R } ^ { d _ { x } }", + "type": "inline_equation" + }, + { + "bbox": [ + 264, + 385, + 317, + 401 + ], + "score": 1.0, + "content": "is invertible,", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 20 + }, + { + "type": "text", + "bbox": [ + 130, + 407, + 371, + 422 + ], + "lines": [ + { + "bbox": [ + 127, + 406, + 371, + 423 + ], + "spans": [ + { + "bbox": [ + 127, + 406, + 142, + 423 + ], + "score": 1.0, + "content": "2.", + "type": "text" + }, + { + "bbox": [ + 142, + 408, + 170, + 421 + ], + "score": 0.9, + "content": "h _ { j - 1 : 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 170, + 406, + 205, + 423 + ], + "score": 1.0, + "content": "satisfies", + "type": "text" + }, + { + "bbox": [ + 206, + 408, + 304, + 421 + ], + "score": 0.89, + "content": "\\left\\| h _ { j - 1 : 1 } ( u ) \\right\\| _ { 2 } \\geq \\epsilon _ { 1 } \\left\\| u \\right\\| _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 304, + 406, + 333, + 423 + ], + "score": 1.0, + "content": "for all", + "type": "text" + }, + { + "bbox": [ + 333, + 407, + 367, + 420 + ], + "score": 0.78, + "content": "u \\in \\mathbb { R } ^ { d _ { x } }", + "type": "inline_equation" + }, + { + "bbox": [ + 368, + 406, + 371, + 423 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 21 + }, + { + "type": "text", + "bbox": [ + 129, + 427, + 446, + 442 + ], + "lines": [ + { + "bbox": [ + 127, + 425, + 444, + 444 + ], + "spans": [ + { + "bbox": [ + 127, + 425, + 142, + 444 + ], + "score": 1.0, + "content": "3.", + "type": "text" + }, + { + "bbox": [ + 143, + 428, + 249, + 441 + ], + "score": 0.89, + "content": "J [ h _ { H + 1 : j + 1 } ] ( z ) \\in \\mathbb { R } ^ { d _ { y } \\times d _ { j } }", + "type": "inline_equation" + }, + { + "bbox": [ + 250, + 425, + 267, + 444 + ], + "score": 1.0, + "content": "has", + "type": "text" + }, + { + "bbox": [ + 268, + 428, + 383, + 441 + ], + "score": 0.73, + "content": "\\sigma _ { \\mathrm { m i n } } ( J [ h _ { H + 1 : j + 1 } ] ( z ) ) \\geq \\epsilon _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 383, + 425, + 410, + 444 + ], + "score": 1.0, + "content": "for all", + "type": "text" + }, + { + "bbox": [ + 411, + 428, + 444, + 439 + ], + "score": 0.66, + "content": "z \\in \\mathbb { R } ^ { d _ { j } }", + "type": "inline_equation" + } + ], + "index": 22 + } + ], + "index": 22 + }, + { + "type": "text", + "bbox": [ + 130, + 447, + 288, + 460 + ], + "lines": [ + { + "bbox": [ + 128, + 447, + 288, + 461 + ], + "spans": [ + { + "bbox": [ + 128, + 447, + 142, + 461 + ], + "score": 1.0, + "content": "4.", + "type": "text" + }, + { + "bbox": [ + 142, + 448, + 195, + 460 + ], + "score": 0.92, + "content": "h _ { H + 1 : j + 1 } ( z )", + "type": "inline_equation" + }, + { + "bbox": [ + 196, + 447, + 288, + 461 + ], + "score": 1.0, + "content": "is twice-differentiable,", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 23 + }, + { + "type": "text", + "bbox": [ + 107, + 468, + 488, + 482 + ], + "lines": [ + { + "bbox": [ + 106, + 468, + 488, + 483 + ], + "spans": [ + { + "bbox": [ + 106, + 468, + 207, + 483 + ], + "score": 1.0, + "content": "then any critical point of", + "type": "text" + }, + { + "bbox": [ + 208, + 469, + 228, + 481 + ], + "score": 0.91, + "content": "L ( h )", + "type": "inline_equation" + }, + { + "bbox": [ + 229, + 468, + 278, + 483 + ], + "score": 1.0, + "content": ", in terms of", + "type": "text" + }, + { + "bbox": [ + 278, + 468, + 399, + 482 + ], + "score": 0.91, + "content": "D _ { h _ { 1 } } [ L ( h ) ] , \\ldots , D _ { h _ { H + 1 } } [ L ( h ) ] ,", + "type": "inline_equation" + }, + { + "bbox": [ + 399, + 468, + 488, + 483 + ], + "score": 1.0, + "content": ", is a global minimum.", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 24 + }, + { + "type": "text", + "bbox": [ + 107, + 489, + 505, + 534 + ], + "lines": [ + { + "bbox": [ + 105, + 488, + 505, + 503 + ], + "spans": [ + { + "bbox": [ + 105, + 488, + 505, + 503 + ], + "score": 1.0, + "content": "Note that these theorems give sufficient conditions, whereas Theorems 2.1 and 2.2 provide necessary", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 500, + 505, + 512 + ], + "spans": [ + { + "bbox": [ + 106, + 500, + 429, + 512 + ], + "score": 1.0, + "content": "and sufficient conditions. So, if the sets we are describing in Theorems 4.1 and", + "type": "text" + }, + { + "bbox": [ + 430, + 501, + 456, + 511 + ], + "score": 0.33, + "content": "4 . 2 \\ \\mathrm { d o }", + "type": "inline_equation" + }, + { + "bbox": [ + 456, + 500, + 505, + 512 + ], + "score": 1.0, + "content": "not contain", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 511, + 506, + 525 + ], + "spans": [ + { + "bbox": [ + 105, + 511, + 506, + 525 + ], + "score": 1.0, + "content": "any critical point, the claims would be vacuous. We ensure that there are critical points in the sets,", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 522, + 448, + 535 + ], + "spans": [ + { + "bbox": [ + 105, + 522, + 448, + 535 + ], + "score": 1.0, + "content": "by presenting the following proposition, whose proof is also deferred to Appendix A.", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 26.5 + }, + { + "type": "text", + "bbox": [ + 106, + 537, + 504, + 560 + ], + "lines": [ + { + "bbox": [ + 105, + 537, + 505, + 550 + ], + "spans": [ + { + "bbox": [ + 105, + 537, + 505, + 550 + ], + "score": 1.0, + "content": "Proposition 4.3. For each of Theorems 4.1 and 4.2, there exists at least one global minimum solution", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 549, + 301, + 561 + ], + "spans": [ + { + "bbox": [ + 106, + 549, + 117, + 561 + ], + "score": 1.0, + "content": "of", + "type": "text" + }, + { + "bbox": [ + 117, + 549, + 138, + 560 + ], + "score": 0.89, + "content": "L ( h )", + "type": "inline_equation" + }, + { + "bbox": [ + 138, + 549, + 301, + 561 + ], + "score": 1.0, + "content": "satisfying the conditions of the theorem.", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 29.5 + }, + { + "type": "text", + "bbox": [ + 106, + 572, + 505, + 638 + ], + "lines": [ + { + "bbox": [ + 105, + 570, + 505, + 586 + ], + "spans": [ + { + "bbox": [ + 105, + 570, + 437, + 586 + ], + "score": 1.0, + "content": "Discussion and Future work. Theorems 4.1 and 4.2 state that in certain sets of", + "type": "text" + }, + { + "bbox": [ + 437, + 572, + 501, + 584 + ], + "score": 0.9, + "content": "\\left( h _ { 1 } , \\ldots , h _ { H + 1 } \\right)", + "type": "inline_equation" + }, + { + "bbox": [ + 501, + 570, + 505, + 586 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 583, + 506, + 595 + ], + "spans": [ + { + "bbox": [ + 105, + 583, + 506, + 595 + ], + "score": 1.0, + "content": "any critical point in function space a global minimum. However, this does not imply that any critical", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 594, + 505, + 607 + ], + "spans": [ + { + "bbox": [ + 105, + 594, + 505, + 607 + ], + "score": 1.0, + "content": "point for a fixed sigmoid or arctan network is a global minimum. As noted in (Bartlett et al., 2017),", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 604, + 506, + 618 + ], + "spans": [ + { + "bbox": [ + 105, + 604, + 506, + 618 + ], + "score": 1.0, + "content": "there is a downhill direction in function space at any suboptimal point, but this direction might", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 616, + 505, + 629 + ], + "spans": [ + { + "bbox": [ + 105, + 616, + 505, + 629 + ], + "score": 1.0, + "content": "be orthogonal to the function space represented by a fixed network, and may hence result in local", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 627, + 330, + 638 + ], + "spans": [ + { + "bbox": [ + 106, + 627, + 330, + 638 + ], + "score": 1.0, + "content": "minima in the parameter space of the fixed architecture.", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 33.5 + }, + { + "type": "text", + "bbox": [ + 107, + 643, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 644, + 505, + 656 + ], + "spans": [ + { + "bbox": [ + 105, + 644, + 505, + 656 + ], + "score": 1.0, + "content": "Understanding the connection between the function space and parameter space of commonly used", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 655, + 505, + 667 + ], + "spans": [ + { + "bbox": [ + 105, + 655, + 505, + 667 + ], + "score": 1.0, + "content": "architectures is an open direction for future research, and we believe that these results can be good", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 666, + 505, + 678 + ], + "spans": [ + { + "bbox": [ + 105, + 666, + 505, + 678 + ], + "score": 1.0, + "content": "initial steps from the theoretical point of view. For example, we can see that one of the sufficient", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 677, + 505, + 689 + ], + "spans": [ + { + "bbox": [ + 105, + 677, + 505, + 689 + ], + "score": 1.0, + "content": "conditions for global optimality is the Jacobian matrix being full rank. Given that a nonlinear func-", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 687, + 505, + 701 + ], + "spans": [ + { + "bbox": [ + 105, + 687, + 505, + 701 + ], + "score": 1.0, + "content": "tion can locally be linearly approximated using Jacobians, this connection is already interesting. An", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 699, + 505, + 711 + ], + "spans": [ + { + "bbox": [ + 105, + 699, + 505, + 711 + ], + "score": 1.0, + "content": "extension of the function space viewpoint to cover different architectures or design new architectures", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 710, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 105, + 710, + 505, + 722 + ], + "score": 1.0, + "content": "(that have “better” properties when viewed via the function space view) should also be possible and", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 719, + 171, + 734 + ], + "spans": [ + { + "bbox": [ + 105, + 719, + 171, + 734 + ], + "score": 1.0, + "content": "worth studying.", + "type": "text" + } + ], + "index": 44 + } + ], + "index": 40.5 + } + ], + "page_idx": 7, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 107, + 27, + 293, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 26, + 294, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 26, + 294, + 38 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2018", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 302, + 751, + 308, + 760 + ], + "lines": [ + { + "bbox": [ + 300, + 750, + 309, + 761 + ], + "spans": [ + { + "bbox": [ + 300, + 750, + 309, + 761 + ], + "score": 1.0, + "content": "8", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 82, + 505, + 149 + ], + "lines": [ + { + "bbox": [ + 105, + 80, + 506, + 96 + ], + "spans": [ + { + "bbox": [ + 105, + 80, + 133, + 96 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 134, + 83, + 145, + 93 + ], + "score": 0.88, + "content": "{ \\mathcal { F } } _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 146, + 80, + 222, + 96 + ], + "score": 1.0, + "content": "are defined for all", + "type": "text" + }, + { + "bbox": [ + 223, + 83, + 297, + 94 + ], + "score": 0.91, + "content": "i = 1 , \\ldots , H + 1", + "type": "inline_equation" + }, + { + "bbox": [ + 297, + 80, + 355, + 96 + ], + "score": 1.0, + "content": ". Assume that", + "type": "text" + }, + { + "bbox": [ + 356, + 83, + 390, + 93 + ], + "score": 0.91, + "content": "h ^ { \\ast } \\in \\mathcal { F }", + "type": "inline_equation" + }, + { + "bbox": [ + 390, + 80, + 506, + 96 + ], + "score": 1.0, + "content": ", and that we are optimizing", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 107, + 92, + 506, + 107 + ], + "spans": [ + { + "bbox": [ + 107, + 93, + 128, + 105 + ], + "score": 0.9, + "content": "L ( h )", + "type": "inline_equation" + }, + { + "bbox": [ + 128, + 92, + 151, + 107 + ], + "score": 1.0, + "content": "with", + "type": "text" + }, + { + "bbox": [ + 151, + 94, + 275, + 106 + ], + "score": 0.91, + "content": "h _ { 1 } \\in \\mathcal { F } _ { 1 } , \\ldots , h _ { H + 1 } \\in \\mathcal { F } _ { H + 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 276, + 92, + 416, + 107 + ], + "score": 1.0, + "content": ". In other words, the functions in", + "type": "text" + }, + { + "bbox": [ + 416, + 94, + 488, + 105 + ], + "score": 0.91, + "content": "\\mathcal { F } , \\mathcal { F } _ { 1 } , \\ldots , \\mathcal { F } _ { H + 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 488, + 92, + 506, + 107 + ], + "score": 1.0, + "content": "are", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 104, + 505, + 117 + ], + "spans": [ + { + "bbox": [ + 105, + 104, + 383, + 117 + ], + "score": 1.0, + "content": "differentiable and show sublinear growth starting from 0. Notice that", + "type": "text" + }, + { + "bbox": [ + 383, + 105, + 467, + 116 + ], + "score": 0.91, + "content": "h _ { H + 1 } \\circ \\cdot \\cdot \\cdot \\circ h _ { 1 } \\in \\mathcal { F }", + "type": "inline_equation" + }, + { + "bbox": [ + 467, + 104, + 505, + 117 + ], + "score": 1.0, + "content": ", because", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 115, + 504, + 127 + ], + "spans": [ + { + "bbox": [ + 105, + 115, + 504, + 127 + ], + "score": 1.0, + "content": "a composition of differentiable functions is also differentiable, and a composition of sublinear func-", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 126, + 506, + 139 + ], + "spans": [ + { + "bbox": [ + 105, + 126, + 286, + 139 + ], + "score": 1.0, + "content": "tions is also sublinear. We also assume that", + "type": "text" + }, + { + "bbox": [ + 286, + 126, + 361, + 138 + ], + "score": 0.92, + "content": "d _ { i } \\geq \\operatorname* { m i n } \\{ d _ { x } , d _ { y } \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 362, + 126, + 390, + 139 + ], + "score": 1.0, + "content": "for all", + "type": "text" + }, + { + "bbox": [ + 391, + 127, + 464, + 138 + ], + "score": 0.92, + "content": "i = 1 , \\ldots , H + 1", + "type": "inline_equation" + }, + { + "bbox": [ + 464, + 126, + 506, + 139 + ], + "score": 1.0, + "content": ", which is", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 137, + 355, + 150 + ], + "spans": [ + { + "bbox": [ + 105, + 137, + 216, + 150 + ], + "score": 1.0, + "content": "identical to the assumption", + "type": "text" + }, + { + "bbox": [ + 217, + 137, + 287, + 149 + ], + "score": 0.94, + "content": "k = \\operatorname* { m i n } \\{ d _ { x } , d _ { y } \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 288, + 137, + 355, + 150 + ], + "score": 1.0, + "content": "in Theorem 2.1.", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 2.5, + "bbox_fs": [ + 105, + 80, + 506, + 150 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 160, + 505, + 218 + ], + "lines": [ + { + "bbox": [ + 106, + 160, + 505, + 174 + ], + "spans": [ + { + "bbox": [ + 106, + 160, + 386, + 174 + ], + "score": 1.0, + "content": "Notation. To simplify multiple composition of functions, we denote", + "type": "text" + }, + { + "bbox": [ + 386, + 162, + 505, + 173 + ], + "score": 0.9, + "content": "h _ { i : j } = h _ { i } \\circ h _ { i - 1 } \\circ \\cdot \\cdot \\cdot \\circ h _ { j + 1 } \\circ", + "type": "inline_equation" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 172, + 506, + 186 + ], + "spans": [ + { + "bbox": [ + 106, + 173, + 117, + 185 + ], + "score": 0.86, + "content": "h _ { j }", + "type": "inline_equation" + }, + { + "bbox": [ + 118, + 172, + 217, + 186 + ], + "score": 1.0, + "content": ". As in the matrix case,", + "type": "text" + }, + { + "bbox": [ + 217, + 173, + 235, + 184 + ], + "score": 0.88, + "content": "h _ { 0 : 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 235, + 172, + 254, + 186 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 254, + 173, + 298, + 185 + ], + "score": 0.91, + "content": "h _ { H + 1 : H + 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 299, + 172, + 393, + 186 + ], + "score": 1.0, + "content": "mean identity maps in", + "type": "text" + }, + { + "bbox": [ + 394, + 173, + 411, + 183 + ], + "score": 0.88, + "content": "\\mathbb { R } ^ { d _ { x } }", + "type": "inline_equation" + }, + { + "bbox": [ + 411, + 172, + 431, + 186 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 431, + 173, + 448, + 183 + ], + "score": 0.88, + "content": "\\mathbb { R } ^ { d _ { y } }", + "type": "inline_equation" + }, + { + "bbox": [ + 448, + 172, + 506, + 186 + ], + "score": 1.0, + "content": ", respectively.", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 183, + 504, + 197 + ], + "spans": [ + { + "bbox": [ + 106, + 183, + 176, + 197 + ], + "score": 1.0, + "content": "Given a function", + "type": "text" + }, + { + "bbox": [ + 176, + 185, + 183, + 196 + ], + "score": 0.81, + "content": "f", + "type": "inline_equation" + }, + { + "bbox": [ + 184, + 183, + 200, + 197 + ], + "score": 1.0, + "content": ", let", + "type": "text" + }, + { + "bbox": [ + 200, + 185, + 232, + 196 + ], + "score": 0.92, + "content": "J [ f ] ( x )", + "type": "inline_equation" + }, + { + "bbox": [ + 232, + 183, + 374, + 197 + ], + "score": 1.0, + "content": "be the Jacobian matrix of function", + "type": "text" + }, + { + "bbox": [ + 374, + 185, + 381, + 196 + ], + "score": 0.86, + "content": "f", + "type": "inline_equation" + }, + { + "bbox": [ + 381, + 183, + 432, + 197 + ], + "score": 1.0, + "content": "evaluated at", + "type": "text" + }, + { + "bbox": [ + 433, + 187, + 439, + 194 + ], + "score": 0.76, + "content": "x", + "type": "inline_equation" + }, + { + "bbox": [ + 439, + 183, + 460, + 197 + ], + "score": 1.0, + "content": ". Let", + "type": "text" + }, + { + "bbox": [ + 460, + 184, + 504, + 196 + ], + "score": 0.89, + "content": "D _ { h _ { i } } [ L ( h ) ]", + "type": "inline_equation" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 194, + 504, + 208 + ], + "spans": [ + { + "bbox": [ + 105, + 194, + 216, + 208 + ], + "score": 1.0, + "content": "be the Frechet derivative of ´", + "type": "text" + }, + { + "bbox": [ + 216, + 196, + 237, + 207 + ], + "score": 0.94, + "content": "L ( h )", + "type": "inline_equation" + }, + { + "bbox": [ + 237, + 194, + 298, + 208 + ], + "score": 1.0, + "content": "with respect to", + "type": "text" + }, + { + "bbox": [ + 298, + 196, + 308, + 206 + ], + "score": 0.88, + "content": "h _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 308, + 194, + 358, + 208 + ], + "score": 1.0, + "content": "evaluated at", + "type": "text" + }, + { + "bbox": [ + 358, + 196, + 365, + 205 + ], + "score": 0.81, + "content": "h", + "type": "inline_equation" + }, + { + "bbox": [ + 366, + 194, + 460, + 208 + ], + "score": 1.0, + "content": ". The Frechet derivative ´", + "type": "text" + }, + { + "bbox": [ + 460, + 196, + 504, + 207 + ], + "score": 0.89, + "content": "D _ { h _ { i } } [ L ( h ) ]", + "type": "inline_equation" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 205, + 505, + 219 + ], + "spans": [ + { + "bbox": [ + 105, + 205, + 314, + 219 + ], + "score": 1.0, + "content": "is a linear functional that maps a function (direction)", + "type": "text" + }, + { + "bbox": [ + 314, + 207, + 343, + 218 + ], + "score": 0.92, + "content": "\\eta \\in \\mathcal { F } _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 344, + 205, + 505, + 219 + ], + "score": 1.0, + "content": "to a real number (directional derivative).", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 8, + "bbox_fs": [ + 105, + 160, + 506, + 219 + ] + }, + { + "type": "title", + "bbox": [ + 106, + 231, + 349, + 243 + ], + "lines": [ + { + "bbox": [ + 106, + 231, + 352, + 244 + ], + "spans": [ + { + "bbox": [ + 106, + 231, + 352, + 244 + ], + "score": 1.0, + "content": "4.2 SUFFICIENT CONDITIONS FOR GLOBAL OPTIMALITY", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 11 + }, + { + "type": "text", + "bbox": [ + 105, + 251, + 505, + 274 + ], + "lines": [ + { + "bbox": [ + 105, + 250, + 504, + 265 + ], + "spans": [ + { + "bbox": [ + 105, + 250, + 438, + 265 + ], + "score": 1.0, + "content": "Here, we present two theorems which give sufficient conditions for a critical point", + "type": "text" + }, + { + "bbox": [ + 439, + 252, + 504, + 264 + ], + "score": 0.91, + "content": "( D _ { h _ { i } } [ L ( h ) ] = 0", + "type": "inline_equation" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 262, + 486, + 275 + ], + "spans": [ + { + "bbox": [ + 106, + 262, + 133, + 275 + ], + "score": 1.0, + "content": "for all", + "type": "text" + }, + { + "bbox": [ + 133, + 264, + 138, + 272 + ], + "score": 0.67, + "content": "i", + "type": "inline_equation" + }, + { + "bbox": [ + 138, + 262, + 486, + 275 + ], + "score": 1.0, + "content": ") in the function space to be a global optimum. The proofs are deferred to Appendix A.", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 12.5, + "bbox_fs": [ + 105, + 250, + 504, + 275 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 277, + 400, + 290 + ], + "lines": [ + { + "bbox": [ + 106, + 277, + 400, + 291 + ], + "spans": [ + { + "bbox": [ + 106, + 277, + 241, + 291 + ], + "score": 1.0, + "content": "Theorem 4.1. Consider the case", + "type": "text" + }, + { + "bbox": [ + 241, + 277, + 275, + 290 + ], + "score": 0.92, + "content": "d _ { x } \\geq d _ { y }", + "type": "inline_equation" + }, + { + "bbox": [ + 276, + 277, + 336, + 291 + ], + "score": 1.0, + "content": ". If there exists", + "type": "text" + }, + { + "bbox": [ + 336, + 278, + 360, + 288 + ], + "score": 0.88, + "content": "\\epsilon > 0", + "type": "inline_equation" + }, + { + "bbox": [ + 360, + 277, + 400, + 291 + ], + "score": 1.0, + "content": "such that", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 14, + "bbox_fs": [ + 106, + 277, + 400, + 291 + ] + }, + { + "type": "index", + "bbox": [ + 128, + 298, + 424, + 331 + ], + "lines": [ + { + "bbox": [ + 129, + 297, + 423, + 313 + ], + "spans": [ + { + "bbox": [ + 129, + 297, + 142, + 313 + ], + "score": 1.0, + "content": "1.", + "type": "text" + }, + { + "bbox": [ + 143, + 298, + 240, + 312 + ], + "score": 0.89, + "content": "J [ h _ { H + 1 : 2 } ] ( z ) \\in \\mathbb { R } ^ { d _ { y } \\times d _ { 1 } }", + "type": "inline_equation" + }, + { + "bbox": [ + 241, + 297, + 257, + 313 + ], + "score": 1.0, + "content": "has", + "type": "text" + }, + { + "bbox": [ + 257, + 298, + 423, + 312 + ], + "score": 0.59, + "content": "\\sigma _ { \\operatorname* { m i n } } ( J [ h _ { H + 1 : 2 } ] ( z ) ) \\geq \\epsilon f o r \\ : a l l \\ : z \\in \\mathbb { R } ^ { d _ { 1 } } ,", + "type": "inline_equation", + "image_path": "aec8497d8822eb241bfff78f4099996eef6556b290f68cbe1ee0e96b4347c0bc.jpg" + } + ], + "index": 15, + "is_list_start_line": true + }, + { + "bbox": [ + 129, + 318, + 278, + 332 + ], + "spans": [ + { + "bbox": [ + 129, + 318, + 142, + 332 + ], + "score": 1.0, + "content": "2.", + "type": "text" + }, + { + "bbox": [ + 142, + 318, + 186, + 331 + ], + "score": 0.92, + "content": "h _ { H + 1 : 2 } ( z )", + "type": "inline_equation" + }, + { + "bbox": [ + 186, + 318, + 278, + 332 + ], + "score": 1.0, + "content": "is twice-differentiable,", + "type": "text" + } + ], + "index": 16, + "is_list_start_line": true + } + ], + "index": 15.5, + "bbox_fs": [ + 129, + 297, + 423, + 332 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 338, + 487, + 352 + ], + "lines": [ + { + "bbox": [ + 106, + 339, + 488, + 353 + ], + "spans": [ + { + "bbox": [ + 106, + 339, + 207, + 353 + ], + "score": 1.0, + "content": "then any critical point of", + "type": "text" + }, + { + "bbox": [ + 208, + 339, + 228, + 352 + ], + "score": 0.91, + "content": "L ( h )", + "type": "inline_equation" + }, + { + "bbox": [ + 229, + 339, + 278, + 353 + ], + "score": 1.0, + "content": ", in terms of", + "type": "text" + }, + { + "bbox": [ + 279, + 339, + 399, + 353 + ], + "score": 0.93, + "content": "D _ { h _ { 1 } } [ L ( h ) ] , \\ldots , D _ { h _ { H + 1 } } [ L ( h ) ] ,", + "type": "inline_equation" + }, + { + "bbox": [ + 399, + 339, + 488, + 353 + ], + "score": 1.0, + "content": ", is a global minimum.", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 17, + "bbox_fs": [ + 106, + 339, + 488, + 353 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 354, + 504, + 378 + ], + "lines": [ + { + "bbox": [ + 105, + 353, + 506, + 367 + ], + "spans": [ + { + "bbox": [ + 105, + 353, + 241, + 367 + ], + "score": 1.0, + "content": "Theorem 4.2. Consider the case", + "type": "text" + }, + { + "bbox": [ + 241, + 355, + 276, + 367 + ], + "score": 0.92, + "content": "d _ { x } \\leq d _ { y }", + "type": "inline_equation" + }, + { + "bbox": [ + 276, + 353, + 402, + 367 + ], + "score": 1.0, + "content": ". Assume that there exists some", + "type": "text" + }, + { + "bbox": [ + 402, + 354, + 483, + 367 + ], + "score": 0.93, + "content": "j \\in \\{ 1 , \\dots , H + 1 \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 483, + 353, + 506, + 367 + ], + "score": 1.0, + "content": "such", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 365, + 360, + 379 + ], + "spans": [ + { + "bbox": [ + 105, + 365, + 124, + 379 + ], + "score": 1.0, + "content": "that", + "type": "text" + }, + { + "bbox": [ + 125, + 366, + 169, + 378 + ], + "score": 0.93, + "content": "d _ { x } = d _ { j - 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 169, + 365, + 187, + 379 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 187, + 366, + 221, + 378 + ], + "score": 0.92, + "content": "d _ { y } \\leq d _ { j }", + "type": "inline_equation" + }, + { + "bbox": [ + 221, + 365, + 277, + 379 + ], + "score": 1.0, + "content": ". If there exist", + "type": "text" + }, + { + "bbox": [ + 278, + 366, + 318, + 377 + ], + "score": 0.91, + "content": "\\epsilon _ { 1 } , \\epsilon _ { 2 } > 0", + "type": "inline_equation" + }, + { + "bbox": [ + 318, + 365, + 360, + 379 + ], + "score": 1.0, + "content": "such that", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 18.5, + "bbox_fs": [ + 105, + 353, + 506, + 379 + ] + }, + { + "type": "text", + "bbox": [ + 131, + 386, + 316, + 400 + ], + "lines": [ + { + "bbox": [ + 128, + 385, + 317, + 401 + ], + "spans": [ + { + "bbox": [ + 128, + 385, + 142, + 401 + ], + "score": 1.0, + "content": "1.", + "type": "text" + }, + { + "bbox": [ + 142, + 387, + 263, + 400 + ], + "score": 0.9, + "content": "h _ { j - 1 : 1 } : \\mathbb { R } ^ { d _ { x } } \\mathbb { R } ^ { d _ { j - 1 } } = \\mathbb { R } ^ { d _ { x } }", + "type": "inline_equation" + }, + { + "bbox": [ + 264, + 385, + 317, + 401 + ], + "score": 1.0, + "content": "is invertible,", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 20, + "bbox_fs": [ + 128, + 385, + 317, + 401 + ] + }, + { + "type": "text", + "bbox": [ + 130, + 407, + 371, + 422 + ], + "lines": [ + { + "bbox": [ + 127, + 406, + 371, + 423 + ], + "spans": [ + { + "bbox": [ + 127, + 406, + 142, + 423 + ], + "score": 1.0, + "content": "2.", + "type": "text" + }, + { + "bbox": [ + 142, + 408, + 170, + 421 + ], + "score": 0.9, + "content": "h _ { j - 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So, if the sets we are describing in Theorems 4.1 and", + "type": "text" + }, + { + "bbox": [ + 430, + 501, + 456, + 511 + ], + "score": 0.33, + "content": "4 . 2 \\ \\mathrm { d o }", + "type": "inline_equation" + }, + { + "bbox": [ + 456, + 500, + 505, + 512 + ], + "score": 1.0, + "content": "not contain", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 511, + 506, + 525 + ], + "spans": [ + { + "bbox": [ + 105, + 511, + 506, + 525 + ], + "score": 1.0, + "content": "any critical point, the claims would be vacuous. We ensure that there are critical points in the sets,", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 522, + 448, + 535 + ], + "spans": [ + { + "bbox": [ + 105, + 522, + 448, + 535 + ], + "score": 1.0, + "content": "by presenting the following proposition, whose proof is also deferred to Appendix A.", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 26.5, + "bbox_fs": [ + 105, + 488, + 506, + 535 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 537, + 504, + 560 + ], + "lines": [ + { + "bbox": [ + 105, + 537, + 505, + 550 + ], + "spans": [ + { + "bbox": [ + 105, + 537, + 505, + 550 + ], + "score": 1.0, + "content": "Proposition 4.3. For each of Theorems 4.1 and 4.2, there exists at least one global minimum solution", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 549, + 301, + 561 + ], + "spans": [ + { + "bbox": [ + 106, + 549, + 117, + 561 + ], + "score": 1.0, + "content": "of", + "type": "text" + }, + { + "bbox": [ + 117, + 549, + 138, + 560 + ], + "score": 0.89, + "content": "L ( h )", + "type": "inline_equation" + }, + { + "bbox": [ + 138, + 549, + 301, + 561 + ], + "score": 1.0, + "content": "satisfying the conditions of the theorem.", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 29.5, + "bbox_fs": [ + 105, + 537, + 505, + 561 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 572, + 505, + 638 + ], + "lines": [ + { + "bbox": [ + 105, + 570, + 505, + 586 + ], + "spans": [ + { + "bbox": [ + 105, + 570, + 437, + 586 + ], + "score": 1.0, + "content": "Discussion and Future work. Theorems 4.1 and 4.2 state that in certain sets of", + "type": "text" + }, + { + "bbox": [ + 437, + 572, + 501, + 584 + ], + "score": 0.9, + "content": "\\left( h _ { 1 } , \\ldots , h _ { H + 1 } \\right)", + "type": "inline_equation" + }, + { + "bbox": [ + 501, + 570, + 505, + 586 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 583, + 506, + 595 + ], + "spans": [ + { + "bbox": [ + 105, + 583, + 506, + 595 + ], + "score": 1.0, + "content": "any critical point in function space a global minimum. However, this does not imply that any critical", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 594, + 505, + 607 + ], + "spans": [ + { + "bbox": [ + 105, + 594, + 505, + 607 + ], + "score": 1.0, + "content": "point for a fixed sigmoid or arctan network is a global minimum. As noted in (Bartlett et al., 2017),", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 604, + 506, + 618 + ], + "spans": [ + { + "bbox": [ + 105, + 604, + 506, + 618 + ], + "score": 1.0, + "content": "there is a downhill direction in function space at any suboptimal point, but this direction might", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 616, + 505, + 629 + ], + "spans": [ + { + "bbox": [ + 105, + 616, + 505, + 629 + ], + "score": 1.0, + "content": "be orthogonal to the function space represented by a fixed network, and may hence result in local", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 627, + 330, + 638 + ], + "spans": [ + { + "bbox": [ + 106, + 627, + 330, + 638 + ], + "score": 1.0, + "content": "minima in the parameter space of the fixed architecture.", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 33.5, + "bbox_fs": [ + 105, + 570, + 506, + 638 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 643, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 644, + 505, + 656 + ], + "spans": [ + { + "bbox": [ + 105, + 644, + 505, + 656 + ], + "score": 1.0, + "content": "Understanding the connection between the function space and parameter space of commonly used", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 655, + 505, + 667 + ], + "spans": [ + { + "bbox": [ + 105, + 655, + 505, + 667 + ], + "score": 1.0, + "content": "architectures is an open direction for future research, and we believe that these results can be good", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 666, + 505, + 678 + ], + "spans": [ + { + "bbox": [ + 105, + 666, + 505, + 678 + ], + "score": 1.0, + "content": "initial steps from the theoretical point of view. For example, we can see that one of the sufficient", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 677, + 505, + 689 + ], + "spans": [ + { + "bbox": [ + 105, + 677, + 505, + 689 + ], + "score": 1.0, + "content": "conditions for global optimality is the Jacobian matrix being full rank. Given that a nonlinear func-", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 687, + 505, + 701 + ], + "spans": [ + { + "bbox": [ + 105, + 687, + 505, + 701 + ], + "score": 1.0, + "content": "tion can locally be linearly approximated using Jacobians, this connection is already interesting. 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IEEE transactions on", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 116, + 720, + 281, + 732 + ], + "spans": [ + { + "bbox": [ + 116, + 720, + 281, + 732 + ], + "score": 1.0, + "content": "automatic control, 11(2):228–238, 1966.", + "type": "text" + } + ], + "index": 44 + } + ], + "index": 43, + "bbox_fs": [ + 105, + 698, + 506, + 732 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "title", + "bbox": [ + 107, + 82, + 352, + 94 + ], + "lines": [ + { + "bbox": [ + 106, + 82, + 353, + 95 + ], + "spans": [ + { + "bbox": [ + 106, + 82, + 353, + 95 + ], + "score": 1.0, + "content": "A ANALYSIS OF DEEP NONLINEAR NETWORKS", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "title", + "bbox": [ + 107, + 106, + 179, + 118 + ], + "lines": [ + { + "bbox": [ + 105, + 104, + 180, + 119 + ], + "spans": [ + { + "bbox": [ + 105, + 104, + 180, + 119 + ], + "score": 1.0, + "content": "A.1 NOTATION", + "type": "text" + } + ], + "index": 1 + } + ], + "index": 1 + }, + { + "type": "text", + "bbox": [ + 106, + 126, + 505, + 171 + ], + "lines": [ + { + "bbox": [ + 106, + 127, + 505, + 138 + ], + "spans": [ + { + "bbox": [ + 106, + 127, + 505, + 138 + ], + "score": 1.0, + "content": "In this section, we introduce additional notation that is used in the proofs. To emphasize that", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 137, + 505, + 150 + ], + "spans": [ + { + "bbox": [ + 105, + 137, + 198, + 150 + ], + "score": 1.0, + "content": "the Frechet derivative ´", + "type": "text" + }, + { + "bbox": [ + 199, + 137, + 242, + 150 + ], + "score": 0.94, + "content": "D _ { h _ { i } } [ L ( h ) ]", + "type": "inline_equation" + }, + { + "bbox": [ + 242, + 137, + 505, + 150 + ], + "score": 1.0, + "content": "is a linear functional that outputs a real number, we will write", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 107, + 148, + 505, + 162 + ], + "spans": [ + { + "bbox": [ + 107, + 148, + 163, + 161 + ], + "score": 0.92, + "content": "D _ { h _ { i } } [ L ( h ) ] ( \\eta )", + "type": "inline_equation" + }, + { + "bbox": [ + 163, + 149, + 264, + 162 + ], + "score": 1.0, + "content": "in an inner-product form", + "type": "text" + }, + { + "bbox": [ + 265, + 149, + 325, + 161 + ], + "score": 0.93, + "content": "\\langle D _ { h _ { i } } [ L ( h ) ] , \\eta \\rangle", + "type": "inline_equation" + }, + { + "bbox": [ + 325, + 149, + 505, + 162 + ], + "score": 1.0, + "content": ". This notation also helps avoiding confusion", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 160, + 329, + 172 + ], + "spans": [ + { + "bbox": [ + 105, + 160, + 329, + 172 + ], + "score": 1.0, + "content": "coming from multiple parentheses and square brackets.", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 3.5 + }, + { + "type": "text", + "bbox": [ + 106, + 176, + 505, + 275 + ], + "lines": [ + { + "bbox": [ + 105, + 175, + 506, + 188 + ], + "spans": [ + { + "bbox": [ + 105, + 175, + 506, + 188 + ], + "score": 1.0, + "content": "There are many different kinds of norms that appear in the proofs. Given a finite-dimensional real", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 185, + 505, + 203 + ], + "spans": [ + { + "bbox": [ + 105, + 185, + 135, + 203 + ], + "score": 1.0, + "content": "vector", + "type": "text" + }, + { + "bbox": [ + 135, + 190, + 141, + 198 + ], + "score": 0.6, + "content": "v", + "type": "inline_equation" + }, + { + "bbox": [ + 142, + 185, + 147, + 203 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 147, + 187, + 168, + 200 + ], + "score": 0.89, + "content": "\\lVert \\boldsymbol { v } \\rVert _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 168, + 185, + 218, + 203 + ], + "score": 1.0, + "content": "denotes its", + "type": "text" + }, + { + "bbox": [ + 218, + 188, + 228, + 199 + ], + "score": 0.87, + "content": "\\ell _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 228, + 185, + 317, + 203 + ], + "score": 1.0, + "content": "norm. For a matrix", + "type": "text" + }, + { + "bbox": [ + 318, + 188, + 326, + 198 + ], + "score": 0.76, + "content": "A", + "type": "inline_equation" + }, + { + "bbox": [ + 326, + 185, + 463, + 203 + ], + "score": 1.0, + "content": ", its operator norm is defined as", + "type": "text" + }, + { + "bbox": [ + 463, + 187, + 505, + 201 + ], + "score": 0.92, + "content": "\\| A \\| _ { \\mathrm { o p } } =", + "type": "inline_equation" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 197, + 506, + 218 + ], + "spans": [ + { + "bbox": [ + 106, + 200, + 154, + 218 + ], + "score": 0.93, + "content": "\\operatorname* { s u p } _ { x } { \\frac { \\| A x \\| _ { 2 } } { \\| x \\| _ { 2 } } }", + "type": "inline_equation" + }, + { + "bbox": [ + 154, + 197, + 177, + 217 + ], + "score": 1.0, + "content": ". Let", + "type": "text" + }, + { + "bbox": [ + 177, + 203, + 207, + 214 + ], + "score": 0.89, + "content": "h \\in { \\mathcal { F } }", + "type": "inline_equation" + }, + { + "bbox": [ + 207, + 197, + 506, + 217 + ], + "score": 1.0, + "content": ". Then define a “generalized” induced norm for nonlinear functions with", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 216, + 508, + 237 + ], + "spans": [ + { + "bbox": [ + 105, + 219, + 240, + 234 + ], + "score": 1.0, + "content": "sublinear growth: khknl = supx", + "type": "text" + }, + { + "bbox": [ + 180, + 217, + 268, + 236 + ], + "score": 0.93, + "content": "\\begin{array} { r } { \\| h \\| _ { \\mathrm { n l } } = \\operatorname* { s u p } _ { x } \\frac { \\| h ( x ) \\| _ { 2 } } { \\| x \\| _ { 2 } } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 235, + 216, + 508, + 237 + ], + "score": 1.0, + "content": "kh(x)k2kxk , where the subscript nl is used to emphasize that this norm", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 234, + 506, + 248 + ], + "spans": [ + { + "bbox": [ + 105, + 234, + 265, + 248 + ], + "score": 1.0, + "content": "is for nonlinear functions. The norm", + "type": "text" + }, + { + "bbox": [ + 266, + 235, + 286, + 247 + ], + "score": 0.87, + "content": "\\lVert \\cdot \\rVert _ { \\mathrm { n l } }", + "type": "inline_equation" + }, + { + "bbox": [ + 287, + 234, + 420, + 248 + ], + "score": 1.0, + "content": "is defined in the same way for", + "type": "text" + }, + { + "bbox": [ + 420, + 235, + 432, + 246 + ], + "score": 0.87, + "content": "{ \\mathcal { F } } _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 432, + 234, + 506, + 248 + ], + "score": 1.0, + "content": "’s. 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For any", + "type": "text" + }, + { + "bbox": [ + 265, + 570, + 295, + 582 + ], + "score": 0.92, + "content": "\\eta \\in \\mathcal { F } _ { 1 }", + "type": "inline_equation" + } + ], + "index": 25 + } + ], + "index": 25 + }, + { + "type": "interline_equation", + "bbox": [ + 157, + 587, + 453, + 603 + ], + "lines": [ + { + "bbox": [ + 157, + 587, + 453, + 603 + ], + "spans": [ + { + "bbox": [ + 157, + 587, + 453, + 603 + ], + "score": 0.87, + "content": "\\langle D _ { h _ { 1 } } [ L ( h ) ] , \\eta \\rangle = \\mathbb { E } \\left[ ( h _ { H + 1 : 1 } ( X ) - h ^ { * } ( X ) ) ^ { T } J [ h _ { H + 1 : 2 } ] ( h _ { 1 } ( X ) ) \\eta ( X ) \\right] .", + "type": "interline_equation", + "image_path": "c39a7571c81244288615d2cff6a0b234a2b4b3891c81a81c9993727432d530e3.jpg" + } + ] + } + ], + "index": 26, + "virtual_lines": [ + { + "bbox": [ + 157, + 587, + 453, + 603 + ], + "spans": [], + "index": 26 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 608, + 504, + 631 + ], + "lines": [ + { + "bbox": [ + 104, + 606, + 506, + 622 + ], + "spans": [ + { + "bbox": [ + 104, + 606, + 123, + 622 + ], + "score": 1.0, + "content": "Let", + "type": "text" + }, + { + "bbox": [ + 124, + 608, + 240, + 621 + ], + "score": 0.93, + "content": "A ( X ) = J [ h _ { H + 1 : 2 } ] ( h _ { 1 } ( X ) )", + "type": "inline_equation" + }, + { + "bbox": [ + 241, + 606, + 273, + 622 + ], + "score": 1.0, + "content": ". Since", + "type": "text" + }, + { + "bbox": [ + 273, + 609, + 298, + 621 + ], + "score": 0.93, + "content": "A ( X )", + "type": "inline_equation" + }, + { + "bbox": [ + 298, + 606, + 438, + 622 + ], + "score": 1.0, + "content": "has full row rank by assumption,", + "type": "text" + }, + { + "bbox": [ + 438, + 608, + 493, + 621 + ], + "score": 0.93, + "content": "A ( X ) A ( X ) ^ { T }", + "type": "inline_equation" + }, + { + "bbox": [ + 494, + 606, + 506, + 622 + ], + "score": 1.0, + "content": "is", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 618, + 285, + 632 + ], + "spans": [ + { + "bbox": [ + 105, + 618, + 285, + 632 + ], + "score": 1.0, + "content": "invertible. Then define a particular direction", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 27.5 + }, + { + "type": "interline_equation", + "bbox": [ + 186, + 636, + 424, + 652 + ], + "lines": [ + { + "bbox": [ + 186, + 636, + 424, + 652 + ], + "spans": [ + { + "bbox": [ + 186, + 636, + 424, + 652 + ], + "score": 0.88, + "content": "\\tilde { \\eta } ( X ) = A ( X ) ^ { T } ( A ( X ) A ( X ) ^ { T } ) ^ { - 1 } ( h _ { H + 1 : 1 } ( X ) - h ^ { * } ( X ) ) ,", + "type": "interline_equation", + "image_path": "04459bd1e6ac806c2ab503bb80bc9ab5e79f6f7a4b67659139899209b73c7f57.jpg" + } + ] + } + ], + "index": 29, + "virtual_lines": [ + { + "bbox": [ + 186, + 636, + 424, + 652 + ], + "spans": [], + "index": 29 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 657, + 135, + 667 + ], + "lines": [ + { + "bbox": [ + 105, + 655, + 136, + 668 + ], + "spans": [ + { + "bbox": [ + 105, + 655, + 136, + 668 + ], + "score": 1.0, + "content": "so that", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 30 + }, + { + "type": "interline_equation", + "bbox": [ + 206, + 664, + 404, + 685 + ], + "lines": [ + { + "bbox": [ + 206, + 664, + 404, + 685 + ], + "spans": [ + { + "bbox": [ + 206, + 664, + 404, + 685 + ], + "score": 0.88, + "content": "\\begin{array} { r } { \\langle D _ { h _ { 1 } } [ L ( h ) ] , \\tilde { \\eta } \\rangle = \\mathbb { E } \\left[ \\| h _ { H + 1 : 1 } ( X ) - h ^ { * } ( X ) \\| _ { 2 } ^ { 2 } \\right] . } \\end{array}", + "type": "interline_equation", + "image_path": "5de7dc757dab9ae5a48b1695c2eb8bfda53a50545d137a7f67f69b1f0356d730.jpg" + } + ] + } + ], + "index": 31, + "virtual_lines": [ + { + "bbox": [ + 206, + 664, + 404, + 685 + ], + "spans": [], + "index": 31 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 687, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 686, + 506, + 701 + ], + "spans": [ + { + "bbox": [ + 105, + 686, + 195, + 701 + ], + "score": 1.0, + "content": "It remains to check if", + "type": "text" + }, + { + "bbox": [ + 195, + 688, + 226, + 699 + ], + "score": 0.92, + "content": "\\tilde { \\eta } \\in \\mathcal { F } _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 226, + 686, + 329, + 701 + ], + "score": 1.0, + "content": ". 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To emphasize that", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 137, + 505, + 150 + ], + "spans": [ + { + "bbox": [ + 105, + 137, + 198, + 150 + ], + "score": 1.0, + "content": "the Frechet derivative ´", + "type": "text" + }, + { + "bbox": [ + 199, + 137, + 242, + 150 + ], + "score": 0.94, + "content": "D _ { h _ { i } } [ L ( h ) ]", + "type": "inline_equation" + }, + { + "bbox": [ + 242, + 137, + 505, + 150 + ], + "score": 1.0, + "content": "is a linear functional that outputs a real number, we will write", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 107, + 148, + 505, + 162 + ], + "spans": [ + { + "bbox": [ + 107, + 148, + 163, + 161 + ], + "score": 0.92, + "content": "D _ { h _ { i } } [ L ( h ) ] ( \\eta )", + "type": "inline_equation" + }, + { + "bbox": [ + 163, + 149, + 264, + 162 + ], + "score": 1.0, + "content": "in an inner-product form", + "type": "text" + }, + { + "bbox": [ + 265, + 149, + 325, + 161 + ], + "score": 0.93, + "content": "\\langle D _ { h _ { i } } [ L ( h ) ] , \\eta \\rangle", + "type": "inline_equation" + }, + { + "bbox": [ + 325, + 149, + 505, + 162 + ], + "score": 1.0, + "content": ". This notation also helps avoiding confusion", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 160, + 329, + 172 + ], + "spans": [ + { + "bbox": [ + 105, + 160, + 329, + 172 + ], + "score": 1.0, + "content": "coming from multiple parentheses and square brackets.", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 3.5, + "bbox_fs": [ + 105, + 127, + 505, + 172 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 176, + 505, + 275 + ], + "lines": [ + { + "bbox": [ + 105, + 175, + 506, + 188 + ], + "spans": [ + { + "bbox": [ + 105, + 175, + 506, + 188 + ], + "score": 1.0, + "content": "There are many different kinds of norms that appear in the proofs. Given a finite-dimensional real", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 185, + 505, + 203 + ], + "spans": [ + { + "bbox": [ + 105, + 185, + 135, + 203 + ], + "score": 1.0, + "content": "vector", + "type": "text" + }, + { + "bbox": [ + 135, + 190, + 141, + 198 + ], + "score": 0.6, + "content": "v", + "type": "inline_equation" + }, + { + "bbox": [ + 142, + 185, + 147, + 203 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 147, + 187, + 168, + 200 + ], + "score": 0.89, + "content": "\\lVert \\boldsymbol { v } \\rVert _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 168, + 185, + 218, + 203 + ], + "score": 1.0, + "content": "denotes its", + "type": "text" + }, + { + "bbox": [ + 218, + 188, + 228, + 199 + ], + "score": 0.87, + "content": "\\ell _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 228, + 185, + 317, + 203 + ], + "score": 1.0, + "content": "norm. For a matrix", + "type": "text" + }, + { + "bbox": [ + 318, + 188, + 326, + 198 + ], + "score": 0.76, + "content": "A", + "type": "inline_equation" + }, + { + "bbox": [ + 326, + 185, + 463, + 203 + ], + "score": 1.0, + "content": ", its operator norm is defined as", + "type": "text" + }, + { + "bbox": [ + 463, + 187, + 505, + 201 + ], + "score": 0.92, + "content": "\\| A \\| _ { \\mathrm { o p } } =", + "type": "inline_equation" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 197, + 506, + 218 + ], + "spans": [ + { + "bbox": [ + 106, + 200, + 154, + 218 + ], + "score": 0.93, + "content": "\\operatorname* { s u p } _ { x } { \\frac { \\| A x \\| _ { 2 } } { \\| x \\| _ { 2 } } }", + "type": "inline_equation" + }, + { + "bbox": [ + 154, + 197, + 177, + 217 + ], + "score": 1.0, + "content": ". Let", + "type": "text" + }, + { + "bbox": [ + 177, + 203, + 207, + 214 + ], + "score": 0.89, + "content": "h \\in { \\mathcal { F } }", + "type": "inline_equation" + }, + { + "bbox": [ + 207, + 197, + 506, + 217 + ], + "score": 1.0, + "content": ". Then define a “generalized” induced norm for nonlinear functions with", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 216, + 508, + 237 + ], + "spans": [ + { + "bbox": [ + 105, + 219, + 240, + 234 + ], + "score": 1.0, + "content": "sublinear growth: khknl = supx", + "type": "text" + }, + { + "bbox": [ + 180, + 217, + 268, + 236 + ], + "score": 0.93, + "content": "\\begin{array} { r } { \\| h \\| _ { \\mathrm { n l } } = \\operatorname* { s u p } _ { x } \\frac { \\| h ( x ) \\| _ { 2 } } { \\| x \\| _ { 2 } } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 235, + 216, + 508, + 237 + ], + "score": 1.0, + "content": "kh(x)k2kxk , where the subscript nl is used to emphasize that this norm", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 234, + 506, + 248 + ], + "spans": [ + { + "bbox": [ + 105, + 234, + 265, + 248 + ], + "score": 1.0, + "content": "is for nonlinear functions. The norm", + "type": "text" + }, + { + "bbox": [ + 266, + 235, + 286, + 247 + ], + "score": 0.87, + "content": "\\lVert \\cdot \\rVert _ { \\mathrm { n l } }", + "type": "inline_equation" + }, + { + "bbox": [ + 287, + 234, + 420, + 248 + ], + "score": 1.0, + "content": "is defined in the same way for", + "type": "text" + }, + { + "bbox": [ + 420, + 235, + 432, + 246 + ], + "score": 0.87, + "content": "{ \\mathcal { F } } _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 432, + 234, + 506, + 248 + ], + "score": 1.0, + "content": "’s. 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Finally,", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 12 + }, + { + "type": "interline_equation", + "bbox": [ + 122, + 334, + 489, + 370 + ], + "lines": [ + { + "bbox": [ + 122, + 334, + 489, + 370 + ], + "spans": [ + { + "bbox": [ + 122, + 334, + 489, + 370 + ], + "score": 0.93, + "content": "\\| D _ { h _ { 1 } } [ L ( h ) ] \\| _ { \\mathrm { o p } } \\ge \\frac { \\langle D _ { h _ { 1 } } [ L ( h ) ] , \\tilde { \\eta } \\rangle } { \\| \\tilde { \\eta } \\| _ { \\mathrm { n l } } } \\ge \\frac { \\epsilon \\mathbb { E } \\left[ \\| h _ { H + 1 : 1 } ( X ) - h ^ { * } ( X ) \\| _ { 2 } ^ { 2 } \\right] } { \\| h _ { H + 1 : 1 } - h ^ { * } \\| _ { \\mathrm { n l } } } = \\frac { \\epsilon ( L ( h ) - L ^ { * } ) } { \\| h _ { H + 1 : 1 } - h ^ { * } \\| _ { \\mathrm { n l } } } ,", + "type": "interline_equation", + "image_path": "10be6a4159ee74d4731aa92da85915fdb7dbf502660bc83fca35e1f2590f4183.jpg" + } + ] + } + ], + "index": 14, + "virtual_lines": [ + { + "bbox": [ + 122, + 334, + 489, + 346.0 + ], + "spans": [], + "index": 13 + }, + { + "bbox": [ + 122, + 346.0, + 489, + 358.0 + ], + "spans": [], + "index": 14 + }, + { + "bbox": [ + 122, + 358.0, + 489, + 370.0 + ], + "spans": [], + "index": 15 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 375, + 159, + 387 + ], + "lines": [ + { + "bbox": [ + 106, + 374, + 160, + 388 + ], + "spans": [ + { + "bbox": [ + 106, + 374, + 160, + 388 + ], + "score": 1.0, + "content": "which yields", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 16 + }, + { + "type": "interline_equation", + "bbox": [ + 201, + 386, + 408, + 402 + ], + "lines": [ + { + "bbox": [ + 201, + 386, + 408, + 402 + ], + "spans": [ + { + "bbox": [ + 201, + 386, + 408, + 402 + ], + "score": 0.88, + "content": "\\begin{array} { r } { \\| D _ { h _ { 1 } } [ L ( h ) ] \\| _ { \\mathrm { o p } } \\| h _ { H + 1 : 1 } - h ^ { * } \\| _ { \\mathrm { n l } } \\ge \\epsilon ( L ( h ) - L ^ { * } ) . } \\end{array}", + "type": "interline_equation", + "image_path": "1a6caa84edd7d98070f8b2297f43819df8f34c7df5ead68d1360f6074f35f20f.jpg" + } + ] + } + ], + "index": 17, + "virtual_lines": [ + { + "bbox": [ + 201, + 386, + 408, + 402 + ], + "spans": [], + "index": 17 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 405, + 505, + 431 + ], + "lines": [ + { + "bbox": [ + 104, + 403, + 506, + 421 + ], + "spans": [ + { + "bbox": [ + 104, + 403, + 338, + 421 + ], + "score": 1.0, + "content": "From this we can see that if we have a critical point of", + "type": "text" + }, + { + "bbox": [ + 339, + 406, + 360, + 418 + ], + "score": 0.92, + "content": "L ( h )", + "type": "inline_equation" + }, + { + "bbox": [ + 360, + 403, + 385, + 421 + ], + "score": 1.0, + "content": ", then", + "type": "text" + }, + { + "bbox": [ + 386, + 405, + 471, + 420 + ], + "score": 0.92, + "content": "\\| D _ { h _ { 1 } } [ L ( h ) ] \\| _ { \\mathrm { o p } } = 0", + "type": "inline_equation" + }, + { + "bbox": [ + 471, + 403, + 506, + 421 + ], + "score": 1.0, + "content": "implies", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 107, + 418, + 417, + 432 + ], + "spans": [ + { + "bbox": [ + 107, + 418, + 152, + 431 + ], + "score": 0.93, + "content": "L ( h ) = L ^ { * }", + "type": "inline_equation" + }, + { + "bbox": [ + 153, + 418, + 392, + 432 + ], + "score": 1.0, + "content": ", which means that the critical point is a global minimum of", + "type": "text" + }, + { + "bbox": [ + 392, + 419, + 413, + 431 + ], + "score": 0.91, + "content": "L ( h )", + "type": "inline_equation" + }, + { + "bbox": [ + 414, + 418, + 417, + 432 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 18.5 + }, + { + "type": "title", + "bbox": [ + 107, + 444, + 239, + 456 + ], + "lines": [ + { + "bbox": [ + 106, + 444, + 239, + 457 + ], + "spans": [ + { + "bbox": [ + 106, + 444, + 239, + 457 + ], + "score": 1.0, + "content": "A.4 PROOF OF THEOREM 4.2", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 20 + }, + { + "type": "text", + "bbox": [ + 106, + 465, + 504, + 489 + ], + "lines": [ + { + "bbox": [ + 105, + 464, + 505, + 479 + ], + "spans": [ + { + "bbox": [ + 105, + 464, + 246, + 479 + ], + "score": 1.0, + "content": "Recall that by assumption we have", + "type": "text" + }, + { + "bbox": [ + 247, + 466, + 327, + 478 + ], + "score": 0.92, + "content": "j \\in \\{ 1 , \\dots , H + 1 \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 327, + 464, + 366, + 479 + ], + "score": 1.0, + "content": "such that", + "type": "text" + }, + { + "bbox": [ + 366, + 466, + 410, + 478 + ], + "score": 0.93, + "content": "d _ { x } = d _ { j - 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Consider", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 107, + 475, + 239, + 491 + ], + "spans": [ + { + "bbox": [ + 107, + 477, + 150, + 490 + ], + "score": 0.92, + "content": "D _ { h _ { j } } [ L ( h ) ]", + "type": "inline_equation" + }, + { + "bbox": [ + 151, + 475, + 205, + 491 + ], + "score": 1.0, + "content": ", then for any", + "type": "text" + }, + { + "bbox": [ + 205, + 477, + 235, + 489 + ], + "score": 0.94, + "content": "\\eta \\in \\mathcal { F } _ { j }", + "type": "inline_equation" + }, + { + "bbox": [ + 235, + 475, + 239, + 491 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 21.5 + }, + { + "type": "interline_equation", + "bbox": [ + 132, + 495, + 479, + 512 + ], + "lines": [ + { + "bbox": [ + 132, + 495, + 479, + 512 + ], + "spans": [ + { + "bbox": [ + 132, + 495, + 479, + 512 + ], + "score": 0.88, + "content": "\\left. D _ { h _ { j } } [ L ( h ) ] , \\eta \\right. = \\mathbb { E } \\left[ \\left( h _ { H + 1 : 1 } ( X ) - h ^ { * } ( X ) \\right) ^ { T } J [ h _ { H + 1 : j + 1 } ] ( h _ { j : 1 } ( X ) ) \\eta ( h _ { j - 1 : 1 } ( X ) ) \\right] .", + "type": "interline_equation", + "image_path": "58a9de56e886daecacc3ac5aa36e001abefa68f1e6de2ec9d791cdd70784b542.jpg" + } + ] + } + ], + "index": 23, + "virtual_lines": [ + { + "bbox": [ + 132, + 495, + 479, + 512 + ], + "spans": [], + "index": 23 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 518, + 503, + 543 + ], + "lines": [ + { + "bbox": [ + 105, + 517, + 504, + 533 + ], + "spans": [ + { + "bbox": [ + 105, + 517, + 269, + 533 + ], + "score": 1.0, + "content": "As done in the previous theorem, for any", + "type": "text" + }, + { + "bbox": [ + 269, + 518, + 314, + 530 + ], + "score": 0.92, + "content": "w \\in \\mathbb { R } ^ { d _ { j - 1 } }", + "type": "inline_equation" + }, + { + "bbox": [ + 314, + 517, + 330, + 533 + ], + "score": 1.0, + "content": ", let", + "type": "text" + }, + { + "bbox": [ + 330, + 519, + 451, + 532 + ], + "score": 0.92, + "content": "A ( w ) = J [ h _ { H + 1 : j + 1 } ] ( h _ { j } ( w ) )", + "type": "inline_equation" + }, + { + "bbox": [ + 451, + 517, + 480, + 533 + ], + "score": 1.0, + "content": ". 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Then define", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 24.5 + }, + { + "type": "interline_equation", + "bbox": [ + 181, + 549, + 428, + 566 + ], + "lines": [ + { + "bbox": [ + 181, + 549, + 428, + 566 + ], + "spans": [ + { + "bbox": [ + 181, + 549, + 428, + 566 + ], + "score": 0.88, + "content": "\\tilde { \\eta } ( w ) = A ( w ) ^ { T } ( A ( w ) A ( w ) ^ { T } ) ^ { - 1 } ( h _ { H + 1 : 1 } - h ^ { * } ) \\circ h _ { j - 1 : 1 } ^ { - 1 } ( w ) ,", + "type": "interline_equation", + "image_path": "0a9ab33bd78b11469f3c9aa999f5f18368558a29df86352d58853ae390bf602d.jpg" + } + ] + } + ], + "index": 26, + "virtual_lines": [ + { + "bbox": [ + 181, + 549, + 428, + 566 + ], + "spans": [], + "index": 26 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 571, + 135, + 582 + ], + "lines": [ + { + "bbox": [ + 105, + 569, + 136, + 582 + ], + "spans": [ + { + "bbox": [ + 105, + 569, + 136, + 582 + ], + "score": 1.0, + "content": "so that", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 27 + }, + { + "type": "interline_equation", + "bbox": [ + 205, + 580, + 405, + 601 + ], + "lines": [ + { + "bbox": [ + 205, + 580, + 405, + 601 + ], + "spans": [ + { + "bbox": [ + 205, + 580, + 405, + 601 + ], + "score": 0.9, + "content": "\\begin{array} { r } { \\left. D _ { h _ { j } } [ L ( h ) ] , \\tilde { \\eta } \\right. = \\mathbb { E } \\left[ \\left\\| h _ { H + 1 : 1 } ( X ) - h ^ { * } ( X ) \\right\\| _ { 2 } ^ { 2 } \\right] . } \\end{array}", + "type": "interline_equation", + "image_path": "bd7275c241b21e4ffd73140a9f0170127064a663d5d17a44496921fd80d1775c.jpg" + } + ] + } + ], + "index": 28, + "virtual_lines": [ + { + "bbox": [ + 205, + 580, + 405, + 601 + ], + "spans": [], + "index": 28 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 604, + 505, + 653 + ], + "lines": [ + { + "bbox": [ + 105, + 604, + 506, + 619 + ], + "spans": [ + { + "bbox": [ + 105, + 604, + 188, + 619 + ], + "score": 1.0, + "content": "We need to check if", + "type": "text" + }, + { + "bbox": [ + 188, + 605, + 218, + 617 + ], + "score": 0.91, + "content": "\\tilde { \\eta } \\in \\mathcal { F } _ { j }", + "type": "inline_equation" + }, + { + "bbox": [ + 218, + 604, + 319, + 619 + ], + "score": 1.0, + "content": ". 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1 } \\big \\| _ { \\mathrm { o p } } = \\sigma _ { \\operatorname* { m a x } } ( { A ( X ) } ^ { T } ( A ( X ) { A ( X ) } ^ { T } ) ^ { - 1 } ) \\leq 1 / \\epsilon , } \\end{array}", + "type": "interline_equation", + "image_path": "8a569f45b23113bb8adcec96d75e14fab0f364d26764c0991bb5695e142428ba.jpg" + } + ] + } + ], + "index": 5, + "virtual_lines": [ + { + "bbox": [ + 155, + 166, + 456, + 185 + ], + "spans": [], + "index": 5 + } + ] + }, + { + "type": "text", + "bbox": [ + 105, + 192, + 300, + 204 + ], + "lines": [ + { + "bbox": [ + 105, + 191, + 300, + 205 + ], + "spans": [ + { + "bbox": [ + 105, + 191, + 258, + 205 + ], + "score": 1.0, + "content": "by our assumption. Note that, for any", + "type": "text" + }, + { + "bbox": [ + 259, + 191, + 296, + 203 + ], + "score": 0.91, + "content": "X \\in \\mathbb { R } ^ { d _ { x } }", + "type": "inline_equation" + }, + { + "bbox": [ + 297, + 191, + 300, + 205 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 6, + "bbox_fs": [ + 105, + 191, + 300, + 205 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 166, + 209, + 444, + 264 + ], + "lines": [ + { + "bbox": [ + 166, + 209, + 444, + 264 + ], + "spans": [ + { + "bbox": [ + 166, + 209, + 444, + 264 + ], + "score": 0.93, + "content": "\\begin{array} { r l } & { \\| \\tilde { \\eta } ( X ) \\| _ { 2 } = \\left\\| A ( X ) ^ { T } ( A ( X ) A ( X ) ^ { T } ) ^ { - 1 } ( h _ { H + 1 : 1 } ( X ) - h ^ { * } ( X ) ) \\right\\| _ { 2 } } \\\\ & { \\qquad \\leq \\left\\| A ( X ) ^ { T } ( A ( X ) A ( X ) ^ { T } ) ^ { - 1 } \\right\\| _ { \\mathrm { o p } } \\left\\| h _ { H + 1 : 1 } ( X ) - h ^ { * } ( X ) \\right\\| _ { 2 } } \\\\ & { \\qquad \\leq \\left\\| A ( X ) ^ { T } ( A ( X ) A ( X ) ^ { T } ) ^ { - 1 } \\right\\| _ { \\mathrm { o p } } \\left\\| h _ { H + 1 : 1 } - h ^ { * } \\right\\| _ { \\mathrm { n l } } \\left\\| X \\right\\| _ { 2 } . } \\end{array}", + "type": "interline_equation", + "image_path": "389d29834d4c5b65f2521b16ee654104ebd142a44a77da22a7ae6bf4d8c09502.jpg" + } + ] + } + ], + "index": 8, + "virtual_lines": [ + { + "bbox": [ + 166, + 209, + 444, + 227.33333333333334 + ], + "spans": [], + "index": 7 + }, + { + "bbox": [ + 166, + 227.33333333333334, + 444, + 245.66666666666669 + ], + "spans": [], + "index": 8 + }, + { + "bbox": [ + 166, + 245.66666666666669, + 444, + 264.0 + ], + "spans": [], + "index": 9 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 268, + 252, + 280 + ], + "lines": [ + { + "bbox": [ + 106, + 266, + 252, + 281 + ], + "spans": [ + { + "bbox": [ + 106, + 266, + 203, + 281 + ], + "score": 1.0, + "content": "Since this holds for any", + "type": "text" + }, + { + "bbox": [ + 203, + 269, + 213, + 278 + ], + "score": 0.84, + "content": "X", + "type": "inline_equation" + }, + { + "bbox": [ + 213, + 266, + 252, + 281 + ], + "score": 1.0, + "content": ", we have", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 10, + "bbox_fs": [ + 106, + 266, + 252, + 281 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 145, + 287, + 465, + 311 + ], + "lines": [ + { + "bbox": [ + 145, + 287, + 465, + 311 + ], + "spans": [ + { + "bbox": [ + 145, + 287, + 465, + 311 + ], + "score": 0.9, + "content": "\\begin{array} { r } { \\| \\tilde { \\eta } \\| _ { \\mathrm { n l } } \\leq \\| A ( X ) ^ { T } ( A ( X ) A ( X ) ^ { T } ) ^ { - 1 } \\| _ { \\mathrm { o p } } \\| h _ { H + 1 : 1 } - h ^ { * } \\| _ { \\mathrm { n l } } \\leq \\frac { \\| h _ { H + 1 : 1 } - h ^ { * } \\| _ { \\mathrm { n l } } } { \\epsilon } , } \\end{array}", + "type": "interline_equation", + "image_path": "7173be7c6ac9ffa981851fa3df4c4baa2a6d42b69b26d50ade5e4105b8fbbbdc.jpg" + } + ] + } + ], + "index": 11, + "virtual_lines": [ + { + "bbox": [ + 145, + 287, + 465, + 311 + ], + "spans": [], + "index": 11 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 316, + 250, + 328 + ], + "lines": [ + { + "bbox": [ + 106, + 315, + 250, + 330 + ], + "spans": [ + { + "bbox": [ + 106, + 315, + 183, + 330 + ], + "score": 1.0, + "content": "which ensures that", + "type": "text" + }, + { + "bbox": [ + 183, + 317, + 213, + 328 + ], + "score": 0.92, + "content": "\\tilde { \\eta } \\in \\mathcal { F } _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 213, + 315, + 250, + 330 + ], + "score": 1.0, + "content": ". Finally,", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 12, + "bbox_fs": [ + 106, + 315, + 250, + 330 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 122, + 334, + 489, + 370 + ], + "lines": [ + { + "bbox": [ + 122, + 334, + 489, + 370 + ], + "spans": [ + { + "bbox": [ + 122, + 334, + 489, + 370 + ], + "score": 0.93, + "content": "\\| D _ { h _ { 1 } } [ L ( h ) ] \\| _ { \\mathrm { o p } } \\ge \\frac { \\langle D _ { h _ { 1 } } [ L ( h ) ] , \\tilde { \\eta } \\rangle } { \\| \\tilde { \\eta } \\| _ { \\mathrm { n l } } } \\ge \\frac { \\epsilon \\mathbb { E } \\left[ \\| h _ { H + 1 : 1 } ( X ) - h ^ { * } ( X ) \\| _ { 2 } ^ { 2 } \\right] } { \\| h _ { H + 1 : 1 } - h ^ { * } \\| _ { \\mathrm { n l } } } = \\frac { \\epsilon ( L ( h ) - L ^ { * } ) } { \\| h _ { H + 1 : 1 } - h ^ { * } \\| _ { \\mathrm { n l } } } ,", + "type": "interline_equation", + "image_path": "10be6a4159ee74d4731aa92da85915fdb7dbf502660bc83fca35e1f2590f4183.jpg" + } + ] + } + ], + "index": 14, + "virtual_lines": [ + { + "bbox": [ + 122, + 334, + 489, + 346.0 + ], + "spans": [], + "index": 13 + }, + { + "bbox": [ + 122, + 346.0, + 489, + 358.0 + ], + "spans": [], + "index": 14 + }, + { + "bbox": [ + 122, + 358.0, + 489, + 370.0 + ], + "spans": [], + "index": 15 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 375, + 159, + 387 + ], + "lines": [ + { + "bbox": [ + 106, + 374, + 160, + 388 + ], + "spans": [ + { + "bbox": [ + 106, + 374, + 160, + 388 + ], + "score": 1.0, + "content": "which yields", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 16, + "bbox_fs": [ + 106, + 374, + 160, + 388 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 201, + 386, + 408, + 402 + ], + "lines": [ + { + "bbox": [ + 201, + 386, + 408, + 402 + ], + "spans": [ + { + "bbox": [ + 201, + 386, + 408, + 402 + ], + "score": 0.88, + "content": "\\begin{array} { r } { \\| D _ { h _ { 1 } } [ L ( h ) ] \\| _ { \\mathrm { o p } } \\| h _ { H + 1 : 1 } - h ^ { * } \\| _ { \\mathrm { n l } } \\ge \\epsilon ( L ( h ) - L ^ { * } ) . } \\end{array}", + "type": "interline_equation", + "image_path": "1a6caa84edd7d98070f8b2297f43819df8f34c7df5ead68d1360f6074f35f20f.jpg" + } + ] + } + ], + "index": 17, + "virtual_lines": [ + { + "bbox": [ + 201, + 386, + 408, + 402 + ], + "spans": [], + "index": 17 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 405, + 505, + 431 + ], + "lines": [ + { + "bbox": [ + 104, + 403, + 506, + 421 + ], + "spans": [ + { + "bbox": [ + 104, + 403, + 338, + 421 + ], + "score": 1.0, + "content": "From this we can see that if we have a critical point of", + "type": "text" + }, + { + "bbox": [ + 339, + 406, + 360, + 418 + ], + "score": 0.92, + "content": "L ( h )", + "type": "inline_equation" + }, + { + "bbox": [ + 360, + 403, + 385, + 421 + ], + "score": 1.0, + "content": ", then", + "type": "text" + }, + { + "bbox": [ + 386, + 405, + 471, + 420 + ], + "score": 0.92, + "content": "\\| D _ { h _ { 1 } } [ L ( h ) ] \\| _ { \\mathrm { o p } } = 0", + "type": "inline_equation" + }, + { + "bbox": [ + 471, + 403, + 506, + 421 + ], + "score": 1.0, + "content": "implies", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 107, + 418, + 417, + 432 + ], + "spans": [ + { + "bbox": [ + 107, + 418, + 152, + 431 + ], + "score": 0.93, + "content": "L ( h ) = L ^ { * }", + "type": "inline_equation" + }, + { + "bbox": [ + 153, + 418, + 392, + 432 + ], + "score": 1.0, + "content": ", which means that the critical point is a global minimum of", + "type": "text" + }, + { + "bbox": [ + 392, + 419, + 413, + 431 + ], + "score": 0.91, + "content": "L ( h )", + "type": "inline_equation" + }, + { + "bbox": [ + 414, + 418, + 417, + 432 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 18.5, + "bbox_fs": [ + 104, + 403, + 506, + 432 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 444, + 239, + 456 + ], + "lines": [ + { + "bbox": [ + 106, + 444, + 239, + 457 + ], + "spans": [ + { + "bbox": [ + 106, + 444, + 239, + 457 + ], + "score": 1.0, + "content": "A.4 PROOF OF THEOREM 4.2", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 20 + }, + { + "type": "text", + "bbox": [ + 106, + 465, + 504, + 489 + ], + "lines": [ + { + "bbox": [ + 105, + 464, + 505, + 479 + ], + "spans": [ + { + "bbox": [ + 105, + 464, + 246, + 479 + ], + "score": 1.0, + "content": "Recall that by assumption we have", + "type": "text" + }, + { + "bbox": [ + 247, + 466, + 327, + 478 + ], + "score": 0.92, + "content": "j \\in \\{ 1 , \\dots , H + 1 \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 327, + 464, + 366, + 479 + ], + "score": 1.0, + "content": "such that", + "type": "text" + }, + { + "bbox": [ + 366, + 466, + 410, + 478 + ], + "score": 0.93, + "content": "d _ { x } = d _ { j - 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Consider", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 107, + 475, + 239, + 491 + ], + "spans": [ + { + "bbox": [ + 107, + 477, + 150, + 490 + ], + "score": 0.92, + "content": "D _ { h _ { j } } [ L ( h ) ]", + "type": "inline_equation" + }, + { + "bbox": [ + 151, + 475, + 205, + 491 + ], + "score": 1.0, + "content": ", then for any", + "type": "text" + }, + { + "bbox": [ + 205, + 477, + 235, + 489 + ], + "score": 0.94, + "content": "\\eta \\in \\mathcal { F } _ { j }", + "type": "inline_equation" + }, + { + "bbox": [ + 235, + 475, + 239, + 491 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 21.5, + "bbox_fs": [ + 105, + 464, + 505, + 491 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 132, + 495, + 479, + 512 + ], + "lines": [ + { + "bbox": [ + 132, + 495, + 479, + 512 + ], + "spans": [ + { + "bbox": [ + 132, + 495, + 479, + 512 + ], + "score": 0.88, + "content": "\\left. D _ { h _ { j } } [ L ( h ) ] , \\eta \\right. = \\mathbb { E } \\left[ \\left( h _ { H + 1 : 1 } ( X ) - h ^ { * } ( X ) \\right) ^ { T } J [ h _ { H + 1 : j + 1 } ] ( h _ { j : 1 } ( X ) ) \\eta ( h _ { j - 1 : 1 } ( X ) ) \\right] .", + "type": "interline_equation", + "image_path": "58a9de56e886daecacc3ac5aa36e001abefa68f1e6de2ec9d791cdd70784b542.jpg" + } + ] + } + ], + "index": 23, + "virtual_lines": [ + { + "bbox": [ + 132, + 495, + 479, + 512 + ], + "spans": [], + "index": 23 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 518, + 503, + 543 + ], + "lines": [ + { + "bbox": [ + 105, + 517, + 504, + 533 + ], + "spans": [ + { + "bbox": [ + 105, + 517, + 269, + 533 + ], + "score": 1.0, + "content": "As done in the previous theorem, for any", + "type": "text" + }, + { + "bbox": [ + 269, + 518, + 314, + 530 + ], + "score": 0.92, + "content": "w \\in \\mathbb { R } ^ { d _ { j - 1 } }", + "type": "inline_equation" + }, + { + "bbox": [ + 314, + 517, + 330, + 533 + ], + "score": 1.0, + "content": ", let", + "type": "text" + }, + { + "bbox": [ + 330, + 519, + 451, + 532 + ], + "score": 0.92, + "content": "A ( w ) = J [ h _ { H + 1 : j + 1 } ] ( h _ { j } ( w ) )", + "type": "inline_equation" + }, + { + "bbox": [ + 451, + 517, + 480, + 533 + ], + "score": 1.0, + "content": ". Since", + "type": "text" + }, + { + "bbox": [ + 480, + 519, + 504, + 531 + ], + "score": 0.91, + "content": "A ( w )", + "type": "inline_equation" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 530, + 397, + 545 + ], + "spans": [ + { + "bbox": [ + 105, + 530, + 239, + 545 + ], + "score": 1.0, + "content": "has full row rank by assumption,", + "type": "text" + }, + { + "bbox": [ + 240, + 531, + 292, + 543 + ], + "score": 0.94, + "content": "A ( w ) A ( w ) ^ { T }", + "type": "inline_equation" + }, + { + "bbox": [ + 292, + 530, + 397, + 545 + ], + "score": 1.0, + "content": "is invertible. Then define", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 24.5, + "bbox_fs": [ + 105, + 517, + 504, + 545 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 181, + 549, + 428, + 566 + ], + "lines": [ + { + "bbox": [ + 181, + 549, + 428, + 566 + ], + "spans": [ + { + "bbox": [ + 181, + 549, + 428, + 566 + ], + "score": 0.88, + "content": "\\tilde { \\eta } ( w ) = A ( w ) ^ { T } ( A ( w ) A ( w ) ^ { T } ) ^ { - 1 } ( h _ { H + 1 : 1 } - h ^ { * } ) \\circ h _ { j - 1 : 1 } ^ { - 1 } ( w ) ,", + "type": "interline_equation", + "image_path": "0a9ab33bd78b11469f3c9aa999f5f18368558a29df86352d58853ae390bf602d.jpg" + } + ] + } + ], + "index": 26, + "virtual_lines": [ + { + "bbox": [ + 181, + 549, + 428, + 566 + ], + "spans": [], + "index": 26 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 571, + 135, + 582 + ], + "lines": [ + { + "bbox": [ + 105, + 569, + 136, + 582 + ], + "spans": [ + { + "bbox": [ + 105, + 569, + 136, + 582 + ], + "score": 1.0, + "content": "so that", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 27, + "bbox_fs": [ + 105, + 569, + 136, + 582 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 205, + 580, + 405, + 601 + ], + "lines": [ + { + "bbox": [ + 205, + 580, + 405, + 601 + ], + "spans": [ + { + "bbox": [ + 205, + 580, + 405, + 601 + ], + "score": 0.9, + "content": "\\begin{array} { r } { \\left. D _ { h _ { j } } [ L ( h ) ] , \\tilde { \\eta } \\right. = \\mathbb { E } \\left[ \\left\\| h _ { H + 1 : 1 } ( X ) - h ^ { * } ( X ) \\right\\| _ { 2 } ^ { 2 } \\right] . } \\end{array}", + "type": "interline_equation", + "image_path": "bd7275c241b21e4ffd73140a9f0170127064a663d5d17a44496921fd80d1775c.jpg" + } + ] + } + ], + "index": 28, + "virtual_lines": [ + { + "bbox": [ + 205, + 580, + 405, + 601 + ], + "spans": [], + "index": 28 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 604, + 505, + 653 + ], + "lines": [ + { + "bbox": [ + 105, + 604, + 506, + 619 + ], + "spans": [ + { + "bbox": [ + 105, + 604, + 188, + 619 + ], + "score": 1.0, + "content": "We need to check if", + "type": "text" + }, + { + "bbox": [ + 188, + 605, + 218, + 617 + ], + "score": 0.91, + "content": "\\tilde { \\eta } \\in \\mathcal { F } _ { j }", + "type": "inline_equation" + }, + { + "bbox": [ + 218, + 604, + 319, + 619 + ], + "score": 1.0, + "content": ". 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Set", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 107, + 475, + 506, + 490 + ], + "spans": [ + { + "bbox": [ + 107, + 477, + 214, + 489 + ], + "score": 0.9, + "content": "h _ { j } ( x ) = ( h ^ { * } ( x ) , 0 , \\ldots , 0 )", + "type": "inline_equation" + }, + { + "bbox": [ + 214, + 475, + 279, + 490 + ], + "score": 1.0, + "content": ", where the first", + "type": "text" + }, + { + "bbox": [ + 280, + 478, + 291, + 489 + ], + "score": 0.88, + "content": "d _ { y }", + "type": "inline_equation" + }, + { + "bbox": [ + 291, + 475, + 358, + 490 + ], + "score": 1.0, + "content": "components are", + "type": "text" + }, + { + "bbox": [ + 359, + 477, + 383, + 488 + ], + "score": 0.89, + "content": "h ^ { \\ast } ( x )", + "type": "inline_equation" + }, + { + "bbox": [ + 384, + 475, + 506, + 490 + ], + "score": 1.0, + "content": "and the rest are zero. All the", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 488, + 505, + 500 + ], + "spans": [ + { + "bbox": [ + 105, + 488, + 134, + 500 + ], + "score": 1.0, + "content": "rest of", + "type": "text" + }, + { + "bbox": [ + 134, + 489, + 144, + 498 + ], + "score": 0.87, + "content": "h _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 145, + 488, + 351, + 500 + ], + "score": 1.0, + "content": "are set as in (7). 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Suppose we are given a data matrix", + "type": "text" + }, + { + "bbox": [ + 309, + 551, + 361, + 563 + ], + "score": 0.92, + "content": "X \\in \\mathbb { R } ^ { d _ { x } \\times m }", + "type": "inline_equation" + }, + { + "bbox": [ + 362, + 550, + 450, + 565 + ], + "score": 1.0, + "content": "and an output matrix", + "type": "text" + }, + { + "bbox": [ + 450, + 551, + 501, + 563 + ], + "score": 0.91, + "content": "Y \\in \\mathbb { R } ^ { d _ { y } \\times m }", + "type": "inline_equation" + }, + { + "bbox": [ + 502, + 550, + 504, + 565 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 562, + 505, + 576 + ], + "spans": [ + { + "bbox": [ + 105, + 562, + 134, + 576 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 134, + 564, + 173, + 576 + ], + "score": 0.91, + "content": "d _ { x } \\ < \\ d _ { y }", + "type": "inline_equation" + }, + { + "bbox": [ + 173, + 562, + 216, + 576 + ], + "score": 1.0, + "content": ". Assume", + "type": "text" + }, + { + "bbox": [ + 216, + 563, + 241, + 574 + ], + "score": 0.9, + "content": "X \\breve { X } ^ { T }", + "type": "inline_equation" + }, + { + "bbox": [ + 241, + 562, + 262, + 576 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 262, + 563, + 287, + 574 + ], + "score": 0.89, + "content": "Y X ^ { T }", + "type": "inline_equation" + }, + { + "bbox": [ + 287, + 562, + 505, + 576 + ], + "score": 1.0, + "content": "have full ranks. Consider minimizing the empirical", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 574, + 185, + 586 + ], + "spans": [ + { + "bbox": [ + 105, + 574, + 185, + 586 + ], + "score": 1.0, + "content": "squared error risk:", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 27 + }, + { + "type": "interline_equation", + "bbox": [ + 192, + 592, + 418, + 615 + ], + "lines": [ + { + "bbox": [ + 192, + 592, + 418, + 615 + ], + "spans": [ + { + "bbox": [ + 192, + 592, + 418, + 615 + ], + "score": 0.91, + "content": "L ( W _ { 1 } , \\dots , W _ { H + 1 } ) : = \\frac { 1 } { 2 } \\left\\| W _ { H + 1 } W _ { H } \\cdot \\cdot \\cdot W _ { 1 } X - Y \\right\\| _ { \\mathrm { F } } ^ { 2 } ,", + "type": "interline_equation", + "image_path": "0b642e49ec2994f3f6a01d80e0bb6a617ee0bf39bf31b45b11fe01d3425f4813.jpg" + } + ] + } + ], + "index": 29, + "virtual_lines": [ + { + "bbox": [ + 192, + 592, + 418, + 615 + ], + "spans": [], + "index": 29 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 621, + 505, + 657 + ], + "lines": [ + { + "bbox": [ + 105, + 620, + 506, + 635 + ], + "spans": [ + { + "bbox": [ + 105, + 620, + 133, + 635 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 134, + 622, + 202, + 633 + ], + "score": 0.87, + "content": "W _ { k } \\in \\mathbb { R } ^ { d _ { k } \\times d _ { k - 1 } }", + "type": "inline_equation" + }, + { + "bbox": [ + 203, + 620, + 207, + 635 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 208, + 623, + 283, + 634 + ], + "score": 0.87, + "content": "k = 1 , \\ldots , H + 1", + "type": "inline_equation" + }, + { + "bbox": [ + 283, + 620, + 506, + 635 + ], + "score": 1.0, + "content": "are weight matrices of the linear neural network, and", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 107, + 633, + 505, + 646 + ], + "spans": [ + { + "bbox": [ + 107, + 634, + 141, + 645 + ], + "score": 0.91, + "content": "d _ { 0 } = d _ { x }", + "type": "inline_equation" + }, + { + "bbox": [ + 142, + 633, + 161, + 646 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 161, + 634, + 208, + 645 + ], + "score": 0.91, + "content": "d _ { H + 1 } = d _ { y }", + "type": "inline_equation" + }, + { + "bbox": [ + 209, + 633, + 349, + 646 + ], + "score": 1.0, + "content": "for simplicity in notation. Also let", + "type": "text" + }, + { + "bbox": [ + 349, + 633, + 361, + 644 + ], + "score": 0.67, + "content": "W", + "type": "inline_equation" + }, + { + "bbox": [ + 361, + 633, + 429, + 646 + ], + "score": 1.0, + "content": "denote the tuple", + "type": "text" + }, + { + "bbox": [ + 430, + 633, + 501, + 645 + ], + "score": 0.89, + "content": "( W _ { 1 } , \\dots , W _ { H + 1 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 501, + 633, + 505, + 646 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 644, + 423, + 657 + ], + "spans": [ + { + "bbox": [ + 106, + 644, + 213, + 657 + ], + "score": 1.0, + "content": "Then, any critical point of", + "type": "text" + }, + { + "bbox": [ + 213, + 645, + 239, + 657 + ], + "score": 0.91, + "content": "L ( W )", + "type": "inline_equation" + }, + { + "bbox": [ + 239, + 644, + 423, + 657 + ], + "score": 1.0, + "content": "that is not a local minimum is a saddle point.", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 31 + }, + { + "type": "text", + "bbox": [ + 106, + 670, + 505, + 704 + ], + "lines": [ + { + "bbox": [ + 106, + 671, + 504, + 683 + ], + "spans": [ + { + "bbox": [ + 106, + 671, + 352, + 683 + ], + "score": 1.0, + "content": "Proof. For this lemma, we separate the proof into two cases:", + "type": "text" + }, + { + "bbox": [ + 353, + 671, + 417, + 682 + ], + "score": 0.92, + "content": "W _ { H } \\cdot \\cdot \\cdot W _ { 1 } \\neq 0", + "type": "inline_equation" + }, + { + "bbox": [ + 418, + 671, + 436, + 683 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 436, + 671, + 501, + 682 + ], + "score": 0.9, + "content": "W _ { H } \\cdot \\cdot \\cdot W _ { 1 } = 0", + "type": "inline_equation" + }, + { + "bbox": [ + 501, + 671, + 504, + 683 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 682, + 505, + 694 + ], + "spans": [ + { + "bbox": [ + 106, + 682, + 505, + 694 + ], + "score": 1.0, + "content": "The crux of the proof is to show that any critical point cannot be a local maximum. 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Moreover, for all", + "type": "text" + }, + { + "bbox": [ + 235, + 437, + 272, + 448 + ], + "score": 0.6, + "content": "z ~ \\in ~ \\mathbb { R } ^ { d _ { 1 } }", + "type": "inline_equation" + }, + { + "bbox": [ + 272, + 435, + 278, + 451 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 278, + 438, + 333, + 450 + ], + "score": 0.7, + "content": "J [ h _ { H + 1 : 2 } ] ( z )", + "type": "inline_equation" + }, + { + "bbox": [ + 333, + 435, + 506, + 451 + ], + "score": 1.0, + "content": "is all 0 except 1’s in diagonal entries, so", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 448, + 363, + 462 + ], + "spans": [ + { + "bbox": [ + 106, + 448, + 208, + 461 + ], + "score": 0.91, + "content": "\\sigma _ { \\operatorname* { m i n } } ( J [ h _ { H + 1 : 2 } ] ( z ) ) \\geq 1", + "type": "inline_equation" + }, + { + "bbox": [ + 208, + 448, + 226, + 462 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 226, + 449, + 269, + 461 + ], + "score": 0.92, + "content": "h _ { H + 1 : 2 } ( z )", + "type": "inline_equation" + }, + { + "bbox": [ + 269, + 448, + 363, + 462 + ], + "score": 1.0, + "content": "is twice-differentiable.", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 19, + "bbox_fs": [ + 104, + 423, + 506, + 462 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 465, + 505, + 510 + ], + "lines": [ + { + "bbox": [ + 105, + 464, + 506, + 479 + ], + "spans": [ + { + "bbox": [ + 105, + 464, + 263, + 479 + ], + "score": 1.0, + "content": "(Theorem 4.2) It is given that we have", + "type": "text" + }, + { + "bbox": [ + 264, + 465, + 345, + 478 + ], + "score": 0.92, + "content": "j \\in \\{ 1 , \\dots , H + 1 \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 346, + 464, + 386, + 479 + ], + "score": 1.0, + "content": "such that", + "type": "text" + }, + { + "bbox": [ + 386, + 466, + 431, + 478 + ], + "score": 0.93, + "content": "d _ { x } = d _ { j - 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 431, + 464, + 450, + 479 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 450, + 466, + 484, + 478 + ], + "score": 0.92, + "content": "d _ { y } \\leq d _ { j }", + "type": "inline_equation" + }, + { + "bbox": [ + 485, + 464, + 506, + 479 + ], + "score": 1.0, + "content": ". Set", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 107, + 475, + 506, + 490 + ], + "spans": [ + { + "bbox": [ + 107, + 477, + 214, + 489 + ], + "score": 0.9, + "content": "h _ { j } ( x ) = ( h ^ { * } ( x ) , 0 , \\ldots , 0 )", + "type": "inline_equation" + }, + { + "bbox": [ + 214, + 475, + 279, + 490 + ], + "score": 1.0, + "content": ", where the first", + "type": "text" + }, + { + "bbox": [ + 280, + 478, + 291, + 489 + ], + "score": 0.88, + "content": "d _ { y }", + "type": "inline_equation" + }, + { + "bbox": [ + 291, + 475, + 358, + 490 + ], + "score": 1.0, + "content": "components are", + "type": "text" + }, + { + "bbox": [ + 359, + 477, + 383, + 488 + ], + "score": 0.89, + "content": "h ^ { \\ast } ( x )", + "type": "inline_equation" + }, + { + "bbox": [ + 384, + 475, + 506, + 490 + ], + "score": 1.0, + "content": "and the rest are zero. All the", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 488, + 505, + 500 + ], + "spans": [ + { + "bbox": [ + 105, + 488, + 134, + 500 + ], + "score": 1.0, + "content": "rest of", + "type": "text" + }, + { + "bbox": [ + 134, + 489, + 144, + 498 + ], + "score": 0.87, + "content": "h _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 145, + 488, + 351, + 500 + ], + "score": 1.0, + "content": "are set as in (7). Then, it can be easily checked that", + "type": "text" + }, + { + "bbox": [ + 351, + 488, + 383, + 498 + ], + "score": 0.91, + "content": "h _ { i } \\in { \\mathcal { F } } _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 384, + 488, + 411, + 500 + ], + "score": 1.0, + "content": "for all", + "type": "text" + }, + { + "bbox": [ + 411, + 488, + 416, + 497 + ], + "score": 0.77, + "content": "i", + "type": "inline_equation" + }, + { + "bbox": [ + 416, + 488, + 505, + 500 + ], + "score": 1.0, + "content": "and all the conditions", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 499, + 219, + 510 + ], + "spans": [ + { + "bbox": [ + 106, + 499, + 219, + 510 + ], + "score": 1.0, + "content": "of the theorem are satisfied.", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 22.5, + "bbox_fs": [ + 105, + 464, + 506, + 510 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 527, + 227, + 540 + ], + "lines": [ + { + "bbox": [ + 105, + 525, + 228, + 541 + ], + "spans": [ + { + "bbox": [ + 105, + 525, + 228, + 541 + ], + "score": 1.0, + "content": "B DEFERRED LEMMA", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 25 + }, + { + "type": "text", + "bbox": [ + 106, + 551, + 505, + 586 + ], + "lines": [ + { + "bbox": [ + 104, + 550, + 504, + 565 + ], + "spans": [ + { + "bbox": [ + 104, + 550, + 309, + 565 + ], + "score": 1.0, + "content": "Lemma B.1. Suppose we are given a data matrix", + "type": "text" + }, + { + "bbox": [ + 309, + 551, + 361, + 563 + ], + "score": 0.92, + "content": "X \\in \\mathbb { R } ^ { d _ { x } \\times m }", + "type": "inline_equation" + }, + { + "bbox": [ + 362, + 550, + 450, + 565 + ], + "score": 1.0, + "content": "and an output matrix", + "type": "text" + }, + { + "bbox": [ + 450, + 551, + 501, + 563 + ], + "score": 0.91, + "content": "Y \\in \\mathbb { R } ^ { d _ { y } \\times m }", + "type": "inline_equation" + }, + { + "bbox": [ + 502, + 550, + 504, + 565 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 562, + 505, + 576 + ], + "spans": [ + { + "bbox": [ + 105, + 562, + 134, + 576 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 134, + 564, + 173, + 576 + ], + "score": 0.91, + "content": "d _ { x } \\ < \\ d _ { y }", + "type": "inline_equation" + }, + { + "bbox": [ + 173, + 562, + 216, + 576 + ], + "score": 1.0, + "content": ". Assume", + "type": "text" + }, + { + "bbox": [ + 216, + 563, + 241, + 574 + ], + "score": 0.9, + "content": "X \\breve { X } ^ { T }", + "type": "inline_equation" + }, + { + "bbox": [ + 241, + 562, + 262, + 576 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 262, + 563, + 287, + 574 + ], + "score": 0.89, + "content": "Y X ^ { T }", + "type": "inline_equation" + }, + { + "bbox": [ + 287, + 562, + 505, + 576 + ], + "score": 1.0, + "content": "have full ranks. Consider minimizing the empirical", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 574, + 185, + 586 + ], + "spans": [ + { + "bbox": [ + 105, + 574, + 185, + 586 + ], + "score": 1.0, + "content": "squared error risk:", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 27, + "bbox_fs": [ + 104, + 550, + 505, + 586 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 192, + 592, + 418, + 615 + ], + "lines": [ + { + "bbox": [ + 192, + 592, + 418, + 615 + ], + "spans": [ + { + "bbox": [ + 192, + 592, + 418, + 615 + ], + "score": 0.91, + "content": "L ( W _ { 1 } , \\dots , W _ { H + 1 } ) : = \\frac { 1 } { 2 } \\left\\| W _ { H + 1 } W _ { H } \\cdot \\cdot \\cdot W _ { 1 } X - Y \\right\\| _ { \\mathrm { F } } ^ { 2 } ,", + "type": "interline_equation", + "image_path": "0b642e49ec2994f3f6a01d80e0bb6a617ee0bf39bf31b45b11fe01d3425f4813.jpg" + } + ] + } + ], + "index": 29, + "virtual_lines": [ + { + "bbox": [ + 192, + 592, + 418, + 615 + ], + "spans": [], + "index": 29 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 621, + 505, + 657 + ], + "lines": [ + { + "bbox": [ + 105, + 620, + 506, + 635 + ], + "spans": [ + { + "bbox": [ + 105, + 620, + 133, + 635 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 134, + 622, + 202, + 633 + ], + "score": 0.87, + "content": "W _ { k } \\in \\mathbb { R } ^ { d _ { k } \\times d _ { k - 1 } }", + "type": "inline_equation" + }, + { + "bbox": [ + 203, + 620, + 207, + 635 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 208, + 623, + 283, + 634 + ], + "score": 0.87, + "content": "k = 1 , \\ldots , H + 1", + "type": "inline_equation" + }, + { + "bbox": [ + 283, + 620, + 506, + 635 + ], + "score": 1.0, + "content": "are weight matrices of the linear neural network, and", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 107, + 633, + 505, + 646 + ], + "spans": [ + { + "bbox": [ + 107, + 634, + 141, + 645 + ], + "score": 0.91, + "content": "d _ { 0 } = d _ { x }", + "type": "inline_equation" + }, + { + "bbox": [ + 142, + 633, + 161, + 646 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 161, + 634, + 208, + 645 + ], + "score": 0.91, + "content": "d _ { H + 1 } = d _ { y }", + "type": "inline_equation" + }, + { + "bbox": [ + 209, + 633, + 349, + 646 + ], + "score": 1.0, + "content": "for simplicity in notation. Also let", + "type": "text" + }, + { + "bbox": [ + 349, + 633, + 361, + 644 + ], + "score": 0.67, + "content": "W", + "type": "inline_equation" + }, + { + "bbox": [ + 361, + 633, + 429, + 646 + ], + "score": 1.0, + "content": "denote the tuple", + "type": "text" + }, + { + "bbox": [ + 430, + 633, + 501, + 645 + ], + "score": 0.89, + "content": "( W _ { 1 } , \\dots , W _ { H + 1 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 501, + 633, + 505, + 646 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 644, + 423, + 657 + ], + "spans": [ + { + "bbox": [ + 106, + 644, + 213, + 657 + ], + "score": 1.0, + "content": "Then, any critical point of", + "type": "text" + }, + { + "bbox": [ + 213, + 645, + 239, + 657 + ], + "score": 0.91, + "content": "L ( W )", + "type": "inline_equation" + }, + { + "bbox": [ + 239, + 644, + 423, + 657 + ], + "score": 1.0, + "content": "that is not a local minimum is a saddle point.", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 31, + "bbox_fs": [ + 105, + 620, + 506, + 657 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 670, + 505, + 704 + ], + "lines": [ + { + "bbox": [ + 106, + 671, + 504, + 683 + ], + "spans": [ + { + "bbox": [ + 106, + 671, + 352, + 683 + ], + "score": 1.0, + "content": "Proof. For this lemma, we separate the proof into two cases:", + "type": "text" + }, + { + "bbox": [ + 353, + 671, + 417, + 682 + ], + "score": 0.92, + "content": "W _ { H } \\cdot \\cdot \\cdot W _ { 1 } \\neq 0", + "type": "inline_equation" + }, + { + "bbox": [ + 418, + 671, + 436, + 683 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 436, + 671, + 501, + 682 + ], + "score": 0.9, + "content": "W _ { H } \\cdot \\cdot \\cdot W _ { 1 } = 0", + "type": "inline_equation" + }, + { + "bbox": [ + 501, + 671, + 504, + 683 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 682, + 505, + 694 + ], + "spans": [ + { + "bbox": [ + 106, + 682, + 505, + 694 + ], + "score": 1.0, + "content": "The crux of the proof is to show that any critical point cannot be a local maximum. Then, any critical", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 693, + 469, + 705 + ], + "spans": [ + { + "bbox": [ + 105, + 693, + 469, + 705 + ], + "score": 1.0, + "content": "point is either a local minimum or a saddle point, so the conclusion of this lemma follows.", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 34, + "bbox_fs": [ + 105, + 671, + 505, + 705 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 709, + 505, + 733 + ], + "lines": [ + { + "bbox": [ + 106, + 709, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 106, + 709, + 152, + 722 + ], + "score": 1.0, + "content": "In case of", + "type": "text" + }, + { + "bbox": [ + 152, + 709, + 222, + 721 + ], + "score": 0.91, + "content": "W _ { H } \\cdot \\cdot \\cdot W _ { 1 } \\neq 0", + "type": "inline_equation" + }, + { + "bbox": [ + 222, + 709, + 505, + 722 + ], + "score": 1.0, + "content": ", we use some of the results in Kawaguchi (2016) and examine the", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 718, + 506, + 735 + ], + "spans": [ + { + "bbox": [ + 105, + 718, + 153, + 735 + ], + "score": 1.0, + "content": "Hessian of", + "type": "text" + }, + { + "bbox": [ + 154, + 721, + 180, + 732 + ], + "score": 0.92, + "content": "L ( W )", + "type": "inline_equation" + }, + { + "bbox": [ + 180, + 718, + 246, + 735 + ], + "score": 1.0, + "content": "with respect to", + "type": "text" + }, + { + "bbox": [ + 247, + 720, + 296, + 734 + ], + "score": 0.89, + "content": "\\mathrm { v e c } ( W _ { H + 1 } ^ { T } )", + "type": "inline_equation" + }, + { + "bbox": [ + 296, + 718, + 329, + 735 + ], + "score": 1.0, + "content": ", where", + "type": "text" + }, + { + "bbox": [ + 329, + 721, + 360, + 732 + ], + "score": 0.78, + "content": "\\operatorname { v e c } ( A )", + "type": "inline_equation" + }, + { + "bbox": [ + 360, + 718, + 492, + 735 + ], + "score": 1.0, + "content": "denotes vectorization of matrix", + "type": "text" + }, + { + "bbox": [ + 493, + 721, + 501, + 730 + ], + "score": 0.79, + "content": "A", + "type": "inline_equation" + }, + { + "bbox": [ + 501, + 718, + 506, + 735 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 37 + } + ], + "index": 36.5, + "bbox_fs": [ + 105, + 709, + 506, + 735 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 106, + 81, + 505, + 108 + ], + "lines": [ + { + "bbox": [ + 104, + 80, + 506, + 98 + ], + "spans": [ + { + "bbox": [ + 104, + 80, + 123, + 98 + ], + "score": 1.0, + "content": "Let", + "type": "text" + }, + { + "bbox": [ + 123, + 82, + 198, + 97 + ], + "score": 0.93, + "content": "D _ { \\mathrm { v e c } ( W _ { H + 1 } ^ { T } ) } L ( W )", + "type": "inline_equation" + }, + { + "bbox": [ + 198, + 80, + 308, + 98 + ], + "score": 1.0, + "content": "be the partial derivative of", + "type": "text" + }, + { + "bbox": [ + 309, + 82, + 335, + 95 + ], + "score": 0.92, + "content": "L ( W )", + "type": "inline_equation" + }, + { + "bbox": [ + 335, + 80, + 399, + 98 + ], + "score": 1.0, + "content": "with respect to", + "type": "text" + }, + { + "bbox": [ + 399, + 82, + 449, + 96 + ], + "score": 0.9, + "content": "\\mathrm { v e c } ( W _ { H + 1 } ^ { T } )", + "type": "inline_equation" + }, + { + "bbox": [ + 449, + 80, + 506, + 98 + ], + "score": 1.0, + "content": "in numerator", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 95, + 426, + 109 + ], + "spans": [ + { + "bbox": [ + 106, + 95, + 426, + 109 + ], + "score": 1.0, + "content": "layout. It was shown by Kawaguchi (2016, Lemma 4.3) that the Hessian matrix", + "type": "text" + } + ], + "index": 1 + } + ], + "index": 0.5 + }, + { + "type": "interline_equation", + "bbox": [ + 127, + 116, + 483, + 158 + ], + "lines": [ + { + "bbox": [ + 127, + 116, + 483, + 158 + ], + "spans": [ + { + "bbox": [ + 127, + 116, + 483, + 158 + ], + "score": 0.93, + "content": "\\begin{array} { r l r } & { } & { \\mathcal { H } ( W ) = D _ { \\mathrm { v e c } ( W _ { H + 1 } ^ { T } ) } \\left( D _ { \\mathrm { v e c } ( W _ { H + 1 } ^ { T } ) } L ( W ) \\right) ^ { T } = \\left( I \\otimes ( W _ { H } \\cdot \\cdot \\cdot W _ { 1 } X ) ( W _ { H } \\cdot \\cdot \\cdot W _ { 1 } X ) ^ { T } \\right) } \\\\ & { } & { \\quad = \\left( I \\otimes W _ { H } \\cdot \\cdot \\cdot W _ { 1 } X X ^ { T } W _ { 1 } ^ { T } \\cdot \\cdot \\cdot W _ { H } ^ { T } \\right) , } \\end{array}", + "type": "interline_equation", + "image_path": "f88a4fcdb95d0f09921555acbf0b09a713fc7ef01eb47519a4f759b8f1cce629.jpg" + } + ] + } + ], + "index": 3, + "virtual_lines": [ + { + "bbox": [ + 127, + 116, + 483, + 130.0 + ], + "spans": [], + "index": 2 + }, + { + "bbox": [ + 127, + 130.0, + 483, + 144.0 + ], + "spans": [], + "index": 3 + }, + { + "bbox": [ + 127, + 144.0, + 483, + 158.0 + ], + "spans": [], + "index": 4 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 165, + 505, + 200 + ], + "lines": [ + { + "bbox": [ + 105, + 165, + 504, + 178 + ], + "spans": [ + { + "bbox": [ + 105, + 165, + 133, + 178 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 133, + 167, + 142, + 176 + ], + "score": 0.83, + "content": "\\otimes", + "type": "inline_equation" + }, + { + "bbox": [ + 143, + 165, + 380, + 178 + ], + "score": 1.0, + "content": "denotes the Kronecker product of two matrices. Notice that", + "type": "text" + }, + { + "bbox": [ + 380, + 166, + 408, + 178 + ], + "score": 0.93, + "content": "\\mathcal { H } ( W )", + "type": "inline_equation" + }, + { + "bbox": [ + 408, + 165, + 504, + 178 + ], + "score": 1.0, + "content": "is positive semidefinite.", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 176, + 505, + 190 + ], + "spans": [ + { + "bbox": [ + 105, + 176, + 132, + 190 + ], + "score": 1.0, + "content": "Since", + "type": "text" + }, + { + "bbox": [ + 132, + 177, + 158, + 187 + ], + "score": 0.9, + "content": "X X ^ { T }", + "type": "inline_equation" + }, + { + "bbox": [ + 158, + 176, + 254, + 190 + ], + "score": 1.0, + "content": "is full rank, whenever", + "type": "text" + }, + { + "bbox": [ + 254, + 177, + 323, + 189 + ], + "score": 0.92, + "content": "W _ { H } \\cdot \\cdot \\cdot W _ { 1 } \\neq 0", + "type": "inline_equation" + }, + { + "bbox": [ + 323, + 176, + 505, + 190 + ], + "score": 1.0, + "content": "there exists a strictly positive eigenvalue in", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 107, + 187, + 496, + 201 + ], + "spans": [ + { + "bbox": [ + 107, + 188, + 135, + 200 + ], + "score": 0.92, + "content": "\\mathcal { H } ( W )", + "type": "inline_equation" + }, + { + "bbox": [ + 135, + 187, + 367, + 201 + ], + "score": 1.0, + "content": ", which means that there exists an increasing direction. So", + "type": "text" + }, + { + "bbox": [ + 367, + 188, + 380, + 198 + ], + "score": 0.41, + "content": "W", + "type": "inline_equation" + }, + { + "bbox": [ + 380, + 187, + 496, + 201 + ], + "score": 1.0, + "content": "cannot be a local maximum.", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 6 + }, + { + "type": "text", + "bbox": [ + 107, + 204, + 503, + 228 + ], + "lines": [ + { + "bbox": [ + 106, + 204, + 505, + 217 + ], + "spans": [ + { + "bbox": [ + 106, + 204, + 173, + 217 + ], + "score": 1.0, + "content": "The case where", + "type": "text" + }, + { + "bbox": [ + 173, + 205, + 241, + 216 + ], + "score": 0.91, + "content": "W _ { H } \\cdot \\cdot \\cdot W _ { 1 } = 0", + "type": "inline_equation" + }, + { + "bbox": [ + 241, + 204, + 505, + 217 + ], + "score": 1.0, + "content": "requires a bit more careful treatment. Note that this case corre-", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 216, + 505, + 228 + ], + "spans": [ + { + "bbox": [ + 105, + 216, + 505, + 228 + ], + "score": 1.0, + "content": "sponds to where we have degenerate critical points, which are in many cases much harder to handle.", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 8.5 + }, + { + "type": "text", + "bbox": [ + 106, + 232, + 505, + 278 + ], + "lines": [ + { + "bbox": [ + 104, + 231, + 505, + 247 + ], + "spans": [ + { + "bbox": [ + 104, + 231, + 180, + 247 + ], + "score": 1.0, + "content": "For any arbitrary", + "type": "text" + }, + { + "bbox": [ + 180, + 234, + 208, + 244 + ], + "score": 0.89, + "content": "\\epsilon > 0", + "type": "inline_equation" + }, + { + "bbox": [ + 208, + 231, + 426, + 247 + ], + "score": 1.0, + "content": ", we describe a procedure that perturbs the matrices", + "type": "text" + }, + { + "bbox": [ + 426, + 233, + 490, + 245 + ], + "score": 0.92, + "content": "W _ { 1 } , \\dots , W _ { H + 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 490, + 231, + 505, + 247 + ], + "score": 1.0, + "content": "by", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 243, + 506, + 257 + ], + "spans": [ + { + "bbox": [ + 105, + 243, + 343, + 257 + ], + "score": 1.0, + "content": "perturbations sampled from Frobenius norm balls of radius", + "type": "text" + }, + { + "bbox": [ + 344, + 246, + 350, + 254 + ], + "score": 0.71, + "content": "\\epsilon", + "type": "inline_equation" + }, + { + "bbox": [ + 350, + 243, + 506, + 257 + ], + "score": 1.0, + "content": "centered at 0, which we will denote as", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 254, + 505, + 267 + ], + "spans": [ + { + "bbox": [ + 106, + 254, + 129, + 267 + ], + "score": 0.64, + "content": "\\bar { \\boldsymbol { B } } _ { i } ( \\epsilon )", + "type": "inline_equation" + }, + { + "bbox": [ + 129, + 254, + 133, + 267 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 133, + 255, + 201, + 266 + ], + "score": 0.61, + "content": "i = 1 , \\ldots , H + 1", + "type": "inline_equation" + }, + { + "bbox": [ + 202, + 254, + 221, + 267 + ], + "score": 1.0, + "content": ". Let", + "type": "text" + }, + { + "bbox": [ + 221, + 254, + 259, + 267 + ], + "score": 0.93, + "content": "\\mathcal { U } ( B _ { i } ( \\epsilon ) )", + "type": "inline_equation" + }, + { + "bbox": [ + 259, + 254, + 419, + 267 + ], + "score": 1.0, + "content": "be the uniform distribution over the ball", + "type": "text" + }, + { + "bbox": [ + 419, + 254, + 442, + 267 + ], + "score": 0.91, + "content": "B _ { i } ( \\epsilon )", + "type": "inline_equation" + }, + { + "bbox": [ + 442, + 254, + 505, + 267 + ], + "score": 1.0, + "content": ". The algorithm", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 266, + 197, + 279 + ], + "spans": [ + { + "bbox": [ + 105, + 266, + 197, + 279 + ], + "score": 1.0, + "content": "goes as the following:", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 11.5 + }, + { + "type": "text", + "bbox": [ + 131, + 299, + 240, + 313 + ], + "lines": [ + { + "bbox": [ + 129, + 298, + 239, + 315 + ], + "spans": [ + { + "bbox": [ + 129, + 298, + 158, + 315 + ], + "score": 1.0, + "content": "1. For", + "type": "text" + }, + { + "bbox": [ + 159, + 300, + 239, + 313 + ], + "score": 0.75, + "content": "i \\in \\{ 1 , \\ldots , H + 1 \\}", + "type": "inline_equation" + } + ], + "index": 14 + } + ], + "index": 14 + }, + { + "type": "text", + "bbox": [ + 144, + 321, + 366, + 334 + ], + "lines": [ + { + "bbox": [ + 142, + 320, + 365, + 335 + ], + "spans": [ + { + "bbox": [ + 142, + 320, + 194, + 335 + ], + "score": 1.0, + "content": "1.1. Sample", + "type": "text" + }, + { + "bbox": [ + 195, + 321, + 257, + 334 + ], + "score": 0.93, + "content": "\\Delta _ { i } \\sim \\mathcal { U } ( B _ { i } ( \\epsilon ) )", + "type": "inline_equation" + }, + { + "bbox": [ + 257, + 320, + 305, + 335 + ], + "score": 1.0, + "content": ", and define", + "type": "text" + }, + { + "bbox": [ + 305, + 322, + 365, + 333 + ], + "score": 0.91, + "content": "V _ { i } = W _ { i } + \\Delta _ { i }", + "type": "inline_equation" + } + ], + "index": 15 + } + ], + "index": 15 + }, + { + "type": "text", + "bbox": [ + 142, + 336, + 385, + 349 + ], + "lines": [ + { + "bbox": [ + 142, + 336, + 386, + 350 + ], + "spans": [ + { + "bbox": [ + 142, + 336, + 171, + 350 + ], + "score": 1.0, + "content": "1.2. If", + "type": "text" + }, + { + "bbox": [ + 172, + 336, + 290, + 349 + ], + "score": 0.91, + "content": "W _ { H + 1 } \\cdot \\cdot \\cdot W _ { i + 1 } V _ { i } \\cdot \\cdot \\cdot V _ { 1 } \\neq 0", + "type": "inline_equation" + }, + { + "bbox": [ + 290, + 336, + 356, + 350 + ], + "score": 1.0, + "content": ", stop and return", + "type": "text" + }, + { + "bbox": [ + 356, + 337, + 382, + 347 + ], + "score": 0.9, + "content": "i ^ { * } = i", + "type": "inline_equation" + }, + { + "bbox": [ + 382, + 336, + 386, + 350 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 16 + }, + { + "type": "text", + "bbox": [ + 106, + 371, + 506, + 426 + ], + "lines": [ + { + "bbox": [ + 105, + 371, + 505, + 384 + ], + "spans": [ + { + "bbox": [ + 105, + 371, + 505, + 384 + ], + "score": 1.0, + "content": "First, recall that the set of rank-deficient matrices have Lebesgue measure zero, so for any sample", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 107, + 382, + 505, + 395 + ], + "spans": [ + { + "bbox": [ + 107, + 382, + 170, + 394 + ], + "score": 0.83, + "content": "\\Delta _ { i } \\sim \\mathcal { U } ( B _ { i } ( \\epsilon ) )", + "type": "inline_equation" + }, + { + "bbox": [ + 171, + 382, + 176, + 395 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 176, + 383, + 239, + 393 + ], + "score": 0.82, + "content": "V _ { i } = W _ { i } + \\Delta _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 239, + 382, + 505, + 395 + ], + "score": 1.0, + "content": "has full rank with probability 1. If we proceed the for loop until", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 107, + 393, + 506, + 406 + ], + "spans": [ + { + "bbox": [ + 107, + 394, + 148, + 404 + ], + "score": 0.89, + "content": "i = H + 1", + "type": "inline_equation" + }, + { + "bbox": [ + 149, + 393, + 230, + 406 + ], + "score": 1.0, + "content": ", we have a full-rank", + "type": "text" + }, + { + "bbox": [ + 231, + 394, + 280, + 405 + ], + "score": 0.91, + "content": "V _ { H + 1 } \\cdots V _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 280, + 393, + 506, + 406 + ], + "score": 1.0, + "content": "with probability 1, which means that the algorithm must", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 104, + 403, + 506, + 417 + ], + "spans": [ + { + "bbox": [ + 104, + 403, + 133, + 417 + ], + "score": 1.0, + "content": "return", + "type": "text" + }, + { + "bbox": [ + 133, + 405, + 220, + 416 + ], + "score": 0.93, + "content": "i ^ { * } \\in \\{ 1 , \\ldots , H + 1 \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 221, + 403, + 430, + 417 + ], + "score": 1.0, + "content": "with probability 1. Notice that before and after the", + "type": "text" + }, + { + "bbox": [ + 430, + 405, + 439, + 414 + ], + "score": 0.86, + "content": "i ^ { * }", + "type": "inline_equation" + }, + { + "bbox": [ + 439, + 403, + 506, + 417 + ], + "score": 1.0, + "content": "-th iteration, we", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 414, + 129, + 429 + ], + "spans": [ + { + "bbox": [ + 105, + 414, + 129, + 429 + ], + "score": 1.0, + "content": "have", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 19 + }, + { + "type": "interline_equation", + "bbox": [ + 147, + 435, + 464, + 465 + ], + "lines": [ + { + "bbox": [ + 147, + 435, + 464, + 465 + ], + "spans": [ + { + "bbox": [ + 147, + 435, + 464, + 465 + ], + "score": 0.93, + "content": "\\begin{array} { r l } & { W _ { H + 1 } \\cdots W _ { i ^ { * } } V _ { i ^ { * } - 1 } \\cdots V _ { 1 } = 0 , } \\\\ & { W _ { H + 1 } \\cdots W _ { i ^ { * } + 1 } V _ { i ^ { * } } \\cdots V _ { 1 } = W _ { H + 1 } \\cdot \\cdot \\cdot W _ { i ^ { * } + 1 } ( W _ { i ^ { * } } + \\Delta _ { i ^ { * } } ) V _ { i ^ { * } - 1 } \\cdot \\cdot \\cdot V _ { 1 } \\not = 0 . } \\end{array}", + "type": "interline_equation", + "image_path": "8a9c6fbff9950abaac2af6e9b89fa98e3a907815f9b2f541c5f450bef7c1deed.jpg" + } + ] + } + ], + "index": 23, + "virtual_lines": [ + { + "bbox": [ + 147, + 435, + 464, + 445.0 + ], + "spans": [], + "index": 22 + }, + { + "bbox": [ + 147, + 445.0, + 464, + 455.0 + ], + "spans": [], + "index": 23 + }, + { + "bbox": [ + 147, + 455.0, + 464, + 465.0 + ], + "spans": [], + "index": 24 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 474, + 501, + 488 + ], + "lines": [ + { + "bbox": [ + 106, + 474, + 502, + 489 + ], + "spans": [ + { + "bbox": [ + 106, + 474, + 222, + 489 + ], + "score": 1.0, + "content": "This means that if we define", + "type": "text" + }, + { + "bbox": [ + 222, + 474, + 378, + 488 + ], + "score": 0.93, + "content": "\\hat { \\Delta } = { \\cal W } _ { H + 1 } \\cdot \\cdot \\cdot { \\cal W } _ { i ^ { * } + 1 } \\Delta _ { i ^ { * } } V _ { i ^ { * } - 1 } \\cdot \\cdot \\cdot V _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 378, + 474, + 401, + 489 + ], + "score": 1.0, + "content": ", then", + "type": "text" + }, + { + "bbox": [ + 402, + 474, + 429, + 487 + ], + "score": 0.92, + "content": "\\hat { \\Delta } \\neq 0", + "type": "inline_equation" + }, + { + "bbox": [ + 429, + 474, + 502, + 489 + ], + "score": 1.0, + "content": ". Also, notice that", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 25 + }, + { + "type": "interline_equation", + "bbox": [ + 202, + 497, + 408, + 513 + ], + "lines": [ + { + "bbox": [ + 202, + 497, + 408, + 513 + ], + "spans": [ + { + "bbox": [ + 202, + 497, + 408, + 513 + ], + "score": 0.86, + "content": "W _ { H + 1 } \\cdot \\cdot \\cdot W _ { i ^ { * } + 1 } ( W _ { i ^ { * } } - \\Delta _ { i ^ { * } } ) V _ { i ^ { * } - 1 } \\cdot \\cdot \\cdot V _ { 1 } = - \\hat { \\Delta } .", + "type": "interline_equation", + "image_path": "0ab829fdf667e451c4a56f155c26b0c1b81e22b7fd70164b84b3f2494dfdc95b.jpg" + } + ] + } + ], + "index": 26, + "virtual_lines": [ + { + "bbox": [ + 202, + 497, + 408, + 513 + ], + "spans": [], + "index": 26 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 522, + 201, + 533 + ], + "lines": [ + { + "bbox": [ + 105, + 520, + 201, + 535 + ], + "spans": [ + { + "bbox": [ + 105, + 520, + 201, + 535 + ], + "score": 1.0, + "content": "Now, define two points", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 27 + }, + { + "type": "interline_equation", + "bbox": [ + 190, + 541, + 421, + 576 + ], + "lines": [ + { + "bbox": [ + 190, + 541, + 421, + 576 + ], + "spans": [ + { + "bbox": [ + 190, + 541, + 421, + 576 + ], + "score": 0.92, + "content": "\\begin{array} { r l } & { U ^ { ( 1 ) } = ( V _ { 1 } , \\ldots , V _ { i ^ { * } - 1 } , W _ { i ^ { * } } + \\Delta _ { i ^ { * } } , W _ { i ^ { * } + 1 } , \\ldots , W _ { H + 1 } ) , } \\\\ & { U ^ { ( 2 ) } = ( V _ { 1 } , \\ldots , V _ { i ^ { * } - 1 } , W _ { i ^ { * } } - \\Delta _ { i ^ { * } } , W _ { i ^ { * } + 1 } , \\ldots , W _ { H + 1 } ) , } \\end{array}", + "type": "interline_equation", + "image_path": "bc7fe630950f6912ea4f6c64d0d22462d3799ceaf87a697a139e056aac7ee859.jpg" + } + ] + } + ], + "index": 28.5, + "virtual_lines": [ + { + "bbox": [ + 190, + 541, + 421, + 558.5 + ], + "spans": [], + "index": 28 + }, + { + "bbox": [ + 190, + 558.5, + 421, + 576.0 + ], + "spans": [], + "index": 29 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 583, + 504, + 607 + ], + "lines": [ + { + "bbox": [ + 106, + 582, + 505, + 595 + ], + "spans": [ + { + "bbox": [ + 106, + 582, + 307, + 595 + ], + "score": 1.0, + "content": "and notice that they are all in the neighborhood of", + "type": "text" + }, + { + "bbox": [ + 307, + 584, + 319, + 594 + ], + "score": 0.82, + "content": "W", + "type": "inline_equation" + }, + { + "bbox": [ + 319, + 582, + 450, + 595 + ], + "score": 1.0, + "content": ", that is, the Cartesian product of", + "type": "text" + }, + { + "bbox": [ + 450, + 586, + 455, + 594 + ], + "score": 0.79, + "content": "\\epsilon", + "type": "inline_equation" + }, + { + "bbox": [ + 455, + 582, + 505, + 595 + ], + "score": 1.0, + "content": "-radius balls", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 595, + 300, + 607 + ], + "spans": [ + { + "bbox": [ + 105, + 595, + 153, + 607 + ], + "score": 1.0, + "content": "centered at", + "type": "text" + }, + { + "bbox": [ + 153, + 595, + 217, + 606 + ], + "score": 0.91, + "content": "W _ { 1 } , \\dots , W _ { H + 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 217, + 595, + 300, + 607 + ], + "score": 1.0, + "content": ". Moreover, we have", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 30.5 + }, + { + "type": "interline_equation", + "bbox": [ + 165, + 614, + 444, + 691 + ], + "lines": [ + { + "bbox": [ + 165, + 614, + 444, + 691 + ], + "spans": [ + { + "bbox": [ + 165, + 614, + 444, + 691 + ], + "score": 0.94, + "content": "\\begin{array} { l } { \\displaystyle { L ( W ) = \\frac { 1 } { 2 } \\left\\| 0 \\cdot X - Y \\right\\| _ { \\mathrm { F } } ^ { 2 } = \\frac { 1 } { 2 } \\left\\| Y \\right\\| _ { \\mathrm { F } } ^ { 2 } , } } \\\\ { \\displaystyle { L ( U ^ { ( 1 ) } ) = \\frac { 1 } { 2 } \\left\\| \\hat { \\Delta } X - Y \\right\\| _ { \\mathrm { F } } ^ { 2 } = \\frac { 1 } { 2 } \\left\\| Y \\right\\| _ { \\mathrm { F } } ^ { 2 } + \\frac { 1 } { 2 } \\left\\| \\hat { \\Delta } X \\right\\| _ { \\mathrm { F } } ^ { 2 } - \\left. \\hat { \\Delta } X , Y \\right. , } } \\\\ { \\displaystyle { L ( U ^ { ( 2 ) } ) = \\frac { 1 } { 2 } \\left\\| - \\hat { \\Delta } X - Y \\right\\| _ { \\mathrm { F } } ^ { 2 } = \\frac { 1 } { 2 } \\left\\| Y \\right\\| _ { \\mathrm { F } } ^ { 2 } + \\frac { 1 } { 2 } \\left\\| \\hat { \\Delta } X \\right\\| _ { \\mathrm { F } } ^ { 2 } + \\left. \\hat { \\Delta } X , Y \\right. , } } \\end{array}", + "type": "interline_equation", + "image_path": "bdb9d7935b5a4689dc480f1774ba6db1e49bd9a1b6d738ab3754c74f82fc1336.jpg" + } + ] + } + ], + "index": 33, + "virtual_lines": [ + { + "bbox": [ + 165, + 614, + 444, + 639.6666666666666 + ], + "spans": [], + "index": 32 + }, + { + "bbox": [ + 165, + 639.6666666666666, + 444, + 665.3333333333333 + ], + "spans": [], + "index": 33 + }, + { + "bbox": [ + 165, + 665.3333333333333, + 444, + 690.9999999999999 + ], + "spans": [], + "index": 34 + } + ] + }, + { + "type": "text", + "bbox": [ + 108, + 698, + 504, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 697, + 506, + 712 + ], + "spans": [ + { + "bbox": [ + 105, + 697, + 278, + 712 + ], + "score": 1.0, + "content": "from which we can see that at least one of", + "type": "text" + }, + { + "bbox": [ + 279, + 698, + 352, + 711 + ], + "score": 0.92, + "content": "L ( W ) < L ( U ^ { ( 1 ) } )", + "type": "inline_equation" + }, + { + "bbox": [ + 352, + 697, + 364, + 712 + ], + "score": 1.0, + "content": "or", + "type": "text" + }, + { + "bbox": [ + 365, + 698, + 437, + 711 + ], + "score": 0.93, + "content": "L ( W ) < L ( U ^ { ( 2 ) } )", + "type": "inline_equation" + }, + { + "bbox": [ + 438, + 697, + 506, + 712 + ], + "score": 1.0, + "content": "must hold. 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So", + "type": "text" + }, + { + "bbox": [ + 367, + 188, + 380, + 198 + ], + "score": 0.41, + "content": "W", + "type": "inline_equation" + }, + { + "bbox": [ + 380, + 187, + 496, + 201 + ], + "score": 1.0, + "content": "cannot be a local maximum.", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 6, + "bbox_fs": [ + 105, + 165, + 505, + 201 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 204, + 503, + 228 + ], + "lines": [ + { + "bbox": [ + 106, + 204, + 505, + 217 + ], + "spans": [ + { + "bbox": [ + 106, + 204, + 173, + 217 + ], + "score": 1.0, + "content": "The case where", + "type": "text" + }, + { + "bbox": [ + 173, + 205, + 241, + 216 + ], + "score": 0.91, + "content": "W _ { H } \\cdot \\cdot \\cdot W _ { 1 } = 0", + "type": "inline_equation" + }, + { + "bbox": [ + 241, + 204, + 505, + 217 + ], + "score": 1.0, + "content": "requires a bit more careful treatment. 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Let", + "type": "text" + }, + { + "bbox": [ + 221, + 254, + 259, + 267 + ], + "score": 0.93, + "content": "\\mathcal { U } ( B _ { i } ( \\epsilon ) )", + "type": "inline_equation" + }, + { + "bbox": [ + 259, + 254, + 419, + 267 + ], + "score": 1.0, + "content": "be the uniform distribution over the ball", + "type": "text" + }, + { + "bbox": [ + 419, + 254, + 442, + 267 + ], + "score": 0.91, + "content": "B _ { i } ( \\epsilon )", + "type": "inline_equation" + }, + { + "bbox": [ + 442, + 254, + 505, + 267 + ], + "score": 1.0, + "content": ". The algorithm", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 266, + 197, + 279 + ], + "spans": [ + { + "bbox": [ + 105, + 266, + 197, + 279 + ], + "score": 1.0, + "content": "goes as the following:", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 11.5, + "bbox_fs": [ + 104, + 231, + 506, + 279 + ] + }, + { + "type": "text", + "bbox": [ + 131, + 299, + 240, + 313 + ], + "lines": [ + { + "bbox": [ + 129, + 298, + 239, + 315 + ], + "spans": [ + { + "bbox": [ + 129, + 298, + 158, + 315 + ], + "score": 1.0, + "content": "1. For", + "type": "text" + }, + { + "bbox": [ + 159, + 300, + 239, + 313 + ], + "score": 0.75, + "content": "i \\in \\{ 1 , \\ldots , H + 1 \\}", + "type": "inline_equation" + } + ], + "index": 14 + } + ], + "index": 14, + "bbox_fs": [ + 129, + 298, + 239, + 315 + ] + }, + { + "type": "text", + "bbox": [ + 144, + 321, + 366, + 334 + ], + "lines": [ + { + "bbox": [ + 142, + 320, + 365, + 335 + ], + "spans": [ + { + "bbox": [ + 142, + 320, + 194, + 335 + ], + "score": 1.0, + "content": "1.1. Sample", + "type": "text" + }, + { + "bbox": [ + 195, + 321, + 257, + 334 + ], + "score": 0.93, + "content": "\\Delta _ { i } \\sim \\mathcal { U } ( B _ { i } ( \\epsilon ) )", + "type": "inline_equation" + }, + { + "bbox": [ + 257, + 320, + 305, + 335 + ], + "score": 1.0, + "content": ", and define", + "type": "text" + }, + { + "bbox": [ + 305, + 322, + 365, + 333 + ], + "score": 0.91, + "content": "V _ { i } = W _ { i } + \\Delta _ { i }", + "type": "inline_equation" + } + ], + "index": 15 + } + ], + "index": 15, + "bbox_fs": [ + 142, + 320, + 365, + 335 + ] + }, + { + "type": "text", + "bbox": [ + 142, + 336, + 385, + 349 + ], + "lines": [ + { + "bbox": [ + 142, + 336, + 386, + 350 + ], + "spans": [ + { + "bbox": [ + 142, + 336, + 171, + 350 + ], + "score": 1.0, + "content": "1.2. If", + "type": "text" + }, + { + "bbox": [ + 172, + 336, + 290, + 349 + ], + "score": 0.91, + "content": "W _ { H + 1 } \\cdot \\cdot \\cdot W _ { i + 1 } V _ { i } \\cdot \\cdot \\cdot V _ { 1 } \\neq 0", + "type": "inline_equation" + }, + { + "bbox": [ + 290, + 336, + 356, + 350 + ], + "score": 1.0, + "content": ", stop and return", + "type": "text" + }, + { + "bbox": [ + 356, + 337, + 382, + 347 + ], + "score": 0.9, + "content": "i ^ { * } = i", + "type": "inline_equation" + }, + { + "bbox": [ + 382, + 336, + 386, + 350 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 16, + "bbox_fs": [ + 142, + 336, + 386, + 350 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 371, + 506, + 426 + ], + "lines": [ + { + "bbox": [ + 105, + 371, + 505, + 384 + ], + "spans": [ + { + "bbox": [ + 105, + 371, + 505, + 384 + ], + "score": 1.0, + "content": "First, recall that the set of rank-deficient matrices have Lebesgue measure zero, so for any sample", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 107, + 382, + 505, + 395 + ], + "spans": [ + { + "bbox": [ + 107, + 382, + 170, + 394 + ], + "score": 0.83, + "content": "\\Delta _ { i } \\sim \\mathcal { U } ( B _ { i } ( \\epsilon ) )", + "type": "inline_equation" + }, + { + "bbox": [ + 171, + 382, + 176, + 395 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 176, + 383, + 239, + 393 + ], + "score": 0.82, + "content": "V _ { i } = W _ { i } + \\Delta _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 239, + 382, + 505, + 395 + ], + "score": 1.0, + "content": "has full rank with probability 1. If we proceed the for loop until", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 107, + 393, + 506, + 406 + ], + "spans": [ + { + "bbox": [ + 107, + 394, + 148, + 404 + ], + "score": 0.89, + "content": "i = H + 1", + "type": "inline_equation" + }, + { + "bbox": [ + 149, + 393, + 230, + 406 + ], + "score": 1.0, + "content": ", we have a full-rank", + "type": "text" + }, + { + "bbox": [ + 231, + 394, + 280, + 405 + ], + "score": 0.91, + "content": "V _ { H + 1 } \\cdots V _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 280, + 393, + 506, + 406 + ], + "score": 1.0, + "content": "with probability 1, which means that the algorithm must", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 104, + 403, + 506, + 417 + ], + "spans": [ + { + "bbox": [ + 104, + 403, + 133, + 417 + ], + "score": 1.0, + "content": "return", + "type": "text" + }, + { + "bbox": [ + 133, + 405, + 220, + 416 + ], + "score": 0.93, + "content": "i ^ { * } \\in \\{ 1 , \\ldots , H + 1 \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 221, + 403, + 430, + 417 + ], + "score": 1.0, + "content": "with probability 1. Notice that before and after the", + "type": "text" + }, + { + "bbox": [ + 430, + 405, + 439, + 414 + ], + "score": 0.86, + "content": "i ^ { * }", + "type": "inline_equation" + }, + { + "bbox": [ + 439, + 403, + 506, + 417 + ], + "score": 1.0, + "content": "-th iteration, we", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 414, + 129, + 429 + ], + "spans": [ + { + "bbox": [ + 105, + 414, + 129, + 429 + ], + "score": 1.0, + "content": "have", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 19, + "bbox_fs": [ + 104, + 371, + 506, + 429 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 147, + 435, + 464, + 465 + ], + "lines": [ + { + "bbox": [ + 147, + 435, + 464, + 465 + ], + "spans": [ + { + "bbox": [ + 147, + 435, + 464, + 465 + ], + "score": 0.93, + "content": "\\begin{array} { r l } & { W _ { H + 1 } \\cdots W _ { i ^ { * } } V _ { i ^ { * } - 1 } \\cdots V _ { 1 } = 0 , } \\\\ & { W _ { H + 1 } \\cdots W _ { i ^ { * } + 1 } V _ { i ^ { * } } \\cdots V _ { 1 } = W _ { H + 1 } \\cdot \\cdot \\cdot W _ { i ^ { * } + 1 } ( W _ { i ^ { * } } + \\Delta _ { i ^ { * } } ) V _ { i ^ { * } - 1 } \\cdot \\cdot \\cdot V _ { 1 } \\not = 0 . } \\end{array}", + "type": "interline_equation", + "image_path": "8a9c6fbff9950abaac2af6e9b89fa98e3a907815f9b2f541c5f450bef7c1deed.jpg" + } + ] + } + ], + "index": 23, + "virtual_lines": [ + { + "bbox": [ + 147, + 435, + 464, + 445.0 + ], + "spans": [], + "index": 22 + }, + { + "bbox": [ + 147, + 445.0, + 464, + 455.0 + ], + "spans": [], + "index": 23 + }, + { + "bbox": [ + 147, + 455.0, + 464, + 465.0 + ], + "spans": [], + "index": 24 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 474, + 501, + 488 + ], + "lines": [ + { + "bbox": [ + 106, + 474, + 502, + 489 + ], + "spans": [ + { + "bbox": [ + 106, + 474, + 222, + 489 + ], + "score": 1.0, + "content": "This means that if we define", + "type": "text" + }, + { + "bbox": [ + 222, + 474, + 378, + 488 + ], + "score": 0.93, + "content": "\\hat { \\Delta } = { \\cal W } _ { H + 1 } \\cdot \\cdot \\cdot { \\cal W } _ { i ^ { * } + 1 } \\Delta _ { i ^ { * } } V _ { i ^ { * } - 1 } \\cdot \\cdot \\cdot V _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 378, + 474, + 401, + 489 + ], + "score": 1.0, + "content": ", then", + "type": "text" + }, + { + "bbox": [ + 402, + 474, + 429, + 487 + ], + "score": 0.92, + "content": "\\hat { \\Delta } \\neq 0", + "type": "inline_equation" + }, + { + "bbox": [ + 429, + 474, + 502, + 489 + ], + "score": 1.0, + "content": ". Also, notice that", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 25, + "bbox_fs": [ + 106, + 474, + 502, + 489 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 202, + 497, + 408, + 513 + ], + "lines": [ + { + "bbox": [ + 202, + 497, + 408, + 513 + ], + "spans": [ + { + "bbox": [ + 202, + 497, + 408, + 513 + ], + "score": 0.86, + "content": "W _ { H + 1 } \\cdot \\cdot \\cdot W _ { i ^ { * } + 1 } ( W _ { i ^ { * } } - \\Delta _ { i ^ { * } } ) V _ { i ^ { * } - 1 } \\cdot \\cdot \\cdot V _ { 1 } = - \\hat { \\Delta } .", + "type": "interline_equation", + "image_path": "0ab829fdf667e451c4a56f155c26b0c1b81e22b7fd70164b84b3f2494dfdc95b.jpg" + } + ] + } + ], + "index": 26, + "virtual_lines": [ + { + "bbox": [ + 202, + 497, + 408, + 513 + ], + "spans": [], + "index": 26 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 522, + 201, + 533 + ], + "lines": [ + { + "bbox": [ + 105, + 520, + 201, + 535 + ], + "spans": [ + { + "bbox": [ + 105, + 520, + 201, + 535 + ], + "score": 1.0, + "content": "Now, define two points", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 27, + "bbox_fs": [ + 105, + 520, + 201, + 535 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 190, + 541, + 421, + 576 + ], + "lines": [ + { + "bbox": [ + 190, + 541, + 421, + 576 + ], + "spans": [ + { + "bbox": [ + 190, + 541, + 421, + 576 + ], + "score": 0.92, + "content": "\\begin{array} { r l } & { U ^ { ( 1 ) } = ( V _ { 1 } , \\ldots , V _ { i ^ { * } - 1 } , W _ { i ^ { * } } + \\Delta _ { i ^ { * } } , W _ { i ^ { * } + 1 } , \\ldots , W _ { H + 1 } ) , } \\\\ & { U ^ { ( 2 ) } = ( V _ { 1 } , \\ldots , V _ { i ^ { * } - 1 } , W _ { i ^ { * } } - \\Delta _ { i ^ { * } } , W _ { i ^ { * } + 1 } , \\ldots , W _ { H + 1 } ) , } \\end{array}", + "type": "interline_equation", + "image_path": "bc7fe630950f6912ea4f6c64d0d22462d3799ceaf87a697a139e056aac7ee859.jpg" + } + ] + } + ], + "index": 28.5, + "virtual_lines": [ + { + "bbox": [ + 190, + 541, + 421, + 558.5 + ], + "spans": [], + "index": 28 + }, + { + "bbox": [ + 190, + 558.5, + 421, + 576.0 + ], + "spans": [], + "index": 29 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 583, + 504, + 607 + ], + "lines": [ + { + "bbox": [ + 106, + 582, + 505, + 595 + ], + "spans": [ + { + "bbox": [ + 106, + 582, + 307, + 595 + ], + "score": 1.0, + "content": "and notice that they are all in the neighborhood of", + "type": "text" + }, + { + "bbox": [ + 307, + 584, + 319, + 594 + ], + "score": 0.82, + "content": "W", + "type": "inline_equation" + }, + { + "bbox": [ + 319, + 582, + 450, + 595 + ], + "score": 1.0, + "content": ", that is, the Cartesian product of", + "type": "text" + }, + { + "bbox": [ + 450, + 586, + 455, + 594 + ], + "score": 0.79, + "content": "\\epsilon", + "type": "inline_equation" + }, + { + "bbox": [ + 455, + 582, + 505, + 595 + ], + "score": 1.0, + "content": "-radius balls", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 595, + 300, + 607 + ], + "spans": [ + { + "bbox": [ + 105, + 595, + 153, + 607 + ], + "score": 1.0, + "content": "centered at", + "type": "text" + }, + { + "bbox": [ + 153, + 595, + 217, + 606 + ], + "score": 0.91, + "content": "W _ { 1 } , \\dots , W _ { H + 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 217, + 595, + 300, + 607 + ], + "score": 1.0, + "content": ". Moreover, we have", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 30.5, + "bbox_fs": [ + 105, + 582, + 505, + 607 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 165, + 614, + 444, + 691 + ], + "lines": [ + { + "bbox": [ + 165, + 614, + 444, + 691 + ], + "spans": [ + { + "bbox": [ + 165, + 614, + 444, + 691 + ], + "score": 0.94, + "content": "\\begin{array} { l } { \\displaystyle { L ( W ) = \\frac { 1 } { 2 } \\left\\| 0 \\cdot X - Y \\right\\| _ { \\mathrm { F } } ^ { 2 } = \\frac { 1 } { 2 } \\left\\| Y \\right\\| _ { \\mathrm { F } } ^ { 2 } , } } \\\\ { \\displaystyle { L ( U ^ { ( 1 ) } ) = \\frac { 1 } { 2 } \\left\\| \\hat { \\Delta } X - Y \\right\\| _ { \\mathrm { F } } ^ { 2 } = \\frac { 1 } { 2 } \\left\\| Y \\right\\| _ { \\mathrm { F } } ^ { 2 } + \\frac { 1 } { 2 } \\left\\| \\hat { \\Delta } X \\right\\| _ { \\mathrm { F } } ^ { 2 } - \\left. \\hat { \\Delta } X , Y \\right. , } } \\\\ { \\displaystyle { L ( U ^ { ( 2 ) } ) = \\frac { 1 } { 2 } \\left\\| - \\hat { \\Delta } X - Y \\right\\| _ { \\mathrm { F } } ^ { 2 } = \\frac { 1 } { 2 } \\left\\| Y \\right\\| _ { \\mathrm { F } } ^ { 2 } + \\frac { 1 } { 2 } \\left\\| \\hat { \\Delta } X \\right\\| _ { \\mathrm { F } } ^ { 2 } + \\left. \\hat { \\Delta } X , Y \\right. , } } \\end{array}", + "type": "interline_equation", + "image_path": "bdb9d7935b5a4689dc480f1774ba6db1e49bd9a1b6d738ab3754c74f82fc1336.jpg" + } + ] + } + ], + "index": 33, + "virtual_lines": [ + { + "bbox": [ + 165, + 614, + 444, + 639.6666666666666 + ], + "spans": [], + "index": 32 + }, + { + "bbox": [ + 165, + 639.6666666666666, + 444, + 665.3333333333333 + ], + "spans": [], + "index": 33 + }, + { + "bbox": [ + 165, + 665.3333333333333, + 444, + 690.9999999999999 + ], + "spans": [], + "index": 34 + } + ] + }, + { + "type": "text", + "bbox": [ + 108, + 698, + 504, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 697, + 506, + 712 + ], + "spans": [ + { + "bbox": [ + 105, + 697, + 278, + 712 + ], + "score": 1.0, + "content": "from which we can see that at least one of", + "type": "text" + }, + { + "bbox": [ + 279, + 698, + 352, + 711 + ], + "score": 0.92, + "content": "L ( W ) < L ( U ^ { ( 1 ) } )", + "type": "inline_equation" + }, + { + "bbox": [ + 352, + 697, + 364, + 712 + ], + "score": 1.0, + "content": "or", + "type": "text" + }, + { + "bbox": [ + 365, + 698, + 437, + 711 + ], + "score": 0.93, + "content": "L ( W ) < L ( U ^ { ( 2 ) } )", + "type": "inline_equation" + }, + { + "bbox": [ + 438, + 697, + 506, + 712 + ], + "score": 1.0, + "content": "must hold. This", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 710, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 106, + 710, + 182, + 722 + ], + "score": 1.0, + "content": "shows that for any", + "type": "text" + }, + { + "bbox": [ + 182, + 710, + 206, + 720 + ], + "score": 0.92, + "content": "\\epsilon > 0", + "type": "inline_equation" + }, + { + "bbox": [ + 206, + 710, + 272, + 722 + ], + "score": 1.0, + "content": ", there is a point", + "type": "text" + }, + { + "bbox": [ + 272, + 712, + 280, + 720 + ], + "score": 0.87, + "content": "U", + "type": "inline_equation" + }, + { + "bbox": [ + 281, + 710, + 292, + 722 + ], + "score": 1.0, + "content": "in", + "type": "text" + }, + { + "bbox": [ + 293, + 714, + 297, + 720 + ], + "score": 0.77, + "content": "\\epsilon", + "type": "inline_equation" + }, + { + "bbox": [ + 298, + 710, + 369, + 722 + ], + "score": 1.0, + "content": "-neighborhood of", + "type": "text" + }, + { + "bbox": [ + 369, + 711, + 380, + 720 + ], + "score": 0.8, + "content": "W", + "type": "inline_equation" + }, + { + "bbox": [ + 380, + 710, + 505, + 722 + ], + "score": 1.0, + "content": "with a strictly greater function", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 106, + 720, + 506, + 733 + ], + "spans": [ + { + "bbox": [ + 106, + 720, + 130, + 733 + ], + "score": 1.0, + "content": "value", + "type": "text" + }, + { + "bbox": [ + 131, + 720, + 154, + 732 + ], + "score": 0.92, + "content": "L ( U )", + "type": "inline_equation" + }, + { + "bbox": [ + 154, + 720, + 225, + 733 + ], + "score": 1.0, + "content": ". 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Also,", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 6.5 + }, + { + "type": "interline_equation", + "bbox": [ + 179, + 231, + 431, + 247 + ], + "lines": [ + { + "bbox": [ + 179, + 231, + 431, + 247 + ], + "spans": [ + { + "bbox": [ + 179, + 231, + 431, + 247 + ], + "score": 0.88, + "content": "( R X - Y X ^ { T } ( X X ^ { T } ) ^ { - 1 } X ) ^ { T } = X ^ { T } ( R ^ { T } - ( X X ^ { T } ) ^ { - 1 } X Y ^ { T } ) .", + "type": "interline_equation", + "image_path": "1bc806a6b60bfbcc9679f4f2534a4ea3995b1b6f44d6ceeaf195f7c4af00cbe0.jpg" + } + ] + } + ], + "index": 8, + "virtual_lines": [ + { + "bbox": [ + 179, + 231, + 431, + 247 + ], + "spans": [], + "index": 8 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 251, + 504, + 275 + ], + "lines": [ + { + "bbox": [ + 105, + 249, + 505, + 265 + ], + "spans": [ + { + "bbox": [ + 105, + 249, + 124, + 265 + ], + "score": 1.0, + "content": "It is", + "type": "text" + }, + { + "bbox": [ + 124, + 251, + 141, + 262 + ], + "score": 0.89, + "content": "X ^ { T }", + "type": "inline_equation" + }, + { + "bbox": [ + 141, + 249, + 388, + 265 + ], + "score": 1.0, + "content": "right-multiplied with some matrix, so its columns must lie in", + "type": "text" + }, + { + "bbox": [ + 389, + 251, + 471, + 264 + ], + "score": 0.91, + "content": "\\operatorname { c o l } ( X ^ { T } ) = \\operatorname { r o w } ( X )", + "type": "inline_equation" + }, + { + "bbox": [ + 471, + 249, + 505, + 265 + ], + "score": 1.0, + "content": ". By the", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 260, + 232, + 276 + ], + "spans": [ + { + "bbox": [ + 105, + 260, + 142, + 276 + ], + "score": 1.0, + "content": "fact that", + "type": "text" + }, + { + "bbox": [ + 142, + 262, + 228, + 275 + ], + "score": 0.9, + "content": "\\mathrm { n u l l } ( X ) ^ { \\perp } \\overset { \\cdot } { = } \\operatorname { r o w } ( X )", + "type": "inline_equation" + }, + { + "bbox": [ + 228, + 260, + 232, + 276 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 9.5 + }, + { + "type": "interline_equation", + "bbox": [ + 182, + 280, + 425, + 295 + ], + "lines": [ + { + "bbox": [ + 182, + 280, + 425, + 295 + ], + "spans": [ + { + "bbox": [ + 182, + 280, + 425, + 295 + ], + "score": 0.82, + "content": "( Y X ^ { T } ( X X ^ { T } ) ^ { - 1 } X - Y ) ( R X - Y X ^ { T } ( X X ^ { T } ) ^ { - 1 } X ) ^ { T } = 0 .", + "type": "interline_equation", + "image_path": "382ec98deed29e6c41b709f6f880ceefb1054b22ca061eadd123e6074acf6f18.jpg" + } + ] + } + ], + "index": 11, + "virtual_lines": [ + { + "bbox": [ + 182, + 280, + 425, + 295 + ], + "spans": [], + "index": 11 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 300, + 126, + 311 + ], + "lines": [ + { + "bbox": [ + 105, + 300, + 126, + 312 + ], + "spans": [ + { + "bbox": [ + 105, + 300, + 126, + 312 + ], + "score": 1.0, + "content": "thus", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 12 + }, + { + "type": "interline_equation", + "bbox": [ + 152, + 307, + 458, + 332 + ], + "lines": [ + { + "bbox": [ + 152, + 307, + 458, + 332 + ], + "spans": [ + { + "bbox": [ + 152, + 307, + 458, + 332 + ], + "score": 0.86, + "content": "L _ { 0 } ( R ) = { \\frac { 1 } { 2 } } \\left\\| R X - Y X ^ { T } ( X X ^ { T } ) ^ { - 1 } X \\right\\| _ { \\mathrm { F } } ^ { 2 } + { \\frac { 1 } { 2 } } \\left\\| Y X ^ { T } ( X X ^ { T } ) ^ { - 1 } X - Y \\right\\| _ { \\mathrm { F } } ^ { 2 }", + "type": "interline_equation", + "image_path": "60dadb429ab12bf420bf00827e81526092d90d15356e70e10a974b063ee74686.jpg" + } + ] + } + ], + "index": 13, + "virtual_lines": [ + { + "bbox": [ + 152, + 307, + 458, + 332 + ], + "spans": [], + "index": 13 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 333, + 132, + 344 + ], + "lines": [ + { + "bbox": [ + 104, + 332, + 134, + 345 + ], + "spans": [ + { + "bbox": [ + 104, + 332, + 134, + 345 + ], + "score": 1.0, + "content": "holds.", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 14 + }, + { + "type": "text", + "bbox": [ + 106, + 348, + 506, + 408 + ], + "lines": [ + { + "bbox": [ + 105, + 347, + 506, + 363 + ], + "spans": [ + { + "bbox": [ + 105, + 347, + 285, + 363 + ], + "score": 1.0, + "content": "Now, (2) becomes a problem of minimizing", + "type": "text" + }, + { + "bbox": [ + 285, + 347, + 407, + 363 + ], + "score": 0.92, + "content": "\\begin{array} { r } { \\left\\| R X - Y X ^ { T } ( X X ^ { T } ) ^ { - 1 } X \\right\\| _ { \\mathrm { F } } ^ { 2 } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 407, + 347, + 506, + 363 + ], + "score": 1.0, + "content": "subject to the rank con-", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 361, + 505, + 375 + ], + "spans": [ + { + "bbox": [ + 105, + 361, + 135, + 375 + ], + "score": 1.0, + "content": "straint", + "type": "text" + }, + { + "bbox": [ + 135, + 362, + 193, + 373 + ], + "score": 0.86, + "content": "\\mathrm { r a n k } ( R ) \\leq k", + "type": "inline_equation" + }, + { + "bbox": [ + 194, + 361, + 393, + 375 + ], + "score": 1.0, + "content": ". 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Therefore,", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 17 + }, + { + "type": "interline_equation", + "bbox": [ + 247, + 405, + 363, + 421 + ], + "lines": [ + { + "bbox": [ + 247, + 405, + 363, + 421 + ], + "spans": [ + { + "bbox": [ + 247, + 405, + 363, + 421 + ], + "score": 0.9, + "content": "R ^ { * } = \\hat { U } \\hat { U } ^ { T } Y X ^ { T } ( X X ^ { T } ) ^ { - 1 }", + "type": "interline_equation", + "image_path": "2c637caf87bf5f7e7c88d70f1b222a800b4779f1de9bde81351cd8420ebf9d0d.jpg" + } + ] + } + ], + "index": 20, + "virtual_lines": [ + { + "bbox": [ + 247, + 405, + 363, + 421 + ], + "spans": [], + "index": 20 + } + ] + }, + { + "type": "text", + "bbox": [ + 105, + 423, + 386, + 435 + ], + "lines": [ + { + "bbox": [ + 104, + 421, + 383, + 438 + ], + "spans": [ + { + "bbox": [ + 104, + 421, + 312, + 438 + ], + "score": 1.0, + "content": "is the unique global minimum solution of (2) when", + "type": "text" + }, + { + "bbox": [ + 312, + 423, + 383, + 436 + ], + "score": 0.9, + "content": "k < \\operatorname* { m i n } \\{ d _ { x } , d _ { y } \\}", + "type": "inline_equation" + } + ], + "index": 21 + } + ], + "index": 21 + }, + { + "type": "title", + "bbox": [ + 107, + 448, + 229, + 460 + ], + "lines": [ + { + "bbox": [ + 105, + 446, + 230, + 461 + ], + "spans": [ + { + "bbox": [ + 105, + 446, + 230, + 461 + ], + "score": 1.0, + "content": "C.2 PROOF OF LEMMA 3.2", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 22 + }, + { + "type": "interline_equation", + "bbox": [ + 122, + 473, + 488, + 572 + ], + "lines": [ + { + "bbox": [ + 122, + 473, + 488, + 572 + ], + "spans": [ + { + "bbox": [ + 122, + 473, + 488, + 572 + ], + "score": 0.94, + "content": "\\begin{array} { l } { { \\displaystyle \\begin{array} { l } { { \\displaystyle L ( W _ { 1 } , \\ldots , W _ { i - 1 } , W _ { i } + \\Delta _ { i } , W _ { i + 1 } , \\ldots , W _ { H + 1 } ) } } \\\\ { { \\displaystyle = \\frac { 1 } { 2 } \\| W _ { H + 1 } \\cdot \\cdot \\cdot W _ { i + 1 } ( W _ { i } + \\Delta _ { i } ) W _ { i - 1 } \\cdot \\cdot \\cdot W _ { 1 } X - Y \\| _ { \\mathrm { F } } ^ { 2 } } } \\\\ { { \\displaystyle = \\frac { 1 } { 2 } \\| W _ { H + 1 } \\cdot \\cdot \\cdot W _ { 1 } X - Y + W _ { H + 1 } \\cdot \\cdot \\cdot W _ { i + 1 } \\Delta _ { i } W _ { i - 1 } \\cdot \\cdot \\cdot W _ { 1 } X \\| _ { \\mathrm { F } } ^ { 2 } } } \\\\ { { \\displaystyle = L ( W ) + \\mathrm { t r } ( ( W _ { H + 1 } \\cdot \\cdot \\cdot W _ { i + 1 } \\Delta _ { i } W _ { i - 1 } \\cdot \\cdot \\cdot W _ { 1 } X ) ^ { T } ( W _ { H + 1 } \\cdot \\cdot \\cdot W _ { 1 } X - Y ) ) + O ( \\| \\Delta _ { i } \\| _ { \\mathrm { F } } ^ { 2 } ) } } \\end{array} } } \\\\ { { \\displaystyle = L ( W ) + \\mathrm { t r } ( W _ { i + 1 } ^ { T } \\cdot \\cdot \\cdot W _ { H + 1 } ^ { T } ( W _ { H + 1 } \\cdot \\cdot \\cdot W _ { 1 } X - Y ) X ^ { T } W _ { 1 } ^ { T } \\cdot \\cdot \\cdot W _ { i - 1 } ^ { T } \\Delta _ { i } ^ { T } ) + O ( \\| \\Delta _ { i } \\| _ { \\mathrm { F } } ^ { 2 } ) . } } \\end{array}", + "type": "interline_equation", + "image_path": "b984de74cb582bbf3157bb91ef7d905c869b5d83ec778aa061324390086f5531.jpg" + } + ] + } + ], + "index": 24, + "virtual_lines": [ + { + "bbox": [ + 122, + 473, + 488, + 506.0 + ], + "spans": [], + "index": 23 + }, + { + "bbox": [ + 122, + 506.0, + 488, + 539.0 + ], + "spans": [], + "index": 24 + }, + { + "bbox": [ + 122, + 539.0, + 488, + 572.0 + ], + "spans": [], + "index": 25 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 575, + 236, + 587 + ], + "lines": [ + { + "bbox": [ + 106, + 575, + 236, + 587 + ], + "spans": [ + { + "bbox": [ + 106, + 575, + 236, + 587 + ], + "score": 1.0, + "content": "From this, we can conclude that", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 26 + }, + { + "type": "interline_equation", + "bbox": [ + 172, + 591, + 439, + 617 + ], + "lines": [ + { + "bbox": [ + 172, + 591, + 439, + 617 + ], + "spans": [ + { + "bbox": [ + 172, + 591, + 439, + 617 + ], + "score": 0.88, + "content": "\\frac { \\partial L } { \\partial W _ { i } } = W _ { i + 1 } ^ { T } \\cdot \\cdot \\cdot W _ { H + 1 } ^ { T } ( W _ { H + 1 } \\cdot \\cdot \\cdot W _ { 1 } X - Y ) X ^ { T } W _ { 1 } ^ { T } \\cdot \\cdot \\cdot W _ { i - 1 } ^ { T } .", + "type": "interline_equation", + "image_path": "945811bff5e4d9a7f126e0391c20bb777c5a03e8e3a8bbe7f0d31967fe4e51ae.jpg" + } + ] + } + ], + "index": 27, + "virtual_lines": [ + { + "bbox": [ + 172, + 591, + 439, + 617 + ], + "spans": [], + "index": 27 + } + ] + }, + { + "type": "title", + "bbox": [ + 107, + 627, + 229, + 640 + ], + "lines": [ + { + "bbox": [ + 105, + 626, + 230, + 641 + ], + "spans": [ + { + "bbox": [ + 105, + 626, + 230, + 641 + ], + "score": 1.0, + "content": "C.3 PROOF OF LEMMA 3.3", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 28 + }, + { + "type": "text", + "bbox": [ + 107, + 646, + 364, + 662 + ], + "lines": [ + { + "bbox": [ + 106, + 646, + 364, + 663 + ], + "spans": [ + { + "bbox": [ + 106, + 646, + 141, + 663 + ], + "score": 1.0, + "content": "1. Since", + "type": "text" + }, + { + "bbox": [ + 141, + 647, + 336, + 662 + ], + "score": 0.88, + "content": "A ^ { T } A \\succeq \\sigma _ { \\operatorname* { m i n } } ^ { 2 } ( A ) I , B ^ { T } A ^ { T } A B \\succeq \\sigma _ { \\operatorname* { m i n } } ^ { 2 } ( A ) B ^ { T } B", + "type": "inline_equation" + }, + { + "bbox": [ + 337, + 646, + 364, + 663 + ], + "score": 1.0, + "content": ". 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Y ) X ^ { T } W _ { 1 } ^ { T } \\cdot \\cdot \\cdot W _ { i - 1 } ^ { T } \\Delta _ { i } ^ { T } ) + O ( \\| \\Delta _ { i } \\| _ { \\mathrm { F } } ^ { 2 } ) . } } \\end{array}", + "type": "interline_equation", + "image_path": "b984de74cb582bbf3157bb91ef7d905c869b5d83ec778aa061324390086f5531.jpg" + } + ] + } + ], + "index": 24, + "virtual_lines": [ + { + "bbox": [ + 122, + 473, + 488, + 506.0 + ], + "spans": [], + "index": 23 + }, + { + "bbox": [ + 122, + 506.0, + 488, + 539.0 + ], + "spans": [], + "index": 24 + }, + { + "bbox": [ + 122, + 539.0, + 488, + 572.0 + ], + "spans": [], + "index": 25 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 575, + 236, + 587 + ], + "lines": [ + { + "bbox": [ + 106, + 575, + 236, + 587 + ], + "spans": [ + { + "bbox": [ + 106, + 575, + 236, + 587 + ], + "score": 1.0, + "content": "From this, we can conclude that", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 26, + "bbox_fs": [ + 106, + 575, + 236, + 587 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 172, + 591, + 439, + 617 + ], + "lines": [ + { + "bbox": [ + 172, + 591, + 439, + 617 + ], + "spans": [ + { + "bbox": [ + 172, + 591, + 439, + 617 + ], + "score": 0.88, + "content": "\\frac { \\partial L } { \\partial W _ { i } } = W _ { i + 1 } ^ { T } \\cdot \\cdot \\cdot W _ { H + 1 } ^ { T } ( W _ { H + 1 } \\cdot \\cdot \\cdot W _ { 1 } X - Y ) X ^ { T } W _ { 1 } ^ { T } \\cdot \\cdot \\cdot W _ { i - 1 } ^ { T } .", + "type": "interline_equation", + "image_path": "945811bff5e4d9a7f126e0391c20bb777c5a03e8e3a8bbe7f0d31967fe4e51ae.jpg" + } + ] + } + ], + "index": 27, + "virtual_lines": [ + { + "bbox": [ + 172, + 591, + 439, + 617 + ], + "spans": [], + "index": 27 + } + ] + }, + { + "type": "title", + "bbox": [ + 107, + 627, + 229, + 640 + ], + "lines": [ + { + "bbox": [ + 105, + 626, + 230, + 641 + ], + "spans": [ + { + "bbox": [ + 105, + 626, + 230, + 641 + ], + "score": 1.0, + "content": "C.3 PROOF OF LEMMA 3.3", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 28 + }, + { + "type": "text", + "bbox": [ + 107, + 646, + 364, + 662 + ], + "lines": [ + { + "bbox": [ + 106, + 646, + 364, + 663 + ], + "spans": [ + { + "bbox": [ + 106, + 646, + 141, + 663 + ], + "score": 1.0, + "content": "1. 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"height": 2200 + } + } +] \ No newline at end of file diff --git a/parse/train/BkgrBgSYDS/BkgrBgSYDS.md b/parse/train/BkgrBgSYDS/BkgrBgSYDS.md new file mode 100644 index 0000000000000000000000000000000000000000..002919d74bc50879ca8d3f46ee19d75213f2bd54 --- /dev/null +++ b/parse/train/BkgrBgSYDS/BkgrBgSYDS.md @@ -0,0 +1,1353 @@ +# KALEIDOSCOPE: AN EFFICIENT, LEARNABLE REPRESENTATION FOR ALL STRUCTURED LINEAR MAPS + +Tri Dao 1, Nimit Sharad Sohoni∗2, Albert $\mathbf { G u } ^ { * 1 }$ , Matthew Eichhorn 3, Amit Blonder 4, Megan Leszczynski 1, Atri Rudra 4, Christopher Ré 1 + +1 Department of Computer Science, Stanford University +2 Institute for Computational and Mathematical Engineering, Stanford University +3 Center for Applied Mathematics, Cornell University +4 Department of Computer Science and Engineering, University at Buffalo, The State University of New York +{trid,nims,albertgu}@stanford.edu, mae226@cornell.edu, amitblon@buffalo.edu, +mleszczy@stanford.edu, atri@buffalo.edu, chrismre@cs.stanford.edu + +# ABSTRACT + +Modern neural network architectures use structured linear transformations, such as low-rank matrices, sparse matrices, permutations, and the Fourier transform, to improve inference speed and reduce memory usage compared to general linear maps. However, choosing which of the myriad structured transformations to use (and its associated parameterization) is a laborious task that requires trading off speed, space, and accuracy. We consider a different approach: we introduce a family of matrices called kaleidoscope matrices (K-matrices) that provably capture any structured matrix with near-optimal space (parameter) and time (arithmetic operation) complexity. We empirically validate that K-matrices can be automatically learned within end-to-end pipelines to replace hand-crafted procedures, in order to improve model quality. For example, replacing channel shuffles in ShuffleNet improves classification accuracy on ImageNet by up to $5 \%$ . K-matrices can also simplify hand-engineered pipelines—we replace filter bank feature computation in speech data preprocessing with a learnable kaleidoscope layer, resulting in only $0 . 4 \%$ loss in accuracy on the TIMIT speech recognition task. In addition, K-matrices can capture latent structure in models: for a challenging permuted image classification task, a K-matrix based representation of permutations is able to learn the right latent structure and improves accuracy of a downstream convolutional model by over $9 \%$ . We provide a practically efficient implementation of our approach, and use K-matrices in a Transformer network to attain $36 \%$ faster end-to-end inference speed on a language translation task. + +# 1 INTRODUCTION + +Structured linear maps are fundamental and ubiquitous in modern machine learning. Their efficiency in speed (fast algorithms) and space (few parameters) can reduce computation and memory usage. The class of structured linear maps includes fixed specialized transforms such as the discrete Fourier transform (DFT) and Hadamard transform used in signal processing (Cooley et al., 1969), convolutions for image, language, and speech modeling (Gu et al., 2018), and low-rank and sparse matrices for efficient storage and inference on edge devices (Yu et al., 2017). Forms of structure such as sparsity have been at the forefront of recent advances in ML (Frankle & Carbin, 2019), and are critical for on-device and energy-efficient models, two application areas of tremendous recent interest (Tsidulko, 2019; Schwartz et al., 2019). + +There are a plethora of classes of structured linear maps, each with a significantly different representation, algorithm, and implementation. They have different tradeoffs in terms of inference speed, training speed, and accuracy, and the conventional wisdom is that no one class works uniformly well across all applications. As a result, ML practitioners currently hand-pick specific classes of structured linear maps for each of their applications. This is a difficult and labor-intensive task. + +Ideally, these problems should be addressed with a universal representation for structured linear maps: (i) Such a parameterization should be expressive enough to capture important classes of structure, with a nearly tight parameter count and runtime: the space required to represent the linear map should be close to optimal, and the resulting algorithm for matrix vector multiplication should be close to the fastest possible algorithm. (ii) The parameterization should be differentiable in order to be learned as a component of end-to-end ML pipelines, enabling it to easily be used as a drop-in replacement for manually engineered structured components. (iii) The parameterization should admit practically efficient algorithms for training and inference, in terms of both speed and memory. + +Currently, no class of structured linear maps satisfies all of these criteria. Most existing classes of structured matrices—such as the class of low-rank matrices—fail to tightly capture other important types of structure. For example, the DFT has an efficient structured representation of size $O ( n \log n )$ , yet cannot be well-approximated by a low-rank transform of size $\dot { \ll } n ^ { 2 }$ . Another important type of structure is sparsity; lots of exciting recent work has focused on the design of sparse neural networks. For instance, sparse networks of comparable quality to their dense counterparts—yet an order of magnitude fewer parameters—may be created via pruning (Han et al., 2016) or by identifying “winning lottery tickets” (Frankle & Carbin, 2019). In parallel, recent theoretical results by De Sa et al. (2018) show that sparsity and the notion of structure in linear maps are fundamentally linked: any given matrix can be factored into a product of sparse matrices with total parameter count equal to the efficiency (i.e. minimum arithmetic circuit complexity) of the matrix. In other words, the representation of linear maps as products of sparse matrices tightly captures all forms of structure. Unfortunately, it is difficult to actually learn these sparse factorizations, because it requires finding the sparsity patterns of the factors—a discrete, nondifferentiable search problem. Thus, current methods for training sparse neural networks are either expensive (Frankle & Carbin, 2019) or rely on highly hand-tuned heuristics for evolving the sparsity patterns throughout training (Dettmers & Zettlemoyer, 2019). + +By contrast, we propose a representation of linear maps as products of sparse matrices with specific predefined sparsity patterns (Section 2), and show that it does satisfy our desiderata: it retains the expressiveness of unstructured sparsity, while being differentiably learnable and efficient like other structured representations. Concretely, our representation is based on products of a particular building block known as a butterfly matrix (Parker, 1995; Dao et al., 2019); we term such products kaleidoscope matrices (K-matrices for short).1 (i) Our main theoretical contribution (Section 2.3) concerns the expressiveness of this representation: we show that any structured linear map (i.e. one that can be applied using $s \ll n ^ { 2 }$ arithmetic operations) can be represented as a K-matrix, with a nearly tight number of parameters and algorithmic complexity (both on the order of $s$ up to logarithmic factors). (ii) The kaleidoscope representation is fully differentiable; thus, all the parameters of a K-matrix can be learned using standard optimization algorithms such as SGD. (iii) Because of their simple, regular structure, K-matrices are practical and easy to use. We provide memory- and runtime-efficient implementations of K-matrix multiplication on CPU and GPU for training and inference, with a simple PyTorch interface. + +We empirically validate that, due to their expressiveness, learnability, and efficiency, we can use K-matrices as a drop-in replacement for linear components in deep learning models. In Section 3.1, we use K-matrices to replace hand-crafted structure in two different settings. We simplify the six steps of filter bank computation in speech preprocessing into a single learnable K-matrix step, with only an $0 . 4 \%$ accuracy drop on the TIMIT speech recognition task. We use K-matrices to replace channel shuffles in ShuffleNet, improving ImageNet classification accuracy by up to $5 \%$ . In Section 3.2, we show that K-matrices can successfully recover latent structure; a K-matrix is used to learn latent permutations in a permuted image dataset (Permuted CIFAR), resulting in 9 points higher accuracy in a downstream CNN model. In Section 3.3, we show that our efficient K-matrix multiplication implementation can be applied to speed up real-world tasks: we replace linear layers with K-matrices in a DynamicConv-Transformer network to attain $36 \%$ faster end-to-end inference speed with a 1.0 drop in BLEU score on the IWSLT14 German English translation task. + +We first present some background on the characterization of all structured matrices (i.e. those with subquadratic multiplication algorithms) as products of sparse factors, along with the definition of butterfly matrices. We then propose a differentiable family of kaleidoscope matrices, composed of products of butterfly matrices, and prove their expressivity: all structured matrices can be represented in this form, with almost optimal parameter count and runtime. + +# 2.1 BACKGROUND: SPARSE FACTORIZATION, BUTTERFLY MATRICES + +Sparse factorization One method of constructing matrices with theoretically fast matrix-vector multiplication algorithms is as a product of sparse matrices, so that multiplication by an arbitrary vector has cost proportional to the total number of nonzeros (NNZ) of the matrices in the product. Surprisingly, the converse is also true. De Sa et al. (2018) introduce the concept of sparse product width (SPW), which roughly corresponds to the total NNZ in a factorization of a matrix, and show that it is an asymptotically optimal descriptor of the algorithmic complexity of matrix-vector multiplication (Bürgisser et al., 2013). We use a similar argument in the proof of our main theorem (Section 2.3). However, attempting to learn such a factorization of a given matrix is difficult, as the sparsity constraint is not continuous. Moreover, because of the possibly irregular sparsity patterns, it is difficult to realize the theoretical speedups in practice (Gray et al., 2017; Gahvari et al., 2007). + +Butterfly matrices Butterfly matrices, encoding the recursive divide-and-conquer structure of the fast Fourier transform (FFT) algorithm, have long been used in numerical linear algebra (Parker, 1995; Li et al., 2015) and machine learning (Mathieu & LeCun, 2014; Jing et al., 2017; Munkhoeva et al., 2018; Dao et al., 2019; Choromanski et al., 2019). Here we define butterfly matrices, which we use as a building block for our hierarchy of kaleidoscope matrices. + +Definition 2.1. A butterfly factor of size $k \geq 2$ (denoted as $\mathbf { B } _ { k }$ ) is a matrix of the form $\mathbf { B } _ { k } \ =$ $\left[ \begin{array} { c c } { \mathbf { D } _ { 1 } } & { \mathbf { D } _ { 2 } } \\ { \mathbf { D } _ { 3 } } & { \mathbf { D } _ { 4 } } \end{array} \right]$ where each $\mathbf { D } _ { i }$ is a ${ \begin{array} { l } { { \frac { k } { 2 } } \times { \frac { k } { 2 } } } \end{array} }$ diagonal matrix. We restrict $k$ to be a power of 2. + +Definition 2.2. A butterfly factor matrix of size n with block size $k$ (denoted as $\mathbf { B } _ { k } ^ { \left( n \right) }$ ) is a block diagonal matrix of $\frac { n } { k }$ (possibly different) butterfly factors of size $k$ : + +$$ +\mathbf B _ { k } ^ { ( n ) } = \operatorname { d i a g } \left( \left[ \mathbf B _ { k } \right] _ { 1 } , \left[ \mathbf B _ { k } \right] _ { 2 } , \ldots , \left[ \mathbf B _ { k } \right] _ { \frac n k } \right) +$$ + +product of butterfly factor matrices: B(n) = B(n)n B(n)n2 . Definition 2.3. A butterfly matrix of size $n$ (denoted as $\mathbf { B } ^ { ( n ) }$ ) is a matrix that can be expressed as a . Equivalently, we may define $\mathbf { B } ^ { ( n ) }$ recursively as a matrix that can be expressed in the following form: + +$$ +\mathbf { B } ^ { ( n ) } = \mathbf { B } _ { n } ^ { ( n ) } \left[ { \big [ } { \mathbf { B } ^ { ( \frac { n } { 2 } ) } } { \big ] } _ { 1 } \right. \qquad 0 \qquad { \big ] } +$$ + +(Note that $\big [ \mathbf { B } ^ { ( \frac { n } { 2 } ) } \big ] _ { 1 }$ and $\big [ \mathbf { B } ^ { ( \frac { n } { 2 } ) } \big ] _ { 2 }$ may be different.) + +# 2.2 THE KALEIDOSCOPE HIERARCHY + +Using the building block of butterfly matrices, we formally define the kaleidoscope $( B B ^ { * } )$ hierarchy and prove its expressiveness. This class of matrices serves as a fully differentiable alternative to products of sparse matrices (Section 2.1), with similar expressivity. In Appendix J, we show where various common structured matrix classes are located within this hierarchy. + +The building block for this hierarchy is the product of a butterfly matrix and the (conjugate) transpose of another butterfly matrix (which is simply a product of butterfly factors taken in the opposite order). Figure 1 visualizes the sparsity patterns of the butterfly factors in $B B ^ { * }$ , where the red and blue dots represent the allowed locations of nonzero entries. + +Definition 2.4 (Kaleidoscope hierarchy, kaleidoscope matrices). + +• Define $\boldsymbol { B }$ as the set of all matrices that can be expressed in the form $\mathbf { B } ^ { ( n ) }$ (for some $n$ ). • Define $B B ^ { * }$ as the set of matrices M of the form $\mathbf { M } = \mathbf { M } _ { 1 } \mathbf { M } _ { 2 } ^ { * }$ for some $\mathbf { M } _ { 1 } , \mathbf { M } _ { 2 } \in B$ + +![](images/f75bf8d4d0ee21f357fac50dfdf80482c8af5cd021491925696495d635d08d3a.jpg) +Figure 1: Visualization of the fixed sparsity pattern of the building blocks in $B B ^ { * }$ , in the case $n = 1 6$ The red and blue dots represent all the possible locations of the nonzero entries. + +• Define $( B B ^ { * } ) ^ { w }$ as the set of matrices M that can be expressed as $\mathbf { M } = \mathbf { M } _ { w } \ldots \mathbf { M } _ { 2 } \mathbf { M } _ { 1 }$ , with each $\mathbf { M } _ { i } \in B B ^ { * }$ $1 \leq i \leq w )$ . (The notation w represents width.) +• Define $( B B ^ { * } ) _ { e } ^ { w }$ as the set of $n \times n$ matrices M that can be expressed as ${ \bf M } = { \bf S } { \bf E } { \bf S } ^ { T }$ for some $e n \times e n$ matrix $\mathbf { E } \in ( B B ^ { \ast } ) ^ { w }$ , where $\mathbf { S } \in \mathbb { F } ^ { n \times e n } = [ \mathbf { I } _ { n } \quad 0 \quad \ldots \quad 0 ]$ (i.e. M is the upper-left corner of $\mathbf { E }$ ). (The notation e represents expansion relative to $n$ .) +• M is a kaleidoscope matrix, abbreviated as $\pmb { K }$ -matrix, if $M \in ( B B ^ { * } ) _ { e } ^ { w }$ for some w and e. + +The kaleidoscope hierarchy, or $( B B ^ { * } )$ hierarchy, refers to the families of matrices $( B B ^ { * } ) _ { e } ^ { 1 } \subseteq$ $( B B ^ { * } ) _ { e } ^ { 2 } \subseteq \dots$ , for a fixed expansion factor $e$ . Each butterfly matrix can represent the identity matrix, so $( B B ^ { * } ) _ { e } ^ { w } ~ \subseteq ~ ( B B ^ { * } ) _ { e } ^ { \bar { w } + 1 }$ . We show that the inclusion is proper in Appendix E. This hierarchy generalizes the $B \mathcal { P }$ hierarchy proposed by Dao et al. (2019), as shown in Appendix J. + +Efficiency in space and speed Each matrix in $( B B ^ { * } ) _ { e } ^ { w }$ is a product of $2 w$ total butterfly matrices and transposes of butterfly matrices, each of which is in turn a product of $\log ( n e )$ factors with $2 n e$ nonzeros (NNZ) each. Therefore, each matrix in $( B B ^ { * } ) _ { e } ^ { w }$ has $4 w n e \log ( n e )$ parameters and a matrix-vector multiplication algorithm of complexity $O ( w n e \log n e )$ (by multiplying the vector with each sparse factor sequentially). We prove this more formally in Appendix E. For the applications in Section 3, $w$ and $e$ are small constants (up to 2), so those $\mathrm { K }$ -matrices have $O ( n \log n )$ parameters and runtime. + +# 2.3 ALL LOW-DEPTH STRUCTURED MATRICES ARE IN THE KALEIDOSCOPE HIERARCHY + +We now present our main theoretical result: the fact that general linear transformations, expressed as low-depth linear arithmetic circuits, are captured in the $B B ^ { * }$ hierarchy with low width. Arithmetic circuits are commonly used to formalize algebraic algorithmic complexity (Bürgisser et al., 2013); we include a primer on this in Appendix M. The quantities of interest are the total number of gates in the circuit, representing the total number of steps required to perform the algorithm for a serial processor, and the depth, representing the minimum number of steps required for a parallel processor. + +Theorem 1. Let M be an $n \times n$ matrix such that multiplication of M times an arbitrary vector v can be represented as a linear arithmetic circuit with s total gates and depth d. Then, M ∈ (BB∗)O(d)O( sn ). + +The representation of such a matrix $\mathbf { M }$ in the $B B ^ { * }$ hierarchy has $O ( d s \log s )$ parameters and yields a $O ( d s \log s )$ multiplication algorithm, compared to the $O ( s )$ parameters and runtime of the circuit representation. To the best of our knowledge, the most general classes of efficient matrices that have been studied (De Sa et al., 2018) have depth $d$ on the order of $\log n$ or poly $\log n$ . In these cases, the representation with K-matrices matches the best known bounds up to polylogarithmic factors. + +The crux of the proof of Theorem 1 (shown in Appendix F) is the construction of an almost tight representation of any sparse matrix as a $\mathbf { K }$ -matrix (i.e. a product of butterfly matrices): specifically, we show that any $n \times n$ sparse matrix with $s$ nonzeros is in $( B B ^ { * } ) _ { O ( 1 ) } ^ { O ( \left\lceil \frac { s } { n } \right\rceil ) }$ (Theorem 3, Appendix I). We then leverage the expressivity result of products of sparse matrices to represent all arithmetic circuits (similar to the sparse product width result of De Sa et al. (2018) referenced in Section 2.1) to complete the proof of Theorem 1. + +This intermediate result is also a novel characterization of sparse matrices. For a matrix with $s$ NNZ, the kaleidoscope representation has $O ( s \log { n } )$ parameters and runtime, instead of the optimal $O ( s )$ parameters and runtime; so, we trade off an extra logarithmic factor in space and time for full differentiability (thanks to the fixed sparsity patterns in the representation). The intuition behind the result is as follows: a sparse matrix with $s$ NNZ can be written as a sum of $\lceil s / n \rceil$ matrices each with at most $n$ NNZ. Any $n \times n$ matrix with at most $n$ NNZ, up to permuting the rows and columns, is a product of two butterfly matrices (Lemma I.1). Sorting networks (Knuth, 1997) imply that permutation matrices are in $( \tilde { B B ^ { * } } ) ^ { O ( \log n ) }$ , but we tighten the result to show that they are in fact in $B B ^ { * }$ (Theorem 2, Appendix $\mathbf { G }$ ). We thus obtain a kaleidoscope representation for each summand matrix with $O ( n \log n )$ parameters. By the addition closure property of the $B B ^ { * }$ hierarchy (Lemma H.5), each sparse matrix with $s$ NNZ then has a kaleidoscope representation with $O ( s \log { n } )$ parameters. + +Tight representation for structured linear maps common in ML Even though Theorem 1 suggests that the kaleidoscope representation can be loose by logarithmic factors, many structured linear maps common in ML can be represented in this hierarchy with an optimal number of parameters and runtime compared to the best known parameterizations, up to constant factors. Appendix J includes several examples such as discrete transforms (the DFT, discrete cosine transform (DCT), discrete sine transform (DST), and Hadamard transform), convolution (i.e. circulant matrices), Toeplitz matrices (Gray, 2006), structured matrices for kernel approximation $( ( H D ) ^ { 3 }$ (Yu et al., 2016)) and compact neural network design (Fastfood (Le et al., 2013), ACDC (Moczulski et al., 2016)). There have been other large classes of structured matrices proposed in the machine learning literature, such as Toeplitz-like (Sindhwani et al., 2015) and low displacement rank (LDR) (Thomas et al., 2018), but they are not known to be able to capture these common structures as tightly as K-matrices can. More detailed discussions are in Appendix A. + +# 2.4 EXTENSIONS + +ReLU networks with low-depth structured weight matrices In Appendix L, we prove that finding an efficient circuit for a ReLU network can be reduced to finding efficient circuits for each of its weight matrices, with at most a constant factor greater size and run-time (i.e. number of gates). We also show that ReLU networks with kaleidoscope weight matrices have near-linear VC dimension in the number of parameters, matching the bound for networks with unconstrained weight matrices (Bartlett et al., 1999; Harvey et al., 2017) and LDR (Thomas et al., 2018). This yields a corresponding sample complexity bound. + +Orthogonal kaleidoscope hierarchy Orthogonal butterfly matrices are one commonly used variant due to their improved stability (Parker, 1995), where each butterfly factor is constrained to be orthogonal: $\left[ \begin{array} { l l } { \mathbf { C } } & { \mathbf { s } } \\ { - \mathbf { S } } & { \mathbf { C } } \end{array} \right]$ with $\mathbf { C } , \mathbf { S }$ being diagonal and $\mathbf { C } ^ { 2 } + \mathbf { S } ^ { 2 } = \mathbf { I }$ . Similar to the $B B ^ { * }$ hierarchy, in Appendix K, we define the OBB hierarchy consisting of products of orthogonal butterfly matrices and diagonal matrices, and show that this hierarchy has the same expressiveness as the $B B ^ { * }$ hierarchy. + +# 3 EMPIRICAL EVALUATION + +We validate three claims that suggest that kaleidoscopes are a promising technique to learn different types of structure in modern architectures. + +1. Section 3.1: for applications in speech and lightweight computer vision relying on highly hand-crafted structured transformations, we show that we can recover—and even improve—the quality of such architectures by simply replacing existing hand-structured components with K-matrices, with only a small overhead in memory and computation. +2. In Section 3.2, for a challenging task with latent structure (Permuted CIFAR-10), a K-matrixbased relaxation of permutations is able to learn the right latent permutation, yielding 9 points better accuracy in a downstream CNN compared to standard RNN and CNN baselines used on such permuted image classification tasks. +3. In Section 3.3, we show that, although not yet highly optimized, our current implementation of K-matrices can improve the inference throughput of DynamicConv Transformer, a state-ofthe-art fast machine translation model, by $36 \%$ , with only a relatively small drop in translation quality. + +In all of the above applications, as K-matrices are fully differentiable, we simply train them jointly with the rest of the model using standard learning algorithms (such as SGD). Full details for all of the experiments (precise architectures, hyperparameters, etc.) are in Appendix $\boldsymbol { \mathrm { ~ B ~ } ^ { 2 } }$ . + +# 3.1 REPLACING HAND-CRAFTED STRUCTURES + +We validate that kaleidoscope matrices can recover or improve on the performance of hand-crafted structure in ML models. For example, a single learnable kaleidoscope layer can be used to replace the hand-engineered filter bank speech preprocessing pipeline with only $0 . 4 \%$ loss in accuracy on the TIMIT speech recognition task (Section 3.1.1). Replacing channel shuffles in ShuffleNet with learnable K-matrices improves classification accuracy on ImageNet by up to $5 . 0 \%$ (Section 3.1.2). + +# 3.1.1 SPEECH PREPROCESSING + +![](images/84b8117d4d1439b40cfd9f8f6af50403ab20bc6079dd9c46a5e034984993d116.jpg) +Figure 2: Comparison of the standard MFSC featurization pipeline with our “kaleidoscope” pipeline. + +We show that K-matrices can remove the need for hand-tuning by significantly simplifying speech recognition data preprocessing pipelines. In particular, we can entirely replace the complex handcrafted MFSC featurization commonly used in speech recognition tasks with a fully learnable kaleidoscope layer, with only $0 . 4 \%$ drop in accuracy on the TIMIT speech recognition benchmark. Results are presented in Table 1. Our approach is competitive with the accuracy of standard models that use hand-crafted features, and significantly outperforms current approaches for learning from raw audio input. + +Table 1: TIMIT phoneme error rate $( \mathrm { P E R } \% )$ for different methods. Our kaleidoscope, raw-input version of the model (row 3) performs competitively with the original model trained on MFSC features (row 1), with only an $0 . 4 \%$ drop in PER. It significantly outperforms existing approaches that learn from raw audio, i.e. without handcrafted featurization (e.g. SincNet [row 2], which to our knowledge attains the previous state-of-the-art for learning from raw audio), and is only $0 . 8 \%$ less accurate than the overall state-of-the-art on TIMIT.3 Additional comparisons are given in Appendix B.1. + +
MethodTest set PER%Raw audio input
MFSC features +LSTM14.2x//
SincNet (Ravanelli et al., 2019)17.2
Kaleidoscope + LSTM14.6
+ +Modern speech recognition models currently rely on carefully hand-crafted features extracted from the audio, which are then fed into an acoustic model. By contrast, learning directly from the raw audio—i.e. end-to-end learning from the audio waveform without any manual featurization—obviates the need for this complicated and often expensive preprocessing step. There have been recent attempts to learn directly from raw audio, such as SincNet (Ravanelli & Bengio, 2018); however, they often rely on specialized architectures designed by domain experts. Instead, we use a standard RNN speech recognition architecture, but use a learnable kaleidoscope layer to replace the featurization steps. + +The baseline architecture takes as input filter bank (MFSC) features, which are a popular standard featurization for speech recognition (Paliwal, 1999) and involve several steps hand-crafted specifically for this domain. These features are extracted from the raw audio waveform, and fed as the input into a Bi-LSTM model. We significantly simplify this pipeline by replacing the featurization step with a trainable kaleidoscope layer that is trained end-to-end together with the Bi-LSTM. The original pipeline and our modified kaleidoscope version are depicted in Figure 2. + +The computation of MFSC features involves a series of painstakingly hand-designed steps (further described in Appendix B.1), each involving their own hyperparameters: (i) the waveform is framed (split into chunks), (ii) the waveform is dithered (noise is added), (iii) pre-emphasis is applied, (iv) the Hamming window is applied, (v) the FFT is applied and the power spectrum is computed, (vi) the result is mapped to the mel scale (which involves applying a particular linear transformation and then taking the logarithm of the result), (vii) cepstral mean and variance normalization is applied. We replace the last six steps (ii-vii) of this featurization process with a learnable kaleidoscope layer; specifically, after windowing, we multiply the input by a K-matrix, and then compute the logarithm of the power spectrum; the output is fed into the Bi-LSTM model. + +# 3.1.2 REPLACING CNN CHANNEL SHUFFLE + +We evaluate how K-matrices can improve the quality of hand-crafted, lightweight architectures for computer vision tasks, without the need for hand-tuning. We select ShuffleNet (Zhang et al., 2018), which is a state-of-the-art lightweight CNN architecture that uses a manually designed “channel shuffle” permutation matrix to improve performance. By replacing this fixed permutation with a learnable K-matrix, we achieve up to $5 \%$ further improvement in classification accuracy, without hand-tuned components and with a modest space penalty of up to $10 \%$ . Results are given in Table 2. + +Table 2: Top-1 classification accuracy of ShuffleNet on ImageNet validation set (parameter counts in parentheses). We compare our approach (col. 3) with our reimplementation of ‘vanilla’ ShuffleNet (col. 1) and a recent approach based on the Hadamard transform (col. 2).4 We report results for different network width multipliers (# channels). The last column shows the differences in accuracy and parameter count between our approach and vanilla ShuffleNet; using a learnable K-matrix in place of each fixed permutation (shuffle) or Hadamard matrix improves accuracy by up to $5 \%$ . + +
ShuffleHadamardKaleidoscope (K.)K. vs. Shuffle
0.25 ShuffleNet g844.1% (0.46M)43.9% (0.46M)49.2% (0.51M)+5.0% (+0.05M)
0.5 ShuffleNet g857.1% (1.0M)56.2% (1.0M)59.5% (1.1M)+2.4% (+0.1M)
1.0 ShuffleNet g865.3% (2.5M)65.0% (2.5M)66.5% (2.8M)+1.2% (+0.2M)
+ +Grouped convolution (Krizhevsky et al., 2012) is often used to reduce parameter count and speed up inference compared to standard convolution, but, by default, channels in different groups cannot exchange information. To remedy this, ShuffleNet uses a permutation matrix to shuffle the channels after each grouped convolution. Zhao et al. (2019) propose to instead use the Hadamard transform before and after each grouped convolution to mix the channels. In place of these hand-engineered solutions, we use a K-matrix before and after each grouped convolution, and learn these end-to-end together with the rest of the network. As shown in Table 2, across a range of sizes, replacing the channel shuffles with K-matrices results in improved performance at comparable parameter counts. + +# 3.2 LEARNING A LATENT PERMUTATION + +We show that K-matrices can be used in a challenging task for which existing classes of structured linear maps have not been found suitable. We investigate the problem of image classification on a permuted image dataset (Permuted CIFAR-10). This problem is challenging due to the discrete nature of learning the latent permutation of the dataset; we present a differentiable relaxation for this using a K-matrix as a key component. Results are presented in Table 3; compared to methods that do not have a permutation learning step, our approach gets 9 points higher accuracy $8 4 . 4 \%$ to $9 3 . 6 \%$ ), coming within 2 points of the accuracy on the un-permuted dataset $( 9 4 . 9 \% )$ . + +Table 3: Permuted CIFAR-10 validation set classification accuracy $( \% )$ . Our kaleidoscope layer is able to nearly perfectly recover the latent structure, allowing a downstream CNN to approach the accuracy of a standard ResNet18 on the unpermuted dataset (last column). + +
ModelFCRNNCNNDense +CNNK +CNN Unpermuted
Accuracy61.257.873.784.493.694.9
+ +In this task, we use a permuted image classification dataset (Permuted CIFAR-10), wherein a fixed global permutation is applied to the pixels of every image in the original input set. Typically, only fully-connected (FC) and recurrent models are applied to such datasets (Le et al., 2015), because the permutation destroys locality in the image, presenting a difficulty for CNNs. However, CNNs are much better-suited for standard image tasks. We thus expect that learning the permutation and then applying a standard CNN should outperform these baselines. As mentioned in Section 2, the kaleidoscope hierarchy provides a nearly tight parameterization of permutations; this makes them a natural fit for the permutation learning step. + +Experimentally, we use a K-matrix to represent a distribution over permutations, which converges to a single permutation at the end of training. The correct latent structure is learned by applying samples from this distribution to the permuted training images, and minimizing an auxiliary smoothness-based loss that encourages the reconstructed images to be more “natural” (i.e. vary smoothly pixel-to-pixel). The learned permutation is evaluated by training a ResNet18 with the K-matrix permutation layer inserted at the beginning. Full details of our approach are provided in Appendix B.3. + +In Table 3, we compare our approach to a ResNet18 without this extra K-matrix layer, a ResNet18 with an extra dense matrix at the beginning instead of a K-matrix, and other baselines. As generic representations such as unstructured matrices do not have the requisite properties to fit in the pipeline, these baselines fail to effectively learn the latent permutation. We emphasize that a K-matrix provides this ability to recover latent structure despite not being specialized for permutations. Figure 3 describes the pipeline and displays examples of permuted and unpermuted images. + +![](images/413753fc1737fb267ca8776a147dae87b59ab479d8f3db8b527eccd5a0dbb78b.jpg) + +![](images/e9a1d672dbf9a7bf87c5651a9c2850de7f5858227bf8566efc1af6180d95ea50.jpg) +Figure 3: (a) (Left) Schematic describing permutation learning approach. The inputs are multiplied by a K-matrix and then fed into a CNN, from which the classification loss is computed. Separately, the input is permuted by a permutation matrix sampled from the distribution described by the Kmatrix, and a “smoothness” loss (Rudin et al., 1992) is computed from the result, as described in Appendix B.3. (b) (Right) Left panel: original (unpermuted) example images. Center panel: the permuted versions. Right panel: these images after then applying the permutation recovered by the K-matrix. The K-matrix is able to nearly unscramble the images into their unpermuted versions. + +# 3.3 SPEEDING UP INFERENCE + +We evaluate the inference speed benefit of using K-matrices on a real language translation model. We choose the state-of-the-art DynamicConv Transformer translation model (Wu et al., 2019), which offers $20 \%$ inference speedup over the standard Transformer model, and replace dense matrices in the decoder’s linear layers with K-matrices, which leads to a further $36 \%$ inference speedup (Table 4). + +As outlined in Section 2.3, K-matrices admit a simple and fast $O ( n \log n )$ matrix-vector multiplication algorithm. We provide fast implementations of this algorithm in $\mathrm { C } { + } { + }$ and CUDA, with an interface to PyTorch (Paszke et al., 2017), and use this implementation in our experiments. + +Table 4: Inference speed on the IWSLT-14 German-English translation task (test set). Using Kmatrices instead of dense matrices in the DynamicConv decoder linear layers results in $36 \%$ faster inference speed (measured on a single-threaded CPU with a batch size of 1 and beam size of 1). + +
Model# paramsBLEUSentences/secTokens/sec
Transformer (Vaswani et al., 2017)43M34.43.066.4
DynamicConv Transformer (Wu et al., 2019)39M35.23.680.2
DynamicConv Transformer w/ K-matrices (ours)30M34.24.9103.4
+ +We use K-matrices to replace all the linear layers in the decoder of DynamicConv (since $90 \%$ of inference time is spent in the decoder). As shown in Table 4, on the IWSLT-14 German-English translation task, this yields a $2 5 \%$ smaller model with $36 \%$ faster inference time on CPU, at the cost of 1.0 drop in BLEU score.5 (Our model also nearly matches the state-of-the-art BLEU performance of 2 years ago obtained by the Transformer model (Vaswani et al., 2017), despite being over $60 \%$ faster for inference than the Transformer.) The majority $( 5 5 \% )$ of inference time is spent in matrix-vector multiplication; our implementation of K-matrix-vector multiplication is about 2 times faster than the optimized implementation of dense matrix-vector multiplication in the Intel MKL library. Direct comparisons of K-matrix multiplication with this and other highly-optimized routines such as the FFT are further detailed in Appendix C. + +# 4 CONCLUSION + +We address the problem of having to manually choose among the numerous classes of structured linear maps by proposing the universal (expressive, efficient, and learnable) family of kaleidoscope matrices. We prove that K-matrices can represent any structured linear maps with near-optimal space and time complexity. Empirical validations suggest that K-matrices are a promising and flexible way to employ structure in modern ML; they can be used to reduce the need for hand-engineering, capture challenging latent structure, and improve efficiency in models. We are excited about future work on further hardware-optimized implementations of K-matrices, to fully realize the size and speed benefits of structured matrices on a broad array of real-world applications. + +# ACKNOWLEDGMENTS + +We thank Avner May and Jian Zhang for their helpful feedback. + +We gratefully acknowledge the support of DARPA under Nos. FA87501720095 (D3M), FA86501827865 (SDH), and FA86501827882 (ASED); NIH under No. U54EB020405 (Mobilize), NSF under Nos. CCF1763315 (Beyond Sparsity), CCF1563078 (Volume to Velocity), and 1937301 (RTML); ONR under No. N000141712266 (Unifying Weak Supervision); the Moore Foundation, NXP, Xilinx, LETI-CEA, Intel, IBM, Microsoft, NEC, Toshiba, TSMC, ARM, Hitachi, BASF, Accenture, Ericsson, Qualcomm, Analog Devices, the Okawa Foundation, American Family Insurance, Google Cloud, Swiss Re, and members of the Stanford DAWN project: Teradata, Facebook, Google, Ant Financial, NEC, VMWare, and Infosys. 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To prune, or not to prune: exploring the efficacy of pruning for model compression. arXiv preprint arXiv:1710.01878, 2017. + +# A RELATED WORK + +# A.1 STRUCTURED MATRICES IN MACHINE LEARNING + +Structured linear maps such as the DFT, the Hadamard transform and convolution are a workhorse of machine learning, with diverse applications including data preprocessing, random projection, featurization, and model compression. For example, the DFT is a crucial step in the standard filter bank speech preprocessing pipeline (Jurafsky & Martin, 2014), and is commonly used when dealing with time series data in general (Panaretos & Tavakoli, 2013). Fast random projection and kernel approximation methods rely on the fast Hadamard transform (Le et al., 2013; Yu et al., 2016) and convolution (Yu et al., 2015), and convolution is a critical component of modern image processing architectures (Krizhevsky et al., 2012) as well as being useful in speech recognition (Zeghidour et al., 2018) and natural language processing (Wu et al., 2019). Large learnable classes of structured matrices such as Toeplitz-like matrices (Sindhwani et al., 2015) and low-displacement rank (LDR) matrices (Thomas et al., 2018) have been used for model compression. However, despite their theoretical speedup, these structured matrix classes lack efficient implementations, especially on GPUs. Therefore, their use has largely been confined to small models (e.g. single hidden layer neural nets) and small datasets (e.g. CIFAR-10). + +Butterfly matrices encode the recursive divide-and-conquer structure of the fast Fourier transform (FFT) algorithm. They were first used in numerical linear algebra for fast preconditioning (Parker, 1995). The butterfly factorization is then generalized to encompass complementary low-rank matrices commonly encountered in solving differential and integral equations (Rokhlin & Tygert, 2006; Tygert, 2008; 2010b;a; Li et al., 2015; 2018). In machine learning, butterfly matrices have been use to approximate the Hessian for fast optimization (Mathieu & LeCun, 2014), and to perform fast random projection (Jing et al., 2017; Munkhoeva et al., 2018; Choromanski et al., 2019). Dao et al. (2019) show that butterfly matrices can be used to learn fast algorithms for discrete transforms such as the Fourier transform, cosine/sine transform, Hadamard transform, and convolution. + +# A.2 SPARSE MATRICES + +Several classes of structured linear transforms are ubiquitous in modern deep learning architectures; particularly widespread examples include convolution and multiheaded attention. Recently, attempts to impose sparsity on the neural network weights have been gaining traction. State-of-the art approaches of this type typically accomplish this by pruning small weights (either gradually during training (Zhu & Gupta, 2017), or post-training (Han et al., 2016)) or by training a dense network and then identifying “winning lottery tickets”—sparse subnetworks which may then be retrained from scratch with appropriate initialization (Frankle & Carbin, 2019). Importantly, these approaches start from a dense network, and therefore training is expensive. There is also a more nascent line of work that aims to train unstructured sparse neural networks directly (Mocanu et al., 2018; Mostafa & Wang, 2019; Dettmers & Zettlemoyer, 2019; Evci et al., 2019). These approaches maintain a constant network sparsity level throughout training, and use heuristics to evolve the sparsity pattern during training. One drawback is that the indices of the nonzero entries need to be stored in addition to the entry values themselves, which increases the memory required to store the sparse weight tensors. Another drawback is that these approaches to learn the sparsity pattern are based on intricate heuristics, which can be brittle. We note that these heuristic sparsification techniques could potentially be combined with our approach, to further sparsify the K-matrix factors. + +# A.3 SPEECH RECOGNITION FROM RAW AUDIO + +Numerous works focus on the problem of speech recognition from raw audio input, i.e. without manual featurization. SincNet (Ravanelli & Bengio, 2018) is a CNN-based architecture parameterized with sinc functions, designed so that the first convolutional layer imitates a band-pass filter. Zeghidour et al. (2018) formulate a learnable version of a filter bank featurization; their filters are initialized as an approximation of MFSC features and then fine-tuned jointly with the rest of the model. Sainath et al. (2015) proposed a powerful combined convolutional LSTM (CLDNN)-based model for learning from raw audio, using a large amount of training data. The WaveNet generative architecture (van den Oord et al., 2016), based on dilated convolutions, has been adapted to speech recognition and can be trained on raw audio. Other approaches that can learn from raw audio can be found in (Palaz et al., + +2013; Collobert et al., 2016; Ghahremani et al., 2016). To our knowledge, the $1 4 . 6 \%$ PER achieved by our kaleidoscope $+ \mathrm { L S T M }$ model on the TIMIT test set is the lowest error rate obtained by a model trained directly on the raw audio. + +# A.4 LEARNING PERMUTATIONS + +Permutation matrices find use in tasks such as matching and sorting (among many others). Techniques to obtain posterior distributions over permutations have been developed, such as the exponential weights algorithm (Helmbold & Warmuth, 2009) and the Gumbel-Sinkhorn network (Mena et al., 2018). + +Classifying images with permuted pixels is a standard task to benchmark the ability of RNNs to learn long range dependencies. Le et al. (2015) propose the Permuted MNIST task, in which the model has to classify digit images with all the pixels permuted. Many new RNN architectures, with unitary or orthogonal weight matrices to avoid gradient explosion or vanishing, have been proposed and tested on this task (Le et al., 2015; Arjovsky et al., 2016; Wisdom et al., 2016; Mhammedi et al., 2017; Trinh et al., 2018). Standard gated RNN architectures such as LSTM and GRU have also been found to be competitive with these new RNN architectures on this task (Bai et al., 2018). + +# B ADDITIONAL EXPERIMENTAL DETAILS + +# B.1 SPEECH PREPROCESSING + +In this section, we fully describe our settings and procedures for the speech preprocessing experiments in Section 3.1.1, and present additional auxiliary baselines and results. + +# B.1.1 EXPERIMENTAL SETUP + +We evaluate our speech recognition models on the TIMIT speech corpus (Garofolo et al., 1993), a standard benchmark for speech recognition. The input is audio (16-bit, $1 6 \mathrm { k H z }$ .wav format), and the target is the transcription into a sequence of phonemes (units of spoken sound). Our evaluation metric is the phoneme error rate (PER) between the true phoneme sequence and the phoneme sequence predicted by our model. We use PyTorch (Paszke et al., 2017), the Kaldi speech recognition toolkit (Povey et al., 2011), and the PyTorch-Kaldi toolkit (Ravanelli et al., 2019) for developing PyTorch speech recognition models for all our experiments and evaluations. + +# B.1.2 MODEL AND EVALUATION + +Our baseline Bi-LSTM architecture is taken from the PyTorch-Kaldi repository.6 This is a strong baseline model that, to the best of our knowledge, matches state-of-the-art performance for models that use a single type of input featurization (Ravanelli et al., 2019). The original Bi-LSTM model takes as input filter bank features. These are computed as follows: (i) the waveform is framed (split into chunks of $2 5 ~ \mathrm { m s }$ each that overlap by $1 0 \mathrm { m s }$ each), (ii) the waveform is dithered (zero-mean Gaussian random noise is added), (iii) pre-emphasis is applied to amplify high frequencies, (iv) the Hamming window function (Harris, 1978) is applied, (v) the FFT is applied, and the power spectrum of the resulting (complex-valued) output is computed, (vi) the power spectrum (which has dimension 512) is mapped to the “mel scale” (which is a scale intended to mimic human auditory perception (Stevens et al., 1937)) by multiplication with a specific banded matrix of dimension $5 1 2 \times 2 3$ , and the entrywise logarithm of the output is taken (the 23 outputs are called the filters), and (vii) cepstral mean and variance normalization (Liu et al., 1993) is applied. Numerical hyperparameters of this procedure include the dither noise scale, the pre-emphasis coefficient, the Hamming window size, the number of mel filters, and more; we kept all these the same as the Kaldi/PyTorch-Kaldi defaults. + +In contrast, our “K-matrix version” of the model takes as input the raw waveform, split into chunks the same way as before but with no normalization, dithering, or other preprocessing, which is then fed into a complex-valued kaleidoscope $[ ( B B ^ { * } ) ^ { 2 } ]$ matrix. Similarly to the nonlinear steps in computing filter bank features, the logarithm of the power spectrum of the output (which has dimension 512) + +is then computed. This output is fed into the Bi-LSTM; the Bi-LSTM and kaleidoscope layer are trained together in standard end-to-end fashion. The Bi-LSTM architecture is not modified aside from changing the input dimension from 23 to 512; this (along with the $\approx 7 5 \mathrm { K }$ parameters in the kaleidoscope layer itself) results in approximately a 1.1M increase in the total number of parameters compared to the model that takes in MFSC features (a modest $8 \%$ relative increase). Total training time for our kaleidoscope-based architecture is $7 \%$ greater than that required for the model that uses MFSC features, not counting the time required to precompute the MFSC features; the FLOPs for inference-time are approximately $15 \%$ greater (mostly due to the larger dimension of the input to the Bi-LSTM; the kaleidoscope layer accounts for less than $0 . 5 \%$ of the total FLOPs). + +As baselines, we also compare to inserting other types of linear transformations before the Bi-LSTM: fixed linear transformations (such as the fixed FFT, or no transform at all [i.e. the identity]), other trainable structured layers (low-rank, circulant, and sparse [using the sparse training algorithm of Dettmers & Zettlemoyer (2019)]), and a trainable unstructured (dense) linear layer. The kaleidoscope layer performs the best out of all such approaches. The fact that it outperforms even a dense linear layer with more parameters is particularly notable, as it suggests that the structural bias imposed by the K-matrix representation is beneficial for performance on this task. Full results are given in Table 5. + +Table 5: TIMIT phoneme error rate $( \mathrm { P E R } \%$ , $\pm$ standard deviation across 5 random seeds). + +
ModelTest set PER%#Parameters
Low rank + LSTM23.6 ± 0.915.5M
Sparse +LSTM21.7 ± 0.915.5M
Circulant +LSTM23.9 ± 0.915.4M
Dense +LSTM15.4 ± 0.615.9M
FFT+LSTM15.7 ± 0.115.4M
Identity +LSTM20.7 ± 0.315.4M
Kaleidoscope + LSTM14.6 ± 0.315.4M
MFSC features + LSTM14.2 ± 0.214.3M
SincNet (Ravanelli et al., 2019)17.210.0M
LiGRU (Ravanelli et al., 2018)13.812.3M
+ +In our experiments, we grid search the initial learning rate for the “preprocessing layer” (if applicable) in {5e-5, 1e-4, 2e-4, 4e-4, 8e-4, 1.6e-3}, and fix all other hyperparameters (including the initial learning rates for the other parts of the network) to their default values in the PyTorch-Kaldi repository. The model and any preprocessing layers are trained end-to-end with the RMSProp optimizer for 24 epochs (as per the defaults in PyTorch-Kaldi). For each model, we use the validation set to select the best preprocessing learning rate, while the final error rates are reported on the separate held-out test set. For all structured matrix baselines except circulant (which always has $n$ parameters for an $n \times n$ matrix), the number of parameters in the structured matrices is set to equal the number of parameters in the butterfly layer, while the unconstrained matrix is simply a standard dense complexvalued square matrix. For all experiments with a trainable “preprocessing layer,” we initialize the preprocessing matrix to represent the FFT (or approximate it as closely as possible [i.e. minimize the Frobenius error to the true FFT matrix], in the case of low-rank, sparse, and circulant), which we found to outperform random initialization. + +# B.1.3 EXTENSION: COMBINING MFSC AND KALEIDOSCOPE + +As an additional experiment, we sought to investigate whether combining the hand-engineered MFSC featurization pipeline and a learnable kaleidoscope layer (instead of replacing the former with the latter) could lead to accuracy gains. Specifically, in this experiment we first used the standard filter bank featurization pipeline described above, and trained end-to-end as usual. Then, we replaced the FFT step with a K-matrix initialized to the FFT, and made the weights of the Hamming window function and the mel filter bank matrix learnable as well (similarly to (Zeghidour et al., 2018)). We fine-tuned the resulting architecture for an additional 10 epochs. The final test $\mathrm { P E R } \%$ attained by this “hybrid” model is ${ \bf 1 4 . 0 \pm 0 . 3 }$ ; the model has 14.4M parameters—a negligible increase over the $1 4 . 3 \mathbf { M }$ in the original architecture. Thus, by combining the manually encoded domain knowledge in the filter bank featurization and allowing this structure to be learnable rather than fixed, we are able to nearly match the state-of-the-art $1 3 . 8 \%$ accuracy on TIMIT. While this “hybrid” model certainly involves some hand-engineering, the state-of-the-art results use a concatenation of three different speech audio featurizations—MFSC, MFCC, and fMLLR—as the neural network input, along with a customized RNN architecture (LiGRU) specifically designed for speech recognition, and thus require a more complicated pipeline that is arguably even more hand-crafted. + +# B.2 REPLACING CNN CHANNEL SHUFFLE + +# B.2.1 MODEL ARCHITECTURES + +ShuffleNet is a convolutional neural network with residual (skip) connections that uses a permutation matrix to shuffle the channels after each grouped 1x1 convolution, sending the $i \cdot$ -th channel to the $( i \bmod g )$ -th group, where $g$ is the total number of groups. The architecture for each residual block in ShuffleNet is: 1x1 group conv Batch norm, ReLU Permutation $\to 3 { \bf x } 3$ depthwise conv Batch norm $ 1 \mathrm { x } 1$ group conv. The permutation is fixed. + +Zhao et al. (2019) propose to instead use the Hadamard transform before and after each grouped 1x1 convolution to mix the channels. Note that the Hadamard transforms are placed before the batch normalization and ReLU layer (unlike the permutation matrix in the original ShuffleNet design). In particular, the architecture for each block is: Hadamard $\to 1 \mathbf { x } 1$ group conv Hadamard Batch norm, $\mathrm { R e L U } \to 3 \mathrm { x } 3$ depthwise conv Batch norm $ 1 \mathrm { x } 1$ group conv. The Hadamard transform is fixed. + +In our architecture, we use a kaleidoscope matrix in OBB (product of an orthogonal butterfly matrix, a diagonal matrix, and the transpose of another butterfly matrix) before and after each grouped 1x1 convolution. We place the second K-matrix after the batch norm and ReLU, to more closely mimic the original ShuffleNet design. The structure for each block is: K-matrix $ 1 \mathrm { x } 1$ group conv Batch norm, $\mathrm { R e L U } \mathrm { K } \mathrm { - m a t r i x } 3 \mathrm { x } 3$ depthwise conv Batch norm $ 1 \mathrm { x } 1$ group conv. The K-matrices are trained along with the rest of the network, rather than being fixed. + +# B.2.2 EXPERIMENTAL SETUP + +We evaluate the CNN architectures on the image classification task of the standard ImageNet dataset (Russakovsky et al., 2015). We use the standard data augmentation, training, and evaluation pipeline as in (Xie et al., 2017). We train with SGD on 8 GPUs for 90 epochs, with a total batch size of 2048 and initial learning rate 0.8. For the 1.0 ShuffleNet g8 architecture, we reduce the total batch size to 1792 to fit into GPU memory, and correspondingly linearly scale the initial learning rate to 0.7. Other hyperparameters (e.g. learning rate schedule, weight decay, etc.) are kept the same as in the ShuffleNet paper (Zhang et al., 2018). We use the training script from NVIDIA’s deep learning examples repository.7 + +# B.2.3 ADDITIONAL RESULTS + +In Table 6, we report top-5 classification accuracy on ImageNet, to complement the top-1 accuracies in Table 2. + +Table 6: Top-5 classification accuracy of ShuffleNet on ImageNet validation set. We report results for different network width multipliers (number of channels), and for different kinds of matrices used for channel mixing. Using a learnable K-matrix in place of each fixed permutation (shuffle) or Hadamard matrix improves top-5 accuracy by up to $4 . 8 \%$ . Parameter counts are the same as in Table 2. + +
ShuffleHadamardKaleidoscope (K.)|K. vs. Shuffle
0.25 ShuffleNet g868.6%68.4%73.4%+4.8%
0.5 ShuffleNet g879.9%79.2%81.7%+1.8%
1.0 ShuffleNet g886.0%85.8%86.8%+0.8%
+ +7https://github.com/NVIDIA/DeepLearningExamples/tree/master/PyTorch/ Classification/RN50v1.5 + +In each setting, the total training time of our K-matrix approach is within $20 \%$ of the total training time of vanilla ShuffleNet. + +In Figure 4, we plot the loss and accuracy on the training set and validation set when we train 1.0 ShuffleNet $^ { \mathrm { g 8 } }$ , with either a fixed permutation (Shuffle) or a K-matrix for channel shuffling. Even though each K-matrix is a product of multiple (sparse) matrices, the model with K-matrices takes about the same number of training steps to converge as the baseline model does. One possible reason is that we constrain the K-matrices to be orthogonal (Section 2.4), thus avoiding vanishing or exploding gradients. + +![](images/667c482d9c16deb50edde5ee298ef0038ca2d217e93dcabc91f95f10b2abadc8.jpg) +Figure 4: Loss and top-1 accuracy of 1.0 ShuffleNet $\mathrm { g } 8$ with either a fixed permutation (Shuffle) or a K-matrix for channel shuffling. The K-matrix model takes about the same number of training steps to converge as does the baseline model. + +# B.3 LEARNING PERMUTATIONS + +# B.3.1 DATASET + +The permuted CIFAR-10 dataset is constructed by applying a fixed permutation to every input. We choose to use the 2-D bit-reversal permutation,8 i.e., the bit reversal permutation on 32 elements is applied to the rows and to the columns. This permutation was chosen because it is locality-destroying: if two indices $i , j$ are close, they must differ in a lower-order bit, so that the bit-reversed indices $i ^ { \prime } , j ^ { \prime }$ are far. This makes it a particularly challenging test case for architectures that rely on spatial locality such as “vanilla” CNNs. + +# B.3.2 MODEL AND TRAINING + +We describe the model architectures used in Section 3.1 (those reported in Table 3). + +Our model $\mathbf { ( K + C N N ) }$ The model represents a fixed permutation $P$ , parametrized as a K-matrix, to learn to recover the true permutation, followed by a standard ResNet18 architecture (He et al., 2016). Because of the simple decomposable nature of the butterfly factors (Section 2.1), our parameterization is easily extensible with additional techniques: + +(i) We constrain each butterfly factor matrix in the K-matrix to be doubly-stochastic. For example, each $2 \times 2$ block in the butterfly factor matrix of block size 2 has the form $\left[ { \begin{array} { c c } { a } & { 1 - a } \\ { 1 - a } & { a } \end{array} } \right]$ , where $a \in [ 0 , 1 ]$ . We treat this block as a distribution over permutations, generating the identity $\left[ { \begin{array} { l l } { 1 } & { 0 } \\ { 0 } & { 1 } \end{array} } \right]$ with probability $a$ and the swap $\left[ { \begin{array} { c c } { 0 } & { 1 } \\ { 1 } & { 0 } \end{array} } \right]$ with probability $1 - a$ . Butterfly factor matrices with larger block sizes are constrained to be doubly-stochastic in a similar manner. In this way, a permutation is sampled for each butterfly factor matrix, and these permutations are composed to get the final permutation that is applied to the image. + +(ii) For each minibatch, the examples $P x$ by applying permutation samples on the (permuted) inputs are fed into an additional unsupervised reconstruction loss + +$$ +\sum _ { 0 \leq i , j < n } \left\| \left[ { \binom { P x } { ( P x ) [ i , j + 1 ] - ( P x ) [ i , j ] } } \right] \right\| _ { 2 } +$$ + +measuring total variation smoothness of the de-noised inputs. Such loss functions are often used in image denoising (Rudin et al., 1992). A final regularization loss was placed on the entropy of $P$ , which was annealed over time to encourage $P$ to converge toward a sharper doubly-stochastic matrix (in other words, a permutation). + +The model is trained with just the reconstruction loss to convergence before the standard ResNet is trained on top. + +These techniques are applicable to the K-matrix as well as specialized methods for representing permutations such as Gumbel-Sinkhorn (Mena et al., 2018) and are important for recovering the true permutation. However, they are not applicable to a general linear layer, which showcases the flexibility of K-matrices for representing generic structure despite not being specially tailored for this task. We also remark that other classes of structured linear maps such as low-rank, circulant, and so on, are even less suited to this task than dense matrices, as they are incapable of representing all permutations. + +# Baseline architectures + +1. Fully connected (FC): This is a 3-layer MLP, with hidden size 1024 and ReLU nonlinearity in-between the fully connected layers. +2. Recurrent neural network (RNN): We use a gated recurrent unit (GRU) model (Cho et al., 2014), with hidden size 1024. Many RNN architectures have been proposed to capture long-range dependency on permuted image dataset such as Permuted MNIST (Arjovsky et al., 2016). Standard gated architectures such as LSTM and GRU have shown competitive performance on the Permuted MNIST dataset, and we choose GRU as a baseline since it has been reported to slightly outperform LSTM (Bai et al., 2018). +3. CNN: We use the standard ResNet18 architecture, adapted to smaller image size of the CIFAR-10 dataset (changing stride from 2 to 1 of the first convolutional layer, and removing max-pooling layer that follows). +4. Dense $^ +$ CNN: We add an additional linear layer (i.e. a dense matrix) of size $1 0 2 4 \times 1 0 2 4$ before the ResNet18 architecture. This dense layer can in theory represent a permutation, but cannot benefit from the additional techniques described above. +5. Baseline CNN (unpermuted): We use the standard ResNet18 architecture applied to the unpermuted CIFAR-10 dataset. + +All models are trained for 200 total epochs, with the Adam optimizer. We use the standard learning rate schedule and weight decay from Mostafa & Wang (2019). We use Hyperband (Li et al., 2017) to tune other hyperparameters such as the initial learning rate and annealing temperature. + +# B.4 SPEEDING UP DYNAMICCONV’S INFERENCE + +# B.4.1 MODEL ARCHITECTURE + +We start with the DynamicConv Transformer architecture (Wu et al., 2019), which is a variant of the Transformer architecture (Vaswani et al., 2017) where the self-attention in each layer is replaced with a light-weight DynamicConv module. We use the implementation from the Fairseq library(Ott et al., 2019),9 with PyTorch version 1.2. + +The architecture of each layer of the decoder is: Linear DynamicConv Linear LayerNorm Encoder-decoder attention LayerNorm Linear ReLU Linear $ \mathrm { R e L U } $ LayerNorm. In every layer of the decoder, we replace the dense weight matrix in each of the four Linear layers with a $\mathrm { K }$ -matrix from the $\boldsymbol { B }$ class (i.e. a butterfly matrix). + +# B.4.2 TRAINING AND EVALUATION + +The models are trained from scratch using the training script from the Fairseq repository, with the same hyperparameters (optimizer, learning rate, number of updates, etc.) used in the DynamicConv paper (Wu et al., 2019). We note that the DynamicConv model with K-matrices in the decoder trains slightly faster than the default DynamicConv model (both models are trained for 50,000 updates, which requires approximately $7 \%$ less time for the K-matrix model than for the default model). + +To evaluate inference speed, we run the decoding script on the IWSLT-14 De-En test set in singlethreaded mode on a server Intel Xeon CPU E5-2690 v4 at 2.60GHz, and measure wall-clock time. The test set contains 6750 sentences, with 149241 tokens. Following Wu et al. (2019), we set the batch size to 1 and beam size to 1 for this evaluation. + +# B.4.3 ADDITIONAL COMPARISON WITH OTHER STRUCTURED MATRICES + +We additionally compare the speed-quality tradeoff of K-matrices with other classes of structured matrices, when used to replace the fully-connected layers of DynamicConv’s decoder. We consider the following additional classes of structured matrices: low-rank, circulant, Toeplitz-like (Sindhwani et al., 2015), ACDC (Moczulski et al., 2016), Fastfood (Le et al., 2013), and sparse. For classes with a variable number of parameters (e.g. low-rank, sparse), we set the number of parameters to match that of K-matrices. For sparse matrices, besides the result for an ensemble of 10 models (the default setting in the Fairseq repository), we also report the result for a single model, as that could have faster inference time (since ensembling/averaging sparse matrices produces a less sparse matrix). + +In Figure 5, we plot the tradeoff between translation quality (measured by BLEU score) and inference speed (sentences per second). Most classes of structured matrices produce similar translation quality (between 34.1 and 34.4 BLEU score). K-matrices have the second fastest inference time, only $7 \%$ slower than low-rank matrices. We note that low-rank matrices benefit from very well-tuned BLAS routines (matrix-matrix multiplication). Even though our implementation of K-matrix multiplication is not yet highly optimized, it is already quite close to the speed of low-rank matrix multiplication at an equivalent parameter count. + +# C SPEED BENCHMARK AND IMPLEMENTATION DETAILS + +Each K-matrix (for fixed width and expansion), has an $O ( n \log n )$ matrix-vector multiplication algorithm: sequentially multiply the input vector with each of the sparse factors. Our implementation of this simple algorithm is surprisingly competitive with optimized subroutines, both on GPU (e.g. for training) and on CPU (e.g. for inference). In Figure 6, we compare the speed of multiplying by a K-matrix in class $\boldsymbol { B }$ (i.e. a butterfly matrix) against a specialized implementation of the FFT. We normalize the speed by the speed of dense matrix-matrix multiply (on GPU) or dense matrix-vector multiply (on CPU). On GPU, with input sizes $n = 1 0 2 4$ and batch size 2048, the training time (forward and backward) of K-matrices matrix is about $3 \mathbf { x }$ faster than dense matrix multiply (GEMM from cuBLAS). For inference on CPU, the kaleidoscope fast multiplication can be one or two orders of magnitude faster than GEMV. Over a range of matrix sizes, our implementation is within a factor of $2 { - } 4 \mathbf { x }$ of specialized implementations of the FFT, a highly optimized kernel. + +Our implementation is also memory-efficient. In the forward pass through the $O ( \log n )$ sparse factors, we do not store the intermediate results, but recompute them during the backward pass. Therefore the activation memory required is $O ( b n )$ for an input batch size of $b$ . + +# D SYNTHETIC MATRIX RECOVERY + +We directly validate Theorem 1 on well-known types of structured matrices used in machine learning. Given a structured matrix $\mathbf { M }$ , we attempt to represent $\mathbf { M }$ as closely as possible using $\mathbf { K }$ -matrices as well as the standard classes of structured matrices: sparse and low-rank. In Table 7, we quantify the expressivity of each of these three methods, as measured by their ability to approximate a range of different structures. Results for “global minimum” of kaleidoscope matrices are obtained from the theoretical expressiveness results in Section I and Section J. Low-rank and sparse approximation have closed form solutions: truncating the SVD and keeping the largest-magnitude entries, respectively. We also report the results using SGD for kaleidoscope matrices to validate that good approximation with K-matrices can be obtained even from standard first-order optimization algorithms. Even with imperfect optimization, kaleidoscope matrices can still capture out-of-class target matrices better than low-rank and sparse matrices. + +![](images/60e8527bd3e997ff76c45727293b2592a1bb95f4efe3920dcade7144a5d1b7fb.jpg) +Figure 5: Tradeoff between translation quality (measured by BLEU score) and inference speed (sentences per second). K-matrices have the second fastest inference speed, only $7 \%$ slower than low-rank matrices. + +![](images/dab94d90fe5fadd003eb3561b1387c255889d8db809628de7ac84a68752cb799.jpg) +Figure 6: Speedup of FFT and Kaleidoscope against dense matrix-matrix multiply (GEMM) for training, and against dense matrix-vector multiply (GEMV) for inference. + +The target matrices are kaleidoscope, low-rank, sparse, convolution (i.e. circulant matrices), Fastfood (Le et al., 2013), and entrywise random IID Gaussian matrix (to show the typical magnitude of the error). All target matrices M were randomly initialized such that $\mathbb { E } [ \mathbf { M } ^ { T } \mathbf { M } ] ^ { - \sum } \mathbf { I }$ . + +Table 7: Expressiveness of different classes of structured matrices: Frobenius error of representing common structured matrices (columns) of dimension 256 using three structured representations of matrices with adjustable numbers of parameters. (Left group: Target matrices in the same class as the methods. Middle group: Target matrices with fixed number of parameters. Right: Random matrix to show typical scale of error.) Each method is allotted the same number of parameters, equal to a $\log n$ factor more than that of the target matrix. Low-rank and sparse matrices are unable to capture any structure outside their own class, while the minima for kaleidoscope matrices found via optimization better capture the actual structure for out-of-class targets better than the baselines. + +
Target MethodKaleidoscopeLow-rankSparseConvolutionFastfoodRandom
Global Min.Kaleidoscope0.00.00.00.00.0
Low-rank14.90.010.814.611.615.5
Sparse11.712.20.013.17.114.1
With SGDKaleidoscope0.00.018.00.05.114.5
+ +To find a kaleidoscope approximation with SGD, we used Hyperband to tune its learning rate (between 0.001 and 0.5). + +# E PROPERTIES OF THE $B B ^ { * }$ HIERARCHY + +Here, we justify why the definitions in Section 2.2 give rise to a hierarchy. We first make some basic observations about the parameterization. + +Observation E.1. An $n \times n$ matrix $\mathbf { M } \in B B ^ { * }$ has $4 n \log n$ parameters. + +Proof. M can be expressed as a product of $2 \log n$ butterfly factor matrices of size $n \times n$ . Each of these factor matrices has 2 parameters per row, for a total of $2 n$ parameters each. Hence, the total number of parameters is $4 n \log n$ . □ + +Observation E.2. Let M be an $n \times n$ matrix in $( B B ^ { * } ) _ { e } ^ { w }$ . Then, given an arbitrary vector v of length $n$ , we can compute Mv with $O ( w n e \log ( n e ) )$ field operations. + +Proof. Since $\mathbf { M } \in ( B B ^ { * } ) _ { e } ^ { w }$ , we can decompose it as $\mathbf { S } \mathbf { E } _ { 1 } \mathbf { E } _ { 2 } \ldots \mathbf { E } _ { w } \mathbf { S } ^ { T }$ , where $\mathbf { S }$ is as given in Definition 2.4, and each $\mathbf { E } _ { i }$ is an $e n \times e n$ matrix in $B B ^ { * }$ . Therefore, to compute Mv, we can use associativity of matrix multiplication to multiply the vector by one of these matrices at a time. + +Since all of these factors are sparse, we use the naïve sparse matrix-vector multiplication algorithm (begin with a 0-vector and perform the corresponding multiplication and addition for each nonzero matrix entry). S (and thus $\mathbf { \vec { S } } ^ { T }$ ) have $n$ NNZ. Therefore, matrix-vector multiplication by S or $\mathbf { S } ^ { T }$ requires $O ( n )$ operations, which is dominated by the butterfly matrix-vector multiplication. Each $\mathbf { E } _ { i }$ can be further decomposed into $2 \log ( n e )$ matrices with at most $2 n e$ non-zero entries each (by Observation E.1). Therefore, matrix vector multiplication by each $\mathbf { E } _ { i }$ requires $O ( n e \log ( n e ) )$ . Since there are $w$ such $\mathbf { E } _ { i }$ , we require a total of $O ( w n \bar { e } \log ( n e ) )$ operations. □ + +Now, we are ready to show that our definition of classes $( B B ^ { * } ) _ { e } ^ { w }$ forms a natural hierarchy. + +First, we must argue that all matrices are contained within the hierarchy. + +Lemma E.3. Let M be an arbitrary $n \times n$ matrix. Then $\mathbf { M } \in ( B B ^ { * } ) ^ { ( 2 n - 2 ) }$ + +Proof. Corollary E.3 in Appendix $\mathbf { K }$ shows that any $n \times n$ matrix can be written in the form $\mathbf { M _ { 1 } } \mathbf { \bar { M } ^ { \prime } } _ { 1 } ^ { * } \dots \mathbf { M } _ { n - 1 } \mathbf { M ^ { \prime } } _ { n - 1 } ^ { * } \mathbf { \bar { M } M _ { n } } \mathbf { M ^ { \prime } } _ { n } ^ { * } \dots \mathbf { M } _ { 2 n - 2 } \mathbf { M ^ { \prime } } _ { n - 2 } ^ { * } \mathbf { \bar { M } M ^ { \prime } } _ { n } ^ { * } \dots \mathbf { M } _ { n } ^ { * } \dots \mathbf { M ^ { \prime } } _ { n } ^ { * }$ , where $\mathbf { M } _ { i } , \mathbf { M } _ { i } ^ { \prime }$ are orthogonal butterfly matrices and M is a diagonal matrix. We can combine $D$ with $M _ { n }$ to form another (possibly not orthogonal) butterfly matrix. This yields a decomposition of $\mathbf { M }$ as products of (possibly not orthogonal) butterfly matrices and their (conjugate) transposes, completing the proof. □ + +Next, we argue that, up to a certain point, this hierarchy is strict. + +Lemma E.4. For every fixed $c \geq 1$ , there is an $n \times n$ matrix ${ { \bf { M } } _ { n } }$ (with n sufficiently large) such that $\mathbf { M } _ { n } \in ( B B ^ { * } ) ^ { c + 1 }$ but $\mathbf { M } _ { n } \notin ( B B ^ { * } ) ^ { c }$ . + +Proof. Given $c$ , fix $n$ to be a power of 2 such that $\begin{array} { r } { c < \frac { n } { 4 \log _ { 2 } n } } \end{array}$ . For sake of contradiction, assume that every $n \times n$ matrix in $( B B ^ { * } ) ^ { c + 1 }$ is also in $( B B ^ { * } ) ^ { c }$ . Let A be an arbitrary $n \times n$ matrix. From Lemma E.3, $\mathbf { A } \in ( B B ^ { \ast } ) ^ { ( 2 n - 2 ) }$ . From our assumption, we can replace the first $c + 1 B B ^ { * }$ factors of $\mathbf { A }$ with $c$ (potentially different) $B B ^ { * }$ factors and still recover A. We can repeat this process until we are left with $c B B ^ { * }$ factors, implying that $\mathbf { A } \in ( B B ^ { * } ) ^ { c }$ . From Observation E.1, we require $4 c n \log n < n ^ { 2 }$ (by our choice of $n$ ) parameters to completely describe A. This is a contradiction since $\mathbf { A }$ is an arbitrary $n \times n$ matrix, and therefore has $n ^ { 2 }$ arbitrary parameters. Hence, there must be some $n \times n$ matrix in $( B B ^ { * } ) ^ { c + 1 }$ that is not in $( B B ^ { * } ) ^ { c }$ . □ + +# F ARITHMETIC CIRCUITS IN $B B ^ { * }$ HIERARCHY + +In this appendix, we prove our main theoretical result, namely, our ability to capture general transformations, expressed as low-depth linear arithmetic circuits, in the $B B ^ { * }$ hierarchy. This result is recorded in Theorem 1. + +Theorem 1. Let M be an $n \times n$ matrix such that matrix-vector multiplication of M times an arbitrary vector v can be represented as a be a linear arithmetic circuit $C$ comprised of $s$ gates (including inputs) and having depth $d .$ . Then, $\mathbf { M } \in \left( B B ^ { * } \right) _ { O ( \frac { s } { n } ) } ^ { O ( d ) }$ . + +To prove Theorem 1, we make use of the following two theorems. + +Theorem 2. Let P be an $n \times n$ permutation matrix (with n a power of 2). Then $\mathbf { P } \in B B ^ { * }$ . + +Theorem 3. Let S be an $n \times n$ matrix of s NNZ. Then $\mathbf { S } \in \left( B B ^ { * } \right) _ { 4 } ^ { 4 \left\lceil \frac { s } { n } \right\rceil }$ . + +Theorem 2 is proven in Appendix $\mathbf { G }$ , and Theorem 3 is proven in Appendix I. + +Proof of Theorem 1. We will represent $C$ as a product of $d$ matrices, each of size $s ^ { \prime } \times s ^ { \prime }$ , where $s ^ { \prime }$ is the smallest power of 2 that is greater than or equal to $s$ . + +To introduce some notation, define $w _ { 1 } , \dots , w _ { d }$ such that $w _ { k }$ represents the number of gates in the $k$ ’th layer of $C$ (note that $\begin{array} { r } { s = n + \sum _ { k = 1 } ^ { d } w _ { k } ) } \end{array}$ . Also, define $z _ { 1 } , \dotsc z _ { d }$ such that $z _ { 1 } ~ = ~ n$ and $z _ { k } = w _ { k - 1 } + z _ { k - 1 }$ ( $z _ { k }$ is the number of gates that have already been used by the time we get to layer $k$ ). + +Let $g _ { i }$ denote the $i ^ { \because }$ ’th gate (and its output) of $C$ $0 \leq i < s$ ), defined such that: + +$$ +g _ { i } = \left\{ { \begin{array} { l l } { v _ { i } } & { 0 \leq i < n } \\ { \alpha _ { j } g _ { i _ { 1 } } + \beta _ { i } g _ { i _ { 2 } } } & { n \leq i < s } \end{array} } \right. +$$ + +where $i _ { 1 } , i _ { 2 }$ are indices of gates in earlier layers. + +For the $k$ ’th layer of $C$ , we define the $s ^ { \prime } \times s ^ { \prime }$ matrix ${ { \bf { M } } _ { k } }$ such that it performs the computations of the gates in that layer. Define the $\Ddot { \iota }$ ’th row of ${ { \bf { M } } _ { k } }$ to be: + +$$ +\begin{array} { r } { \mathbf { M } _ { k } [ i : ] = \left\{ \begin{array} { l l } { \mathbf { e } _ { i } ^ { T } } & { 0 \leq i < z _ { k } } \\ { \alpha _ { i } \mathbf { e } _ { i _ { 1 } } ^ { T } + \beta _ { i } \mathbf { e } _ { i _ { 2 } } ^ { T } } & { z _ { k } \leq i < z _ { k } + w _ { k } } \\ { 0 } & { i \geq z _ { k } + w _ { k } } \end{array} \right. } \end{array} +$$ + +For any $0 \leq k \leq d$ , let $\mathbf { v _ { k } }$ be vector + +$$ +\mathbf { v } _ { k } = \mathbf { M } _ { k } \ldots \mathbf { M } _ { 2 } \mathbf { M } _ { 1 } \left[ \mathbf { v } \right] . +$$ + +We’d like to argue that $\mathbf { v } _ { d }$ contains the outputs of all gates in $C$ (i.e, the $n$ values that make up Mv). To do this we argue, by induction on $k$ , that $\mathbf { v } _ { k }$ is the vector whose first $z _ { k + 1 }$ entries are $g _ { 0 } , g _ { 1 } , \dotsc , g _ { \left( z _ { k } - 1 \right) } .$ , and whose remaining entries are 0. The base case, $k = 0$ is trivial. Assuming this holds for the case $k - 1$ , and consider multiplying $\mathbf { v } _ { k - 1 }$ by ${ { \bf { M } } _ { k } }$ . The first $z _ { k }$ rows of ${ { \bf { M } } _ { k } }$ duplicate the first $z _ { k }$ entries of $\mathbf { v } _ { k - 1 }$ The next $w _ { k }$ rows perform the computation of gates $g _ { z _ { k } } , \ldots , g _ { ( z _ { k + 1 } - 1 ) }$ Finally, the remaining rows pad the output vector with zeros. Therefore, $\mathbf { v } _ { k }$ is exactly as desired. + +The final matrix product will contain all $n$ elements of the output. By left multiplying by some permutation matrix $\mathbf { P }$ , we can reorder this vector such that the first $n$ entries are exactly Mv. Hence, we are left to argue the position of $\mathbf { P M } _ { d } \ldots \mathbf { M } _ { 2 } \mathbf { M } _ { 1 }$ within the $B B ^ { * }$ hierarchy. Each ${ { \bf { M } } _ { k } }$ is a matrix with total $2 w _ { k } + z _ { k } < 2 s ^ { \prime }$ NNZ. From Theorem 3, we can, therefore, represent ${ { \bf { M } } _ { k } }$ as a product of $O ( 1 )$ matrices (of size $2 s ^ { \prime }$ ) in $B B ^ { * }$ . From Theorem 2, $\mathbf { P } \in B B ^ { * }$ . Note that $s \leq s ^ { \prime } < 2 s$ , so $s ^ { \prime } = \Theta ( s )$ . + +Our final decomposition will have $O ( d ) B B ^ { * }$ factors, and requires an expansion from size $n$ to size $2 s ^ { \prime }$ , or an expansion factor of $O ( { \textstyle { \frac { s } { n } } } )$ . Therefore, $\mathbf { M } \in \left( B B ^ { * } \right) _ { O ( \frac { s } { n } ) } ^ { O ( d ) }$ , as desired. □ + +Remark F.1. By applying Observation $E . 2$ , we see that Theorem $I$ gives an $O ( s d \log s )$ matrix vector multiplication algorithm for $\mathbf { M }$ . + +# G PERMUTATIONS IN $B B ^ { * }$ + +In this appendix, we prove Theorem 2. In addition, we will also show that permutations are in $B ^ { * } B$ , where the set $B ^ { * } B$ is defined analogously to $B B ^ { * }$ (i.e. matrices of the form $\mathbf { M } = \mathbf { M } _ { 1 } ^ { * } \mathbf { M } _ { 2 }$ for some $\mathbf { M } _ { 1 } , \mathbf { M } _ { 2 } \in B )$ . + +To prove Theorem 2, we decompose permutation matrix $\mathbf { P }$ into $\mathbf { P } = \mathbf { L R }$ , with $\mathbf { L } \in B$ and $\mathbf { R } \in B ^ { * }$ Throughout the proof, we make use of the following definition. + +Definition G.1. Let L be an $n \times n$ permutation matrix (n a power of 2). We say that L meets the $2 ^ { j }$ balance condition if L can be divided into chunks of $2 ^ { j }$ (with each chunk having all columns $i$ such that $\left\lfloor { \frac { i } { 2 ^ { j } } } \right\rfloor$ has the same value) such that for every $0 \leq m < 2 ^ { j }$ , each chunk has exactly one $\mathbf { L } [ : , k ] = \mathbf { e } _ { \pi _ { k } }$ with $\pi _ { k } \equiv m$ ( mod $2 ^ { j }$ ). We say that L is modular-balanced if it meets the $2 ^ { j }$ balance condition for each $2 \leq 2 ^ { \overset { . } { j } } \leq n$ . + +![](images/e1a749d098c79becb5562ba1896bfb4d508efdd4f05072b76cda0be851a989ca.jpg) +Figure 7: First step of decomposition of modular-balanced matrix L. Here, the red entries must be permuted into the main diagonal blocks. + +Lemma G.1. Let L be an $n \times n$ modular-balanced matrix. Then $\mathbf { L } \in B$ . + +Proof. We proceed by induction on $n$ . The base case $n = 2$ is trivial. As our inductive hypothesis, we assume that all modular-balanced matrices of size ${ \frac { n } { 2 } } \times { \frac { n } { 2 } }$ are butterfly matrices of size $\frac { n } { 2 }$ . From Definition 2.3, it is sufficient to show that can be decomposed as: + +$$ +\begin{array} { r } { \mathbf { L } = \mathbf { B } _ { n } \underbrace { \left[ \begin{array} { c c } { \mathbf { L } _ { 1 } } & { 0 } \\ { 0 } & { \mathbf { L } _ { 2 } } \end{array} \right] } _ { \mathbf { L } ^ { \prime } } , } \end{array} +$$ + +where $\mathbf { B } _ { n }$ is a butterfly factor of size $n$ and each $\mathbf { L } _ { j }$ is an ${ \frac { n } { 2 } } \times { \frac { n } { 2 } }$ modular-balanced matrix. + +Define $\mathbf { L } _ { 1 }$ and $\mathbf { L } _ { 2 }$ such that: + +$$ +{ \bf L } _ { 2 } [ i , j ] = { \bf L } \left[ i , j + \frac { n } { 2 } \right] + { \bf L } \left[ i + \frac { n } { 2 } , j + \frac { n } { 2 } \right] . +$$ + +Note that since $\mathbf { L }$ is a permutation matrix (and thus has exactly one non-zero entry per column), at most one term of each of these sums can be non-zero. + +For sake of contradiction, assume $\mathbf { L } _ { 1 }$ is not modular-balanced. Then, for some $2 ^ { j } \leq { \frac { n } { 2 } }$ , there are two columns $c _ { 1 } , c _ { 2 }$ such that $\textstyle \left\lfloor { \frac { c _ { 1 } } { 2 ^ { j } } } \right\rfloor = \left\lfloor { \frac { c _ { 2 } } { 2 ^ { j } } } \right\rfloor$ and such that indices of the non-zero entries of $\mathbf { L } _ { 1 }$ in columns $c _ { 1 }$ and $c _ { 2 }$ are the same modulo $2 ^ { j }$ . However, from the definition of $\mathbf { L } _ { 1 }$ , this implies that the indices of the non-zero entries of $\mathbf { L }$ in columns $c _ { 1 }$ and $c _ { 2 }$ are also the same modulo $2 ^ { j }$ , contradicting $\mathbf { L }$ being modular-balanced. Hence, $\mathbf { L } _ { 1 }$ is modular-balanced. An analogous argument (that instead considers columns $\begin{array} { r } { c _ { 1 } + \frac { n } { 2 } , c _ { 2 } + \frac { n } { 2 } } \end{array}$ of $\mathbf { L }$ ) shows that $\mathbf { L } _ { 2 }$ is also modular-balanced. + +To complete the proof, we must argue that $\mathbf { B } _ { n }$ is a butterfly factor of size $n$ . Since each $\mathbf { L } _ { i }$ is modular-balanced, it is a permutation matrix. Therefore, $\mathbf { L } ^ { \prime }$ has exactly 1 non-zero entry in each of the first $\frac { n } { 2 }$ rows and columns from $\mathbf { L } _ { 1 }$ and exactly 1 non-zero entry in each of the second $\frac { n } { 2 }$ rows and columns from $\mathbf { L } _ { 2 }$ . Hence, $\mathbf { L } ^ { \prime }$ is a permutation matrix. Since both $\mathbf { L }$ and $\mathbf { L } ^ { \prime }$ are permutation matrices, $\mathbf { B } = \mathbf { L } \left( \mathbf { L } ^ { \prime } \right) ^ { - 1 }$ must also be a permutation matrix. Therefore, we can view $\mathbf { B }$ as performing a permutation of the rows of $\mathbf { L } ^ { \prime }$ to get $\mathbf { L }$ . + +Consider the $\overrightarrow { \imath } ^ { \prime }$ ’th row of $\mathbf { L } ^ { \prime }$ , with $0 \leq i < \frac { n } { 2 }$ . There are two possible cases. + +Case $1 \colon \mathbf { L } ^ { \prime } [ i , : ] = \mathbf { L } [ i , : ]$ + +In this case, the column of $\mathbf { L }$ with a non-zero entry in row $i$ is in the left $\frac { n } { 2 }$ columns. The column of $\mathbf { L }$ with a non-zero entry in row $\begin{array} { r } { i + \frac { n } { 2 } } \end{array}$ must, therefore, be in the right $\frac { n } { 2 }$ columns, otherwise $\mathbf { L }$ would not satisfy the $\frac { n } { 2 }$ balance condition. Therefore, $\begin{array} { r } { \mathbf { L } ^ { \prime } \left[ i + \frac { n } { 2 } , : \right] = \bar { \mathbf { L } } \left[ i + \frac { n } { 2 } , : \right] } \end{array}$ , so we set $\begin{array} { r } { \mathbf { B } [ i , i ] = \mathbf { B } \left[ i + \frac { n } { 2 } , i + \frac { n } { 2 } \right] = 1 } \end{array}$ . + +$$ +{ \mathsf { C a s e 2 : L ^ { \prime } } } [ i , : ] \neq \mathbf { L } [ i , : ] +$$ + +By the definition of $\mathbf { L } ^ { \prime }$ , $\begin{array} { r } { { \bf L } ^ { \prime } [ i , : ] = v L \left[ i + \frac { n } { 2 } , : \right] } \end{array}$ . In this case, the column of $\mathbf { L }$ with a non-zero entry in row $\begin{array} { r } { i + \frac { n } { 2 } } \end{array}$ must be in the left $\frac { n } { 2 }$ columns. By the $\frac { n } { 2 }$ balance condition of $\mathbf { L }$ , the column of $\mathbf { L }$ with a non-zero entry in row $i$ must be in the right $\frac { n } { 2 }$ columns. Therefore, $\begin{array} { r } { \mathbf { L } ^ { \prime } \left[ i + \frac { n } { 2 } , : \right] = \mathbf { L } \left[ i , : \right] } \end{array}$ , so we set $\begin{array} { r } { \mathbf { B } \left[ i , i + \frac { n } { 2 } \right] = \mathbf { B } \left[ i + \frac { n } { 2 } , i \right] = 1 . } \end{array}$ + +In both cases, the non-zero entries of $\mathbf { B }$ fall into the correct diagonal bands (the main diagonal, and the bands $\frac { n } { 2 }$ away). Hence, $\mathbf { B }$ is a butterfly factor of size $n$ . □ + +Now, we consider the process of transforming $\mathbf { P }$ into a modular-balanced matrix. We make use of the following lemma. + +![](images/a11fbc418e35ae750d3922a0ae3f160b759972fa724560634f39157fd9066c93.jpg) +Figure 8: First step of balancing $8 \times 8$ bit reversal permutation (a component of the $8 \times 8$ DFT). Red signifies edges that must be flipped. + +Lemma G.2. Let M be a $k \times k$ matrix with $I$ non-zero entry per column, such that for each $\textstyle 0 \leq m < { \frac { k } { 2 } }$ , there are exactly 2 columns with non-zero entry in a row with index $\equiv m$ mod $\frac { k } { 2 }$  . Then, there is a butterfly factor $\mathbf { B } _ { k }$ such that $\mathbf { M } \mathbf { B } _ { k } = \mathbf { M } ^ { \prime }$ , where $\mathbf { M } ^ { \prime }$ meets the $\frac { k } { 2 }$ balance condition. + +Proof. We construct a directed graph $G$ with nodes in $\left[ \frac { k } { 2 } \right]$ . For each $\begin{array} { r } { 0 \leq i < \frac { k } { 2 } } \end{array}$ we add a directed edge from node $\left( s \mod \frac { k } { 2 } \right)$  to node $\left( t \mod \frac { k } { 2 } \right)$ if $\mathbf { M } [ : , i ] = \mathbf { e } _ { s }$ and $\begin{array} { r } { \mathbf { M } \left[ : , i + \frac { k } { 2 } \right] = \mathbf { e } _ { t } } \end{array}$ . Each node has (undirected) degree exactly 2 by the structure of M. Hence, $G$ is a union of disjoint (undirected) cycles. + +If M met the $\frac { k } { 2 }$ balance condition, then each node would additionally have in-degree exactly 1 and out-degree exactly 1. By reversing edges of $G$ such that each (undirected) cycle becomes a directed cycle, we can achieve this. However, reversing edges corresponds to swapping columns of $\mathbf { M }$ that are $\frac { k } { 2 }$ apart. Let $\mathbf { B } _ { k }$ be the permutation matrix that performs all such swaps. $\mathbf { B } _ { k }$ has non-zero entries only along the main diagonal and the diagonal bands $\frac { k } { 2 }$ away, and thus is a butterfly factor of size $k$ . □ + +We are ready to present the decomposition of $\mathbf { P }$ + +Lemma G.3. Let $\mathbf { P }$ be an $n \times n$ permutation matrix. Then we can decompose $\mathbf { P }$ into $\mathbf { P } = \mathbf { L R }$ , where L is modular-balanced and $\mathbf { R } \in B ^ { * }$ . + +Proof. We repeatedly apply Lemma G.2. First, we conclude that there is a butterfly factor $\mathbf { B } _ { n }$ such that + +$$ +\mathbf { P B } _ { n } = \mathbf { P } ^ { \prime } , +$$ + +where $\mathbf { P ^ { \prime } }$ meets the $\frac { n } { 2 }$ balance condition. Now, we consider the first and last $\frac { n } { 2 }$ columns of $\mathbf { P ^ { \prime } }$ independently. We can again apply Lemma G.2 (twice) to conclude that there are butterfly factors $\left[ \mathbf { B } _ { \frac { n } { 2 } } \right] _ { 1 } , \left[ \mathbf { B } _ { \frac { n } { 2 } } \right] _ { 2 }$ such that + +$$ +\mathbf { P B } _ { n } \left[ \begin{array} { c c } { \left[ \mathbf { B } _ { \frac { n } { 2 } } \right] _ { 1 } } & { 0 } \\ { 0 } & { \left[ \mathbf { B } _ { \frac { n } { 2 } } \right] _ { 2 } } \end{array} \right] = \mathbf { P B } _ { n } ^ { ( n ) } \mathbf { B } _ { \frac { n } { 2 } } ^ { ( n ) } = \mathbf { P } ^ { \prime \prime } , +$$ + +where $\mathbf { P } ^ { \prime \prime }$ meets the $\frac { n } { 2 }$ and $\frac { n } { 4 }$ balance conditions. + +We continue this process until we obtain a matrix that meets all of the balance conditions. Our final equation is of the form: + +$$ +\mathbf { P } \cdot \mathbf { B } _ { n } ^ { \left( n \right) } \mathbf { B } _ { \frac { n } { 2 } } ^ { \left( n \right) } \cdot \ldots \mathbf { B } _ { 2 } ^ { \left( n \right) } = \mathbf { P B } = \mathbf { L } , +$$ + +where $\boldsymbol { B }$ is a butterfly matrix and $\mathbf { L }$ is a modular-balanced matrix. Let $\mathbf { R } = \mathbf { B } ^ { - 1 } = \mathbf { B } ^ { * }$ (since $\mathbf { B }$ is a permutation matrix, and thus is orthogonal) and hence $\mathbf { R } \in B ^ { * }$ . Then $\mathbf { P } = \mathbf { L R }$ , as desired. □ + +Theorem 2 follows immediately from Lemmas G.3 and G.1. + +We now show that permutations are also in $B ^ { * } B$ . We start with the relationship between butterfly matrices and the bit-reversal permutation. + +Lemma G.4. Let $\mathbf { P } _ { \mathrm { b r } }$ be the $n \times n$ bit-reversal permutation matrix where n is some power of 2, and let ${ \bf M } _ { 1 } \in { B }$ be an $n \times n$ butterfly matrix. Then there is some butterfly matrix ${ \bf M } _ { 2 } \in { B }$ such that + +$$ +\mathbf { M } _ { 1 } ^ { * } = \mathbf { P } _ { \mathrm { b r } } \mathbf { M } _ { 2 } \mathbf { P } _ { \mathrm { b r } } . +$$ + +Proof sketch. For any input vector $\mathbf { x }$ of length $n$ , to perform $\mathbf { M } _ { 1 } ^ { * } \mathbf { x }$ , we trace through $\log _ { 2 } n$ steps of the multiplication algorithm. At each step, we perform $2 \times 2$ matrix multiplication on elements of $\mathbf { x }$ whose indices are $n / 2$ apart (e.g. indices 0 and $n / 2 , 1$ and $n / 2 + 1$ , etc.), then $n / 4$ apart, and so on, till indices are that 1 apart. If we apply the bit-reversal permutation on $\mathbf { x }$ , then indices that are $n / 2$ apart will become 1 apart, indices that are $n / 4$ apart will become 2 apart, and so on. So the multiplication algorithm $\mathbf { M } _ { 1 } ^ { * } \mathbf { x }$ is equivalent to applying bit-reversal, then multiplying the permuted vector with another butterfly matrix (i.e. $2 \times 2$ matrix multiplication on indices that are 1 apart, then 2 apart, and so on, till indices that are $n / 2$ apart). Finally we need to do another bit-reversal permutation to put all the indices back to the original order. If we call this other butterfly matrix $\mathbf { M } _ { 2 }$ , then we have shown that $\mathbf { M } _ { 1 } ^ { * } \mathbf { x } = \mathbf { P } _ { \mathrm { b r } } \mathbf { M } _ { 2 } \mathbf { P } _ { \mathrm { b r } } \mathbf { x }$ . This holds for all $\mathbf { x }$ (for the same matrix $\mathbf { M } _ { 2 }$ ), so we have $\mathbf { M } _ { 1 } ^ { * } = \mathbf { P } _ { \mathrm { b r } } \mathbf { M } _ { 2 } \mathbf { P } _ { \mathrm { b r } }$ . □ + +Remark G.5. Lemma G.4 explains the connection between the two most common fast Fourier transform algorithm, decimation in time and decimation in frequency. Using the decimation-in-time + +FFT, we can write the DFT matrix $\mathbf { F }$ as product of a butterfly matrix ${ { \bf { M } } _ { 1 } }$ and the bit-reversal permutation (see Section J): + +$$ +\mathbf { F } = \mathbf { M } _ { 1 } \mathbf { P } _ { \mathrm { b r } } . +$$ + +Taking conjugate transpose, we obtain $\mathbf { F } ^ { * } = \mathbf { P } _ { \mathrm { b r } } \mathbf { M } _ { 1 } ^ { * }$ (recall that $\mathbf { P } _ { \mathrm { b r } }$ is its own transpose/inverse). On the other hand, $\mathbf { F } ^ { * }$ is just a scaled version of the inverse DFT matrix, so apply decimation-in-time FFT to the inverse DFT, we can write $\mathbf { F } ^ { * } = \mathbf { M } _ { 2 } \mathbf { P } _ { \mathrm { b r } }$ for some other butterfly matrix $\mathbf { M } _ { 2 }$ . Hence $\mathbf { P } _ { \mathrm { b r } } \mathbf { M } _ { 1 } ^ { * } = \mathbf { M } _ { 2 } \mathbf { P } _ { \mathrm { b r } } ,$ , and thus $\mathbf { P } _ { \mathrm { b r } } \mathbf { M } _ { 1 } ^ { * } \mathbf { P } _ { \mathrm { b r } } = \mathbf { M } _ { 2 }$ (for these particular butterfly matrices ${ { \bf { M } } _ { 1 } }$ and ${ { \bf { M } } _ { 2 } }$ ). Note that this yields another decomposition of the DFT matrix, $\mathbf { F } = \mathbf { P } _ { \mathrm { b r } } \mathbf { M } _ { 2 } ^ { * }$ , which is exactly the decimation-in-frequency FFT algorithm. + +We are ready to show that permutations are in $B ^ { * } B$ . + +Lemma G.6. Let $\mathbf { P }$ be an $n \times n$ permutation matrix (with n a power of 2). Then there are butterfly matrices $\mathbf { M } _ { 1 } , \mathbf { M } _ { 2 } \in B$ such that $\mathbf { P } = \mathbf { M } _ { 1 } ^ { * } \mathbf { M } _ { 2 }$ . + +Proof. Consider the permutation $\tilde { \mathbf { P } } = \mathbf { P } _ { \mathrm { b r } } \mathbf { P } \mathbf { P } _ { \mathrm { b r } }$ . By Theorem 2, there are some butterfly matrices $\tilde { \mathbf { M } } _ { 1 } , \tilde { \mathbf { M } } _ { 2 } \in B$ such that $\tilde { \mathbf { P } } = \tilde { \mathbf { M } } _ { 1 } \tilde { \mathbf { M } } _ { 2 } ^ { * }$ . Applying Lemma G.4, we can replace ${ { \tilde { \mathbf { M } } } _ { 2 } } ^ { * }$ with $\mathbf { P } _ { \mathrm { b r } } \mathbf { M } _ { 2 } \mathbf { P } _ { \mathrm { b r } }$ for some butterfly matrix ${ \bf M } _ { 2 } \in { B }$ . We thus have: + +$$ +\mathbf { P } _ { \mathrm { b r } } \mathbf { P } \mathbf { P } _ { \mathrm { b r } } = \tilde { \mathbf { M } } _ { 1 } \mathbf { P } _ { \mathrm { b r } } \mathbf { M } _ { 2 } \mathbf { P } _ { \mathrm { b r } } . +$$ + +Pre- and post-multiply both sides by $\mathbf { P } _ { \mathrm { b r } }$ (which is its own inverse): + +$$ +\mathbf { P } = \mathbf { P } _ { \mathrm { b r } } \tilde { \mathbf { M } } _ { 1 } \mathbf { P } _ { \mathrm { b r } } \mathbf { M } _ { 2 } . +$$ + +Applying Lemma G.4 again, we can replace $\mathbf { P } _ { \mathrm { b r } } \mathbf { \tilde { M } } _ { 1 } \mathbf { P } _ { \mathrm { b r } }$ with $\mathbf { M } _ { 1 } ^ { * }$ for some butterfly matrix ${ \bf M } _ { 1 } \in { B }$ . Thus: + +$$ +\mathbf { P } = \mathbf { M } _ { 1 } ^ { * } \mathbf { M } _ { 2 } . +$$ + +# H $B B ^ { * }$ CLOSURE LEMMAS + +Here, we present some basic facts of the $B B ^ { * }$ hierarchy that will be useful for later constructions. For simplicity, we assume (WLOG via 0-padding) that all matrices are square matrices with size that is a power of 2. + +Lemma H.1. If $\mathbf { M } \in B$ (or $\mathbf { M } \in B ^ { * }$ ), then DM, $\mathbf { M D } \in B$ ( $B ^ { * }$ resp.) for any diagonal matrix D. + +Proof. Left multiplication by a diagonal matrix scales the rows of M by the corresponding diagonal entries. The same can be achieved by scaling all entries the leftmost butterfly factor matrix. Similarly, right multiplication by a diagonal matrix scales the columns of $\mathbf { M }$ , which can be achieved by scaling all entries in the columns of the rightmost butterfly factor matrix. + +Lemma H.2. Let A, $\mathbf { B } \in \mathbb { F } ^ { n \times n }$ . If $\mathbf { \dot { A } } \in ( B B ^ { * } ) _ { e } ^ { w _ { 1 } }$ and $\mathbf { B } \in ( B B ^ { * } ) _ { e } ^ { w _ { 2 } }$ then $\mathbf { A B } \in ( B B ^ { * } ) _ { e } ^ { w _ { 1 } + w _ { 2 } } .$ + +Proof. Let $\mathbf { E _ { A } } , \mathbf { E _ { B } } \in \mathbb { F } ^ { e n \times e n }$ be defined such that $\mathbf { A } = \mathbf { S } \mathbf { E } _ { \mathbf { A } } \mathbf { S } ^ { T }$ , ${ \bf B } = { \bf S } { \bf E } _ { \bf B } { \bf S } ^ { T }$ (with S as in Definition 2.4). Then + +$$ +\mathbf { A } \mathbf { B } = \mathbf { S } \underbrace { \left[ \mathbf { I } _ { n } \mathbf { \Gamma } _ { 0 } \right] } _ { e n \times e n } \mathbf { E } _ { \mathbf { A } } \underbrace { \left[ \mathbf { I } _ { n } \mathbf { \Gamma } _ { 0 } \right] } _ { e n \times e n } \mathbf { E } _ { \mathbf { B } } \mathbf { S } ^ { T } +$$ + +$\begin{array} { r l } { \bigg [ \mathbf { I } _ { n } } & { { } 0 \bigg ] \mathbf { E _ { A } } \in ( B B ^ { * } ) ^ { w _ { 1 } } , \bigg [ \mathbf { I } _ { n } \quad 0 \bigg ] \mathbf { E _ { B } } \in ( B B ^ { * } ) ^ { w _ { 2 } } } \end{array}$ by Lemma H.1. Hence, $\mathbf { A B } \in ( B B ^ { * } ) _ { e } ^ { w _ { 1 } + w _ { 2 } }$ by Definition 2.4. + +Lemma H.3. Let $\mathbf { A } _ { 1 } , \ldots , \mathbf { A } _ { m } \in \mathbb { F } ^ { k \times k }$ . If $\mathbf { A } _ { 1 } , \ldots , \mathbf { A } _ { m } \in ( B B ^ { * } ) _ { e } ^ { w }$ then $\mathrm { D i a g } ( \mathbf { A } _ { 1 } , \ldots , \mathbf { A } _ { m } ) \in$ $( B B ^ { * } ) _ { e } ^ { w + 2 }$ . + +Proof. For each $1 \leq i \leq m$ , let $\mathbf { E } _ { \mathbf { A } _ { i } } \in \mathbb { F } ^ { e k \times e k }$ be defined such that $\mathbf { A } _ { i } = \mathbf { S } \mathbf { E } _ { \mathbf { A } _ { i } } \mathbf { S } ^ { T }$ (with $\mathbf { S }$ as in Definition 2.4). Then + +$$ +\left[ \begin{array} { c c c c } { \mathbf { A } _ { 1 } } & { 0 } & { . . . } & { 0 } \\ { 0 } & { \mathbf { A } _ { 2 } } & { . . . } & { 0 } \\ { \vdots } & { \vdots } & { \ddots } & { 0 } \\ { 0 } & { 0 } & { . . . } & { \mathbf { A } _ { m } } \end{array} \right] = \mathbf { S } \mathbf { P } \left[ \begin{array} { c c c c } { \mathbf { E } _ { \mathbf { A } _ { 1 } } } & { 0 } & { . . . } & { 0 } \\ { 0 } & { \mathbf { E } _ { \mathbf { A } _ { 2 } } } & { . . . } & { 0 } \\ { \vdots } & { \vdots } & { \ddots } & { 0 } \\ { 0 } & { 0 } & { . . . } & { \mathbf { E } _ { \mathbf { A } _ { m } } } \end{array} \right] \mathbf { P } ^ { T } \mathbf { S } ^ { T } +$$ + +where $\mathbf { P }$ is a permutation that that moves the first $k$ rows of each $\mathbf { E } _ { \mathbf { A } _ { i } }$ (in order) into the top mk rows. From Theorem 2, $\mathbf { P } \in B B ^ { * }$ , (and so is $\mathbf { P } ^ { T }$ , also a permutation). Within the RHS block matrix, the decompositions of each $\mathbf { E } _ { \mathbf { A } _ { i } }$ can be done in parallel, requiring total width $w$ . Hence, $\mathrm { D i a g } ( \mathbf { A } _ { 1 } , \ldots , \mathbf { A } _ { m } ) \in ( B B ^ { * } ) _ { e } ^ { w + 2 }$ , as desired. □ + +Remark H.4. If $\textit { e } = \ 1$ in Lemma H.3, then $\mathbf { P }$ is unnecessary. Hence, $\mathrm { D i a g } ( \mathbf { A } _ { 1 } , \ldots , \mathbf { A } _ { m } ) \in$ $( B B ^ { * } ) ^ { w }$ . + +Lemma H.5. Let $\mathbf { A } _ { 1 } , \ldots , \mathbf { A } _ { m }$ be $k \times k$ matrices in $( B B ^ { * } ) _ { e } ^ { w }$ then $\begin{array} { r } { \sum _ { i = 1 } ^ { m } { \bf A } _ { i } \in ( B B ^ { * } ) _ { 4 e } ^ { m w } . } \end{array}$ + +Proof. For each $1 \leq i \leq m$ , let $\mathbf { E } _ { \mathbf { A } _ { i } } \in \mathbb { F } ^ { e k \times e k }$ be defined such that $\mathbf { A } _ { i } = \mathbf { S } \mathbf { E } _ { \mathbf { A } _ { i } } \mathbf { S } ^ { T }$ (with $\mathbf { S }$ as in Definition 2.4). Note that $\mathbf { E } _ { \mathbf { A } _ { i } } \in ( B B ^ { * } ) ^ { w }$ . Consider matrices of the form: + +$$ +\underbrace { \left[ \begin{array} { c c c c } { \mathbf { I } _ { e k } } & { \mathbf { E } _ { \mathbf { A } _ { i } } } & { 0 } & { 0 } \\ { 0 } & { \mathbf { I } _ { e k } } & { 0 } & { 0 } \\ { 0 } & { 0 } & { 0 } & { 0 } \\ { \mathbf { M } _ { i } \in \mathbb { R } ^ { 4 e k \times 4 e k } } & { \mathbf { L } } \end{array} \right] } _ { \mathbf { L } } = \underbrace { \left[ \mathbf { I } _ { 2 e k } \right. } _ { \mathbf { L } } , \mathbf { I } _ { 2 e k } \underbrace { } _ { 0 } \underbrace { \left[ \begin{array} { c c c c } { \mathbf { I } _ { e k } } & { 0 } & { 0 } & { 0 } \\ { 0 } & { \mathbf { I } _ { e k } } & { 0 } & { 0 } \\ { 0 } & { 0 } & { \mathbf { E } _ { \mathbf { A } _ { i } } } & { 0 } \\ { 0 } & { 0 } & { 0 } & { 0 } \end{array} \right] } _ { \mathbf { S } } \underbrace { \left[ \begin{array} { c c c c } { \mathbf { I } _ { e k } } & { 0 } & { 0 } & { 0 } \\ { 0 } & { \mathbf { I } _ { e k } } & { 0 } & { 0 } \\ { 0 } & { 0 } & { 0 } & { \mathbf { I } _ { e k } } \\ { 0 } & { 0 } & { \mathbf { I } _ { e k } } & { 0 } \end{array} \right] } _ { \mathbf { P } _ { 1 } } \underbrace { \left[ \mathbf { I } _ { 2 e k } \right]} _ { \mathbf { R } } , \mathbf { I } _ { \mathbf { R } } . +$$ + +Here, $\mathbf { L }$ and $\mathbf { R }$ compute the sum of the $2 e k \times 2 e k$ matrices on the diagonal of $\mathbf { S P } _ { 1 }$ , where ${ \bf P } _ { 1 }$ is a permutation swapping $\mathbf { E } _ { \mathbf { A } _ { i } }$ to the $4 ^ { t h } ~ e k$ -block column. Note that S is the diagonalization of four matrices in $( B B ^ { * } ) ^ { w }$ , so $\mathbf { S } \in ( B B ^ { * } ) ^ { w }$ by Remark H.4. In addition, since each block in $\mathbf { S }$ is a butterfly matrix of size $e k$ , S only uses butterfly factors up to size $e k$ , so the outer factor matrices of sizes $4 e k$ and and $2 e k$ in S are unused. Also note that is a butterfly factor matrix of s $\mathbf { L }$ e $\mathbf { R }$ ar(or fly factor matrices of size ). This allows us to fold $4 e k$ (or urr (4ek)), g ${ \bf P } _ { 1 }$ $2 e k$ $\mathbf { B } _ { 2 e k } ^ { ( 4 e k ) } .$ +matrices $\mathbf { L } , \mathbf { P } _ { 1 } , \mathbf { R }$ into S, so $\mathbf { M } _ { i } \in ( B B ^ { \ast } ) ^ { w }$ . + +Through repeated application $\textbar { m }$ times) of the identity + +$$ +\left[ { \begin{array} { c c } { \mathbf { I } } & { \mathbf { A } } \\ { 0 } & { \mathbf { I } } \end{array} } \right] \left[ { \begin{array} { c c } { \mathbf { I } } & { \mathbf { B } } \\ { 0 } & { \mathbf { I } } \end{array} } \right] = \left[ { \begin{array} { c c } { \mathbf { I } } & { \mathbf { A } + \mathbf { B } } \\ { 0 } & { \mathbf { I } } \end{array} } \right] , +$$ + +we see that + +$$ +\underbrace { \left[ \begin{array} { c c c c } { \mathbf { I } _ { e k } } & { \sum _ { i = 1 } ^ { m } \mathbf { E } _ { \mathbf { A } _ { i } } } & { 0 } & { 0 } \\ { 0 } & { \mathbf { I } _ { e k } } & { 0 } & { 0 } \\ { 0 } & { 0 } & { 0 } & { 0 } \\ { 0 } & { 0 } & { 0 } & { 0 } \end{array} \right] } _ { \mathbf { M } \in \mathbb { R } ^ { 4 e n \times 4 e n } } = \prod _ { i = 1 } ^ { m } \mathbf { M } _ { i } . +$$ + +From Lemma H.2, $\mathbf { M } \in \mathsf { \Gamma } ( B B ^ { * } ) ^ { m w }$ . Finally, note that $\begin{array} { r } { \sum _ { i = 1 } ^ { m } \mathbf { A } _ { i } = \mathbf { S } \mathbf { M } \mathbf { P } _ { 2 } \mathbf { S } ^ { T } } \end{array}$ , where $\mathbf { P } _ { 2 }$ is a permutation that moves the first $k$ columns of the second block-column of M to the left. $\mathbf { P } _ { 2 }$ can be folded into the final summation factor $M _ { m }$ as follows: + +$$ +\begin{array}{c} \begin{array}{c} \underbrace { \left[ \begin{array} { c c c c } { \mathbf { I } _ { e k } } & { \mathbf { 0 } } & { \mathbf { 0 } } & { \mathbf { 0 } } \\ { \mathbf { 0 } } & { \mathbf { I } _ { e k } } & { \mathbf { 0 } } & { \mathbf { 0 } } \\ { \mathbf { 0 } } & { \mathbf { 0 } } & { \mathbf { I } _ { e k } } \\ { \mathbf { 0 } } & { \mathbf { D } } & { \mathbf { I } _ { e k } } & { \mathbf { 0 } } \end{array} \right] } _ { \mathbf { P } _ { 1 } } \underbrace { \left[ \mathbf { I } _ { 2 e k } \right]} & { \mathbf { 0 } } \\ & & { \mathbf { R } } \end{array} _ { \mathbf { R } } \underbrace { \left[ \begin{array} { c c c c } { \mathbf { 0 } } & { \mathbf { I } _ { e k } } & { \mathbf { 0 } } & { \mathbf { 0 } } \\ { \mathbf { I } _ { e k } } & { \mathbf { 0 } } & { \mathbf { 0 } } & { \mathbf { 0 } } \\ { \mathbf { 0 } } & { \mathbf { 0 } } & { \mathbf { I } _ { e k } } & { \mathbf { 0 } } \\ { \mathbf { 0 } } & { \mathbf { 0 } } & { \mathbf { 0 } } & { \mathbf { I } _ { e k } } \end{array} \right] } _ { \mathbf { P } _ { 2 } } = \underbrace { \left[ \begin{array} { c c c c } { \mathbf { 0 } } & { \mathbf { I } _ { e k } } & { \mathbf { 0 } } & { \mathbf { 0 } } \\ { \mathbf { I } _ { e k } } & { \mathbf { 0 } } & { \mathbf { 0 } } & { \mathbf { 0 } } \\ { \mathbf { 0 } } & { \mathbf { 0 } } & { \mathbf { I } _ { e k } } & { \mathbf { 0 } } \\ { \mathbf { 0 } } & { \mathbf { 0 } } & { \mathbf { 0 } } & { \mathbf { I } _ { e k } } \end{array} \right] } _ { \mathbf { P } _ { 1 } ^ { \prime } } \underbrace { \left[ \mathbf { I } _ { 2 e k } & { \mathbf { 0 } } \\ & & { \mathbf { R } } \end{array} \right] } _ { \mathbf { R } } \underbrace { \left[ \begin{array} { c c c c } { \mathbf { I } _ { e k } } & { \mathbf { 0 } } & { \mathbf { I } _ { e k } } & { \mathbf { 0 } } \\ & & { \mathbf { I } _ { e k } } & { \mathbf { 0 } } \\ & & { \mathbf { P } _ { 2 } ^ { \prime } } & { \mathbf { R } } \end{array} \right] } _ \mathbf +$$ + +Hence, $\begin{array} { r } { \sum _ { i = 1 } ^ { m } \mathbf { A } _ { i } \in ( B B ^ { * } ) _ { 4 e } ^ { m w } } \end{array}$ , as desired. + +Lemma H.6. Let M be an invertible $n \times n$ matrix such that $M \in B$ . Then $\mathbf { M } ^ { - 1 } \in { \mathfrak { B } } ^ { * }$ + +Proof. We prove this in a series of steps. + +First, let $\mathbf { B } _ { k }$ be an invertible butterfly factor of size $k$ . Consider the method of computing $\mathbf { B } _ { k } ^ { - 1 }$ by performing Gaussian elimination on the matrix $[ \mathbf { B } _ { k } | \mathbf { I } _ { k } ]$ to obtain the matrix $\left[ \mathbf { I } _ { k } | \mathbf { B } _ { k } ^ { - 1 } \right]$ . By the form of B, non-zero entries within a row or column are always exactly $\frac { k } { 2 }$ positions apart. Therefore, the only row operations needed for this Gaussian elimination are: + +• Scaling a row by a constant factor $c \neq 0$ • Addition of a row to another row exactly $\frac { k } { 2 }$ rows apart + +Performing these operations on $\mathbf { I } _ { k }$ will only allow non-zeros on the main diagonal and $\frac { k } { 2 }$ diagonals away from the main diagonal. Hence, $\mathbf { B } _ { k } ^ { - 1 }$ is also a butterfly factor of size $k$ . + +Next, let $\mathbf { B } _ { k } ^ { \left( n \right) }$ be an invertible butterfly factor matrix of size $n$ and block size $k$ . Its inverse is the block diagonal matrix formed by the inverses of each of its constituent butterfly factors. From above, $\left( \mathbf { B } _ { k } ^ { \left( n \right) } \right) ^ { - 1 }$ is also a butterfly factor matrix of size $n$ and block size $k$ . + +Finally, consider $\mathbf { M } \in B$ . + +$$ +\mathbf { { \cal { M } } } ^ { - 1 } = \left( \mathbf { B } _ { n } ^ { ( n ) } \mathbf { B } _ { \frac { n } { 2 } } ^ { ( n ) } \dots \mathbf { B } _ { 2 } ^ { ( n ) } \right) ^ { - 1 } = \left( \mathbf { B } _ { 2 } ^ { ( n ) } \right) ^ { - 1 } \left( \mathbf { B } _ { 4 } ^ { ( n ) } \right) ^ { - 1 } \dots \left( \mathbf { B } _ { n } ^ { ( n ) } \right) ^ { - 1 } = \mathbf { B } _ { 2 } ^ { \prime { } ^ { ( n ) } } \mathbf { B } _ { 4 } ^ { \prime { } ^ { ( n ) } } \dots \mathbf { B } _ { n } ^ { \prime { } ^ { ( n ) } } \in \mathcal { B } _ { n } ^ { ( n ) } . +$$ + +Finally, we include a closure result for the Kronecker product, another common matrix composition operation. Although Lemma H.7 is not directly used in the subsequent proofs, it allows for examples the results for the DFT to be lifted to higher-dimensional Fourier transforms. We also note that the closure bound in Lemma H.7 can be tightened in such cases (cfRemark H.4). ˙ + +Lemma H.7. Let A, $\mathbf { B } \in \mathbb { F } ^ { n \times n }$ . I $^ { c } \mathbf { A } \in ( B B ^ { * } ) _ { e } ^ { w _ { 1 } }$ and $\mathbf { B } \in ( B B ^ { * } ) _ { e } ^ { w _ { 2 } }$ then $\mathbf { A } \otimes \mathbf { B } \in ( B B ^ { * } ) _ { e } ^ { w _ { 1 } + w _ { 2 } + 6 }$ + +Proof. Note that + +$$ +\mathbf { A } \otimes \mathbf { B } = ( \mathbf { A } \otimes \mathbf { I } ) ( \mathbf { I } \otimes \mathbf { B } ) = \mathbf { P } ^ { - 1 } ( \mathbf { I } \otimes \mathbf { A } ) \mathbf { P } ( \mathbf { I } \otimes \mathbf { B } ) , +$$ + +for some permutation $\mathbf { P }$ . By Lemma H.3, $\mathbf { I } \otimes \mathbf { A }$ and $\mathbf { I } \otimes \mathbf { B }$ are in $( B B ^ { * } ) _ { e } ^ { w _ { 1 } + 2 } , ( B B ^ { * } ) _ { e } ^ { w _ { 2 } + 2 }$ respectively. The result follows from combining with $\mathbf { P } \in B B ^ { * }$ and Lemma H.2. □ + +# I SPARSE MATRICES IN $B B ^ { * }$ HIERARCHY + +In this appendix, we prove Theorem 3. First, we consider matrices with at most $n$ NNZ. + +Lemma I.1. let S be an $n \times n$ matrix with at most n NNZ. Then, $\mathbf { S } \in ( B B ^ { * } ) ^ { 4 }$ . + +We use this lemma and the addition closure lemma to prove Theorem 3. + +Proof of Theorem 3. We note that any $s$ sparse matrix is the sum of $\textstyle { \left\lceil { \frac { s } { n } } \right\rceil }$ matrices of at most $n$ NNZ, and we appeal to Lemma H.5. □ + +In the rest of the section we will prove Lemma I.1. We begin by defining two classes of matrices that will be used in our decomposition. + +Definition I.1. An $n \times n$ matrix $\mathbf { H }$ is a horizontal step matrix if for every $0 \leq i , i ^ { \prime } < n$ and $0 \leq j \leq j ^ { \prime } < n ,$ , $i f \mathbf { H } [ i , j ] \neq 0$ and $\mathbf { H } [ i ^ { \prime } , j ^ { \prime } ] \neq 0$ , then $j ^ { \prime } - j \geq ( i ^ { \prime } - i )$ mod $n$ . + +An $n \times n$ matrix $\mathbf { V }$ is a vertical step matrix if $\mathbf { V } ^ { * }$ is a horizontal step matrix. + +With this definition, the horizontal step matrix obeys a “Lipschitz-like" condition. Each column of a horizontal step matrix can have at most one non-zero entry, and given two non-zero columns $k$ apart, the non-zero entry in the right column must be between 0 and $k$ rows below the non-zero entry in the left column. Note that to show that a matrix is a horizontal step matrix, it is sufficient to argue that this condition holds for each pair of neighboring non-zero columns. + +Similarly, each row of a vertical step matrix can have at most one non-zero entry, and given two non-zero rows $k$ apart, the non-zero entry in the lower row must be between 0 and $k$ columns to the right of the non-zero entry in the upper row. + +Lemma I.2. Let $\mathbf { H }$ be an $n \times n$ horizontal step matrix. Then $\mathbf { H } \in B$ . + +Proof. We proceed by induction on $n$ . The base case $n = 2$ is trivial. As our inductive hypothesis, we assume that all horizontal step matrices of size ${ \begin{array} { l } { { \frac { n } { 2 } } } \end{array} } \times { \begin{array} { l } { { \frac { n } { 2 } } } \end{array} }$ are butterfly matrices of size $\frac { n } { 2 }$ . From Definition 2.3, it is sufficient to show that $\mathbf { H }$ can be decomposed as: + +$$ +\mathbf { H } = \left[ \mathbf { D } _ { 1 } \quad \mathbf { D } _ { 2 } \right] \left[ \mathbf { H } _ { 1 } \quad \mathbf { \Lambda } _ { \mathbf { H } _ { 2 } } ^ { 0 } \right] = \left[ \mathbf { D } _ { 1 } \mathbf { H } _ { 1 } \quad \mathbf { D } _ { 2 } \mathbf { H } _ { 2 } \right] , +$$ + +where $\mathbf { H } _ { 1 } , \mathbf { H } _ { 2 }$ are ${ \frac { n } { 2 } } \times { \frac { n } { 2 } }$ horizontal step matrices and each $\mathbf { D } _ { k }$ is a ${ \begin{array} { l } { { \frac { n } { 2 } } \times { \frac { n } { 2 } } } \end{array} }$ diagonal matrix. Denote the four, ${ \frac { n } { 2 } } \times { \frac { n } { 2 } }$ 2 2corner submatrices of $\mathbf { H }$ by: + +$$ +\mathbf { H } = \left[ \begin{array} { c c } { \mathbf { H } _ { 1 1 } } & { \mathbf { H } _ { 1 2 } } \\ { \mathbf { H } _ { 2 1 } } & { \mathbf { H } _ { 2 2 } } \end{array} \right] . +$$ + +Then, define $\mathbf { H } _ { 1 }$ and $\mathbf { H } _ { 2 }$ by: + +$$ +\mathbf { H } _ { 1 } = \mathbf { H } _ { 1 1 } + \mathbf { H } _ { 2 1 } \qquad \mathbf { H } _ { 2 } = \mathbf { H } _ { 1 2 } + \mathbf { H } _ { 2 2 } +$$ + +For sake of contradiction, assume that $\mathbf { H } _ { 1 }$ is not a horizontal step matrix. Then, there are $\textstyle { 0 \leq i , i ^ { \prime } < { \frac { n } { 2 } } }$ , $\begin{array} { r } { 0 \leq j \leq j ^ { \prime } < \frac { n } { 2 } } \end{array}$ such that $\mathbf { H } _ { 1 } [ i , j ] \neq 0$ , $\mathbf { H } _ { 1 } [ i ^ { \prime } , j ^ { \prime } ] \neq 0$ , and $j ^ { \prime } - j < ( i ^ { \prime } - i )$ mod $\frac { n } { 2 }$ . From our definition of $\mathbf { H } _ { 1 }$ , the non-zero entries in columns $j$ and $j ^ { \prime }$ of $\mathbf { H }$ are either $\left( \left( i ^ { \prime } - i \right) \mod { \frac { n } { 2 } } \right)$ or $\begin{array} { r } { \left( \frac { n } { 2 } + ( i ^ { \prime } - i ) \right. } \end{array}$ mod ${ \frac { n } { 2 } } .$ , both of which are greater than $j ^ { \prime } - j$ , rows apart. This contradicts $\mathbf { H }$ being a horizontal step matrix. Hence, $\mathbf { H } _ { 1 }$ must be a horizontal step matrix, as must $\mathbf { H } _ { 2 }$ from an analogous argument. + +Next, we define $\mathbf { D } _ { 1 } , \mathbf { D } _ { 2 } , \mathbf { D } _ { 3 } , \mathbf { D } _ { 4 }$ by: + +$$ +\begin{array} { r l r l } & { \mathbf { D } _ { 1 } [ k , k ] = \left\{ \begin{array} { l l } { 1 } & { \mathbf { H } _ { 2 1 } [ k , : ] = \mathbf { 0 } } \\ { 0 } & { \mathrm { o t h e r w i s e } } \end{array} \right. } & & { \mathbf { D } _ { 2 } [ k , k ] = \left\{ \begin{array} { l l } { 1 } & { \mathbf { H } _ { 2 2 } [ k , : ] = \mathbf { 0 } } \\ { 0 } & { \mathrm { o t h e r w i s e } } \end{array} \right. } \\ & { \mathbf { D } _ { 3 } [ k , k ] = \left\{ \begin{array} { l l } { 1 } & { \mathbf { H } _ { 1 1 } [ k , : ] = \mathbf { 0 } } \\ { 0 } & { \mathrm { o t h e r w i s e } . } \end{array} \right. } & & { \mathbf { D } _ { 4 } [ k , k ] = \left\{ \begin{array} { l l } { 1 } & { \mathbf { H } _ { 1 2 } [ k , : ] = \mathbf { 0 } } \\ { 0 } & { \mathrm { o t h e r w i s e } . } \end{array} \right. } \end{array} +$$ + +To finish the proof, we argue the correctness of the decomposition by equating arbitrary entries of each of the 4 corner submatrices. We begin with the upper left submatrix. + +$$ +\begin{array} { r l r } { \mathbf { D } _ { 1 } \mathbf { H } _ { 1 } \big [ i , j \big ] = \displaystyle \sum _ { k = 0 } ^ { \frac { n } { 2 } } \mathbf { D } _ { 1 } [ i , k ] \cdot \mathbf { H } _ { 1 } \big [ k , j \big ] \quad } & { \mathrm { b y ~ d e f i n i t i o n ~ o f ~ m a t r i x ~ m u l t i p l i c a t i o n } } \\ { = \mathbf { D } _ { 1 } \big [ i , i \big ] \cdot \mathbf { H } _ { 1 } \big [ i , j \big ] \quad } & { \mathbf { D } _ { 1 } \mathrm { ~ i s ~ a d i a g o n a l ~ m a t r i x } } \\ { = \mathbb { 1 } _ { ( \mathbf { H } _ { 2 1 } [ i , : ] = 0 ) } \cdot \left( \mathbf { H } _ { 1 1 } [ i , j ] + \mathbf { H } _ { 2 1 } [ i , j ] \right) \quad } & { \mathrm { b y ~ d e f i n i t i o n ~ o f ~ \mathbf { D } _ 1 ~ a n d ~ \mathbf { H } _ 1 ~ } } \end{array} +$$ + +Here, we consider two cases: + +Case $1 : \mathbf { H } _ { 2 1 } [ i , j ] \neq 0$ + +Since $\mathbf { H }$ is a horizontal step matrix (and hence may have at most one non-zero entry per column), it follows that $\mathbf { H } _ { 1 1 } [ i , j ] = 0$ . In this case, the indicator function evaluates to 0, so $\mathbf { D } _ { 1 } \mathbf { H } _ { 1 } [ i , j ] = 0 =$ $\mathbf { H } _ { 1 1 } [ i , j ]$ , as desired. + +Case $2 \colon \mathbf { H } _ { 2 1 } [ i , j ] = 0$ + +If $\mathbf { H } _ { 1 1 } [ i , j ] = 0$ , then ${ \bf D } _ { 1 } { \bf H } _ { 1 } [ i , j ] = 0 = { \bf H } _ { 1 1 } [ i , j ]$ . Otherwise, for sake of contradiction, suppose that $\mathbf { H } _ { 2 1 } [ i , : ] \neq \mathbf { 0 }$ . Then, two of the first $\frac { n } { 2 }$ columns of $\mathbf { H }$ would have non-zero entries $\frac { n } { 2 }$ rows apart, contradicting $\mathbf { H }$ being a horizontal step matrix. Hence, $\mathbf { H } _ { 2 1 } [ i , : ] = \mathbf { 0 }$ , so $\mathbf D _ { 1 } \mathbf H _ { 1 } [ i , j ] = \mathbf { \bar { H } } _ { 1 1 } [ i , j ]$ , as desired. + +In all cases, $\mathbf { D } _ { 1 } \mathbf { H } _ { 1 } [ i , j ] = \mathbf { H } _ { 1 1 } [ i , j ]$ , so our decomposition correctly recovers the upper left corner of $\mathbf { H }$ . Analogous arguments show that the other three corners are also correctly recovered. Hence, our decomposition is correct, and by induction, $\mathbf { H } \in B$ . □ + +Corollary I.3. Let $\mathbf { V }$ be a vertical step matrix. Then $\mathbf { V } \in B ^ { * }$ . + +![](images/d4daf5b9090c82cf9774fe7121e9c63dac335d55f5330daff1bf763ab1367e88.jpg) +Figure 9: Decomposition of $4 \times 4$ sparse matrix S into $\mathbf { P } _ { 1 } \mathbf { H } \mathbf { P } _ { 2 } \mathbf { V } \mathbf { P } _ { 3 }$ + +Now, we use step matrices to prove Lemma I.1. + +Proof of Lemma I.1. Given S, we decompose it as $\mathbf { S } = \mathbf { P } _ { 1 } \mathbf { H } \mathbf { P } _ { 2 } \mathbf { V } \mathbf { P } _ { 3 }$ , where each $P _ { \ell }$ is a permutation matrix, $\mathbf { H }$ is a horizontal step matrix, and $\mathbf { V }$ is a vertical step matrix. For an example of this, see Figure 9. + +We first decompose S as $\mathbf { S } = \mathbf { P } _ { 1 } \mathbf { S } ^ { \prime } \mathbf { P } _ { 3 }$ , where ${ \bf P } _ { 1 }$ is the permutation that moves all 0 rows of S to the bottom and $\mathbf { P } _ { 3 }$ is the permutation that moves all 0 columns of S to the right. + +Next, we further decompose $\mathbf { S } ^ { \prime }$ into $\mathbf { S } ^ { \prime } = \mathbf { H } \mathbf { V } ^ { \prime }$ as follows. Since $\mathbf { S } ^ { \prime }$ has $s \leq n$ NNZ, we can parameterize $\mathbf { S } ^ { \prime }$ by $\theta = \{ ( c _ { k } , i _ { k } , j _ { k } ) : 0 \leq k < s \}$ such that $\mathbf { S } ^ { \prime } [ i _ { k } , j _ { k } ] = c _ { k }$ , with the non-zero entries indexed in row-major order. Define matrix $\mathbf { H }$ by: + +$$ +\mathbf { H } [ : , k ] = { \left\{ \begin{array} { l l } { c _ { k } \cdot \mathbf { e } _ { i _ { k } } } & { 0 \leq k < s } \\ { \mathbf { 0 } } & { { \mathrm { o t h e r w i s e } } . } \end{array} \right. } +$$ + +Define matrix $\mathbf { V } ^ { \prime }$ by: + +$$ +\begin{array} { r } { \mathbf { V } ^ { \prime } [ k , : ] = \left\{ \begin{array} { l l } { \mathbf { e } _ { j _ { k } } ^ { T } } & { 0 \leq k < s } \\ { \mathbf { 0 } } & { \mathrm { o t h e r w i s e } . } \end{array} \right. } \end{array} +$$ + +To show that $\mathbf { S } ^ { \prime } = \mathbf { H } \mathbf { V } ^ { \prime }$ , we consider an arbitrary entry: + +$$ +\begin{array} { r l } { \mathbf { H } \mathbf { V } ^ { \prime } [ i , j ] = \displaystyle \sum _ { k = 0 } ^ { n } \mathbf { H } [ i , k ] \cdot \mathbf { V } ^ { \prime } [ k , j ] \qquad } & { \mathrm { ~ b y ~ d e f i n i t i o n ~ o f ~ m a t r i x ~ m u l t i p l i c a t i o n } } \\ { = \displaystyle \sum _ { k = 0 } ^ { s } \mathbf { H } [ i , k ] \cdot \mathbf { V } ^ { \prime } [ k , j ] \qquad } & { \mathrm { ~ H ~ i s ~ 0 ~ i n ~ a l l ~ b u t ~ f i r s t ~ s ~ c o l u m n s } } \\ { = \displaystyle \sum _ { k = 0 } ^ { s } c _ { k } \cdot \mathbb { I } _ { i = i _ { k } } \cdot \mathbb { I } _ { j = j _ { k } } \qquad } & { \mathrm { ~ b y ~ d e f i n i t i o n ~ o f ~ } \mathbf { H } \mathrm { ~ a n d ~ } \mathbf { V } ^ { \prime } } \end{array} +$$ + +Here, we note that $( i , j )$ can equal $( i _ { k } , j _ { k } )$ for at most one value of $k$ since the locations in $\theta$ are unique. Hence, $\mathbf { H } \mathbf { V } ^ { \prime } [ i , j ] = c _ { k }$ only if $( i , j ) = ( i _ { k } , j _ { k } )$ for some $k$ , which is exactly the definition of $\mathbf { S } ^ { \prime }$ . Hence, $\mathbf { S } ^ { \prime } = \mathbf { H } \mathbf { V } ^ { \prime }$ . + +We argue that $\mathbf { H }$ is a horizontal step matrix through a series of assertions. First, note that $\mathbf { H }$ has exactly one non-zero entry in each of its first $s$ columns. Also, note that since $\theta$ is in row-major order, these non-zero entries are sorted (any column to the right cannot have a non-zero entry in a higher row). Hence, to show that $\mathbf { H }$ is a horizontal step matrix, it is sufficient to argue that adjacent columns of $\mathbf { H }$ have non-zero entries at most one row apart. This is equivalent to $\mathbf { S } ^ { \prime }$ having no zero rows between two non-zero rows, which is guaranteed by ${ \bf P } _ { 1 }$ . Hence, $\mathbf { H }$ is a horizontal step matrix. + +Since $\mathbf { V } ^ { \prime }$ has at most one non-zero entry per row, we may permute the rows of $\mathbf { V } ^ { \prime }$ to obtain a matrix $\mathbf { V }$ , where the non-zero entries of $\mathbf { V }$ are sorted (any lower row below cannot have a non-zero entry in an earlier column). Hence, for some permutation matrix $( \mathbf { P } _ { 2 } ) ^ { - 1 }$ , $\mathbf { V } = ( \mathbf { P } _ { 2 } ) ^ { - 1 } \mathbf { V } ^ { \prime }$ , which implies that $\mathbf { V } ^ { \prime } = \mathbf { P } _ { 2 } \mathbf { V }$ . It has exactly one non-zero entry in each of its first $s$ columns. From the action of $\mathbf { P } _ { 2 }$ , these non-zero entries are sorted. Therefore, by the same argument as for $\mathbf { H }$ above, $\mathbf { V } ^ { T }$ is a horizontal step matrix. Hence, $\mathbf { V }$ is a vertical step matrix. + +In all, we have found a decomposition $\mathbf { S } = \mathbf { P } _ { 1 } \mathbf { H } \mathbf { P } _ { 2 } \mathbf { V } \mathbf { P } _ { 3 }$ , where each $P _ { \ell }$ is a permutation matrix $( \in B B ^ { * }$ by Theorem 2), $\mathbf { H }$ is a horizontal step matrix ( $\in B$ by Lemma I.2), and $\mathbf { V }$ is a vertical step matrix $( \in B ^ { * }$ by Corollary I.3). Moreover, by Lemma G.6, $\mathbf { P } _ { 2 } \in B ^ { \ast } B$ , so $\mathbf { H } , \mathbf { P } _ { 2 } , \mathbf { V }$ can be combined to obtain $\mathbf { H } \mathbf { P } _ { 2 } \mathbf { \bar { V } } \in ( B B ^ { \bar { * } } ) ^ { 2 }$ . By Lemma H.2, $\mathbf { S } \in ( B B ^ { * } ) ^ { 4 }$ . □ + +Corollary I.4. Let R be an $n \times n$ matrix of rank $r$ . Then $\mathbf { R } \in ( B B ^ { * } ) _ { 4 } ^ { 8 r }$ + +Proof. We can decompose $\mathbf { R }$ as $\mathbf { R } = \mathbf { G } \mathbf { H } ^ { * }$ where $\mathbf { G } , \mathbf { H }$ are $n \times r$ matrices. With appropriate zero-padding, both of these can be made into $n \times n$ matrices with at most $r n$ NNZ. The proof follows immediately from Theorem 3 and Lemma H.2. □ + +# J EXAMPLE OF K-MATRIX REPRESENTATION OF STRUCTURED MATRICES ANDCOMPARISON TO $B \mathcal { P }$ HIERARCHY + +In this appendix, we show explicitly how some common structured matrices (e.g. originating from fast transforms) can be represented as K-matrices. We also draw comparisons between the $B B ^ { * }$ hierarchy and the $B \mathcal { P }$ hierarchy introduced by Dao et al. (2019). + +Lemma J.1. Let ${ \bf F } _ { n }$ be the Discrete Fourier Transform of size $n$ . Then $\mathbf { F } _ { n } \in ( B B ^ { * } ) ^ { 2 }$ . + +Proof. From Parker (1995), we can express ${ \bf F } _ { n }$ as ${ \bf F } _ { n } = { \bf B } { \bf P }$ , where $\mathbf { B } \in B$ and $\mathbf { P }$ is a permutation (the bit reversal permutation). From Theorem 2, $\mathbf { P } \in B B ^ { * }$ . Hence, by Lemma H.2, $\mathbf { F } _ { n } ^ { - } \in ( B B ^ { * } ) ^ { 2 }$ . + +Lemma J.2. Let $\mathbf { H } _ { n }$ be the Hadamard Transform of size $n$ . Then $\mathbf { H } _ { n } \in B B ^ { * }$ . + +Proof. ${ \mathbf { H } } _ { n } \in { \mathfrak { B } }$ , so trivially $\mathbf { H } _ { n } \in B B ^ { * }$ + +Lemma J.3. Let $\mathbf { S } _ { n }$ be the Discrete Sine Transform of size $n$ . Then $\mathbf { S } _ { n } \in ( B B ^ { * } ) ^ { 2 }$ . + +Proof. As described in Makhoul (1980), $\mathbf { S } _ { n }$ can be performed as a scaled permutation (separating the even and odd indices of the input, and reversing and negating the odd indices) composed with $\mathbb { F } _ { n }$ . Therefore, we may decompose $\mathbf { S } _ { n }$ as $\mathbf { S } _ { n } = \mathbf { B } ~ \mathbf { P } _ { 2 } ~ \mathbf { D } ~ \mathbf { P } _ { 1 } ,$ , where $\mathbf { P } _ { 1 } , \mathbf { P } _ { 2 }$ are permutations, $\mathbf { B } \in B$ , and $\mathbf { D }$ is a diagonal matrix. $\mathbf { P } _ { 2 }$ D $\mathbf { P } _ { 1 }$ is simply a permutation matrix with scaled entries, which can be equivalently expressed as $\mathbf { D } ^ { \prime } \mathbf { P } ^ { \prime }$ for some diagonal matrix $\mathbf { D ^ { \prime } }$ and permutation $\mathbf { P ^ { \prime } }$ . By Lemma H.1, $\mathbf { B } \bar { \mathbf { D ^ { \prime } } } \in B B ^ { * }$ . By Theorem 2, $\mathbf { P } ^ { \prime } \in B B ^ { * }$ . Hence, by Lemma H.2, $\bar { \mathbf { S } _ { n } } \in ( B B ^ { * } ) ^ { 2 }$ . □ + +Remark J.4. An analogous argument shows that the Discrete Cosine Transform is also in $( B B ^ { * } ) ^ { 2 }$ + +Lemma J.5. Let $\mathbf { C } _ { n }$ be an $n \times n$ circulant (convolution) matrix. Then $\mathbf { C } _ { n } \in B B ^ { * }$ . + +Proof. Using Theorem 2.6.4 of Pan (2001), we can express $\mathbf { C } _ { n }$ as $\mathbf { C } _ { n } = ( \mathbf { F } _ { n } ) ^ { - 1 } \mathbf { D } \mathbf { F } _ { n }$ where ${ \bf F } _ { n }$ is the Discrete Fourier Transform and $\mathbf { D }$ is a diagonal matrix. $\left( \mathbf { F } _ { n } \right) ^ { - 1 } = \mathbf { B } \mathbf { \Lambda } \mathbf { P }$ (with $\mathbf { B } \in B$ , $\mathbf { P }$ a permutation), which implies that ${ \bf F } _ { n } = \left( { \bf P } \right) ^ { - 1 } \left( { \bf B } \right) ^ { - 1 }$ . Therefore + +$$ +{ \bf C } _ { n } = { \bf B } { \bf P } { \bf D } { \bf \Lambda } \left( { \bf P } \right) ^ { - 1 } \left( { \bf B } \right) ^ { - 1 } . +$$ + +The middle three factors have the effect of performing a permutation, scaling each element, and undoing the permutation, which is equivalent to simply scaling by some diagonal matrix $\mathbf { D ^ { \prime } }$ . Hence, we are left with + +$$ +\mathbf { C } _ { n } = \mathbf { B } \mathbf { D } ^ { \prime } \left( \mathbf { B } \right) ^ { - 1 } . +$$ + +By Lemma H.1, B $\mathbf { D } ^ { \prime } \in B$ . By Lemma H.6, $\left( \mathbf { B } \right) ^ { - 1 } \in B ^ { * }$ . Hence, $\mathbf { C } _ { n } \in B B ^ { * }$ + +Remark J.6. We can expand any $n \times n$ Toeplitz matrix $\mathbf { T } _ { n }$ into a $2 n \times 2 n$ circulant matrix (with upper left $n \times n$ submatrix equal to $\mathbf { T } _ { n }$ ). Hence, $\mathbf { T } _ { n } \in ( B B ^ { * } ) _ { 2 } ^ { 1 }$ by Lemma J.5. + +The Fastfood matrix class (Le et al., 2013) can be tightly captured in the $B B ^ { * }$ hierarchy: + +Lemma J.7. The product SHDPHB where $\mathbf { S } , \mathbf { D } , \mathbf { B }$ are diagonal matrices, $\mathbf { H }$ is the Hadamard transform, and $\mathbf { P }$ is a permutation matrix, is in $( B B ^ { * } ) ^ { 3 }$ . + +Proof. We have shown in Lemma J.2 that $\mathbf { H } \in B B ^ { * }$ , and in Theorem 2 that $\mathbf { P } \in B B ^ { * }$ . Since $B B ^ { * }$ is closed under diagonal multiplication (Lemma H.1), we conclude that $\mathbf { S H D P H B } \in ( B B ^ { * } ) ^ { 3 }$ . + +The two classes of matrices introduced in Moczulski et al. (2016), called AFDF and ACDC, are also tightly captured in the $B B ^ { * }$ hierarchy: + +Lemma J.8. Let $\mathbf { A } \mathbf { F } ^ { - 1 } \mathbf { D } \mathbf { F }$ be a product of a diagonal matrix A, the inverse Fourier transform $\mathbf { F } ^ { - 1 }$ another diagonal matrix D, and the Fourier transform F. Then $\mathbf { A } \mathbf { F } ^ { - 1 } \mathbf { D } \mathbf { F } \in B B ^ { * }$ . + +Let $\mathbf { A C } ^ { - 1 } \mathbf { D C }$ be a product of a diagonal matrix A, the inverse cosine transform $\mathbf { C } ^ { - 1 }$ , another diagonal matrix D, and the cosine transform C. Then $\mathbf { A } \mathbf { C } ^ { - 1 } \mathbf { D } \mathbf { C } \in ( B B ^ { * } ) ^ { 4 }$ . + +Proof. We have argued in Lemma J.5 that $\mathbf { F } ^ { - 1 } \mathbf { D } \mathbf { F } \in B B ^ { * }$ . Since $B B ^ { * }$ is closed under diagonal multiplication (Lemma H.1), we conclude that $\mathbf { A } \mathbf { F } ^ { - 1 } \mathbf { D } \mathbf { F } \in B B ^ { * }$ . + +We have shown that $\mathbf { C } \in ( B B ^ { * } ) ^ { 2 }$ , so $\mathbf { C } ^ { - 1 } \in ( B B ^ { * } ) ^ { 2 }$ as well. Since $B B ^ { * }$ is closed under diagonal multiplication (Lemma H.1), we conclude that $\mathbf { A C } ^ { - 1 } \mathbf { D C } \in ( B B ^ { * } ) ^ { 4 }$ . + +Remark J.9. Within each butterfly factor matrix of the DFT (excluding the bit reversal permutation) and the Hadamard transform, the columns are pairwise orthogonal and have norm 2. Hence, we can√ divide all factors by $\sqrt { 2 }$ to make orthogonal factor matrices. To counteract this scaling, we can add a diagonal matrix with 2log2(n) ${ \sqrt { 2 } } ^ { \log _ { 2 } ( n ) } = { \sqrt { n } }$ = n in all entries to the factorization. By doing this we can place all of the above transforms in the OBB hierarchy (defined in Appendix $K _ { \cdot }$ ) with the same width and expansion factor. + +# J.1 MULTI-DIMENSIONAL TRANSFORMS + +Here, we show that, using larger matrices, we are able to similarly capture multi-dimensional versions of the above transforms. + +Lemma J.10. Let ${ \bf F } _ { n } ^ { 2 }$ be the 2-dimensional Discrete Fourier Transform (represented as an $n ^ { 2 } \times n ^ { 2 }$ matrix). Then $\mathbf { F } _ { n } ^ { 2 } \in \tilde { ( B B ^ { \ast } ) } ^ { 2 }$ . + +Proof. The separation property of the 2-D DFT allows us to express its action on an $n \times n$ matrix as the composition of a 1-D DFT on each of its rows and a 1-D DFT on each of its columns. If we view the 2-D DFT as an $n ^ { 2 } \times n ^ { 2 }$ matrix, its input and outputs will both be column vectors of size $n ^ { 2 }$ . As our convention, we list the entries of the input vector in the row-major order corresponding to the $n \times n$ input matrix. Then, we consider the 2-D DFT in four steps, where the first two steps perform the 1-D DFT row-wise, and the second two steps perform the 1-D DFT column-wise: + +Step 1: Permute the columns: + +We permute the columns (with a bit reversal permutation), which performs a bit reversal permutation on each row. Viewing the input as a vector, this step corresponds to left multiplication by a permutation matrix $\mathbf { P } _ { c }$ that permutes the entries of each chunk of size $n$ of the input vector. Step 2: Multiply each row by a butterfly matrix + +Since the entries of the input were listed in row major order, this step is achieved through multiplication by a block diagonal matrix of $n$ butterfly matrices of size $n$ , which can be viewed as a product of butterfly factor matrices $\mathbf { B } _ { n } ^ { ( n ^ { 2 } ) } \dots \mathbf { B } _ { \frac { n } { 2 } } ^ { ( n ^ { 2 } ) } \mathbf { \dot { B } } _ { 2 } ^ { ( n ^ { 2 } ) }$ . + +Step 3: Permute the rows: + +We permute the rows (with a bit reversal permutation), which performs a bit reversal permutation on each column. This corresponds to left multiplication by a permutation matrix $\mathbf { P } _ { r }$ . Since we are permuting the rows, $\mathbf { P } _ { r }$ permutes the entries at the granularity of each $n$ -chunk. Since Steps 1 and 2 each performed an identical computation to each $n$ -chunk we can move this row permutation before Step 2, combining $\mathbf { P } _ { c }$ and $\mathbf { P } _ { r }$ into a single permutation $\mathbf { P }$ . + +Step 4: Multiply each column by a butterfly matrix + +Consider multiplication by the first factor matrix. In each row, this matrix is taking linear combinations of adjacent column entries. In our length- $\cdot n ^ { 2 }$ vector, these entries will be exactly $n$ indices apart. Therefore this multiplication can be handled by a butterfly factor matrix $\mathbf { B } _ { 2 n } ^ { ( n ^ { 2 } ) }$ . Similarly, we find that this butterfly multiplication can be expressed as multiplication by a product of butterfly factor matrices $\mathbf { B } _ { n ^ { 2 } } ^ { ( n ^ { 2 } ) } \cdot \cdot . . \mathbf { B } _ { \frac { n ^ { 2 } } { 2 } } ^ { ( n ^ { 2 } ) } \mathbf { B } _ { 2 n } ^ { ( n ^ { 2 } ) }$ . Combined with the factor matrices from Step 2, these form a butterfly matrix $\mathbf { B }$ of size $n ^ { 2 }$ . + +In all, we see that the 2-D DFT may be realized as multiplication by a permutation matrix $\mathbf { P }$ followed by multiplication by a butterfly matrix B. The same argument as Lemma J.1 shows that $\mathbf { F } _ { n } ^ { 2 } \in ( B B ^ { * } ) ^ { 2 }$ . □ + +Remark J.11. An analogous argument (using the separation property of the respective transforms) can be used to argue that 2-D Discrete Sine and Discrete Cosine transforms are in $( B B ^ { * } ) ^ { 2 }$ , and that 2-D Hadamard Transforms are in $B B ^ { * }$ . + +Lemma J.12. Let ${ \bf C } _ { n } ^ { 2 }$ be a 2-dimensional convolution matrix. Then $\mathbf { C } _ { n } ^ { 2 } \in B B ^ { * }$ + +Proof. We can express a 2-D convis the 2-D Fourier transform and matrix as is the in ${ \bf C } _ { n } ^ { 2 } = ( { \bf F } _ { n } ^ { 2 } ) ^ { - 1 } { \bf D } { \bf F } _ { n } ^ { 2 }$ , where ansform $\mathbf { D }$ is diagonal, From the pr ${ \bf F } _ { n } ^ { 2 }$ $( \mathbf { F } _ { n } ^ { 2 } ) ^ { - 1 }$ of Lemma J.10, we see that that we can express ${ \bf F } _ { n } ^ { 2 }$ (and similarly $( \mathbf { F } _ { n } ^ { 2 } ) ^ { - 1 } )$ as the product of a butterfly matrix and a permutation matrix. The rest of the argument is analogous to the proof of Lemma J.5. □ + +Remark J.13. Using an inductive argument, we can show that all $k$ -dimensional $( k \in \mathbb { Z } )$ variants of the above transforms, expressed as $n ^ { k } \times n ^ { k }$ matrices are contained in $B B ^ { * }$ or $( B B ^ { * } ) ^ { 2 }$ . To do this, we use the separation property of the transforms to break them into a $k - 1$ -dimensional transform (the inductive hypothesis) followed by a $^ { l }$ -dimensional transform. + +# K THE ORTHOGONAL KALEIDOSCOPE HIERARCHY + +Through practical application of the butterfly matrices, it has been found useful to constrain them in orthogonality. In Section K.1 we will modify the existing kaleidoscope hierarchy to create the orthogonal kaleidoscope hierarchy OBB. Then, in Section K.2, we will argue that all orthogonal matrices, and as a result all matrices, can also be expressed in this hierarchy in $O ( n )$ width. Lastly, in Section K.3, we will argue that permutation matrices and sparse matrices also exist in this hierarchy in $O ( 1 )$ width, which in turn implies a corresponding result for matrices with low-depth arithmetic circuits. + +# K.1 DEFINITION + +The definition of the orthogonal butterfly is identical to the original butterfly, with the constraint that all butterfly factors are orthogonal. We specify this definition below: + +Definition K.1 (Analog of Definition 2.1). An orthogonal butterfly factor of size $k \geq 2$ (denoted as $\widetilde { \mathbf B } _ { k }$ ) is a butterfly factor that is also orthogonal. + +Definition K.2 (Analog of Definition 2.3). An orthogonal butterfly matrix of size $n$ (denoted as $\widetilde { \mathbf { B } } ^ { ( n ) }$ ) is a butterfly matrix with all butterfly factor matrices being orthogonal. + +Note that the above definition implies that an orthogonal butterfly matrix, as well as its conjugate transpose, is orthogonal. + +The orthogonal hierarchy definition nearly mimics the original hierarchy Definition 2.4, as follows: + +# Definition K.3. + +• We say that an $n \times n$ matrix $\mathbf { M } \in \widetilde { B }$ if we can express $\mathbf { M } = \widetilde { \mathbf { B } } ^ { ( n ) }$ . • We say that an $n \times n$ matrix $\mathbf { M } \in \widetilde { B } ^ { \ast }$ if we can express $\mathbf { M } = \left[ \widetilde { \mathbf { B } } ^ { ( n ) } \right] ^ { * }$ . • We say that an $n \times n$ matrix $\mathbf { M } \in \mathcal { O B B }$ if we can express $\mathbf { M } = \mathbf { M } _ { 1 } \mathbf { D } \mathbf { M } _ { 2 }$ for some $\mathbf { M } _ { 1 } \in \widetilde { B } , \mathbf { M } _ { 2 } \in \widetilde { B } ^ { * }$ , and diagonal matrix D. Note that $\mathbf { D }$ need not be full rank. • Width w and expansion e in $( O B B ) _ { e } ^ { w }$ mimic the same definition as in the original hierarchy, using OBB instead of $B B ^ { * }$ , such that $\mathbf { E } \in ( \mathcal { O } B B ) ^ { w }$ . + +By padding if necessary, we will assume that $n$ is a power of 2. + +# K.2 EXPRESSIVITY + +In this subsection we prove that all orthogonal (resp. unitary) matrices are contained in $O B B ^ { n }$ . To do this, we consider the class of Householder reflections, given by $\mathbf { I } - 2 \mathbf { u } \mathbf { u } ^ { * }$ for any unit vector u (Householder, 1958): + +Lemma K.1. All Householder reflections are in OBB with inner diagonal matrix I. + +We will prove this lemma shortly. First, we use this lemma to present a decomposition for all orthogonal (resp. unitary) matrices. + +Lemma K.2. Let M be an $n \times n$ orthogonal/unitary matrix. Then $\mathbf { M } \in ( O B B ) ^ { n - 1 }$ . + +Proof. We consider the QR decomposition of M. It is known that we can compose $\mathbf { M }$ into a product of $n - 1$ Householder reflections and an orthogonal/unitary diagonal matrix (Householder, 1958).10 From Lemma K.1, each Householder reflection is in $o s B$ . + +To complete the proof, we argue that $\mathbf { R }$ can be folded into the rightmost butterfly matrix. Let $\mathbf { Q } _ { 1 }$ be the rightmost butterfly factor matrix in $\mathbf { Q } ~ ( \in \widetilde { \mathbf { B } } _ { n } ^ { ( n ) } )$ . Right multiplication of $\mathbf { Q } _ { 1 }$ by $\mathbf { R }$ scales each columns of $\mathbf { Q } _ { 1 }$ by some $c \in \mathbb { C }$ with $| | c | | = 1$ ( $\mathbf { R }$ is unitary diagonal). This preserves both the sparsity + +pattern of $\mathbf { Q } _ { 1 }$ and the orthogonality of its columns. Moreover, the norm of each column of $\mathbf { Q } _ { 1 } \mathbf { R }$ is 1. +Therefore, $\mathbf { Q } _ { 1 } \mathbf { R }$ is an orthogonal butterfly factor matrix, so $\mathbf { M } = \mathbf { Q } \mathbf { R } \in ( { \mathcal { O } } B B ) ^ { n - 1 }$ , as desired. + +We now return to the proof of Lemma K.1 + +Proof of Lemma K.1. Given $\mathbf { u } \in \mathbb { C } ^ { n }$ $\scriptstyle { \dot { n } }$ a power of 2), let $\mathbf { u } _ { 0 } = \mathbf { u } [ : n / 2 ] \in \mathbb { C } ^ { n / 2 } , \mathbf { u } _ { 1 } = \mathbf { u } [ n / 2 : ] \in$ $\mathbb { C } ^ { n / 2 }$ denote the first and second halves of $\mathbf { u }$ . + +To show that $\mathbf { H } \in \mathcal { O } B B$ with inner diagonal matrix $\mathbf { I }$ , we proceed by induction. The base case for $n = 2$ is trivial. It suffices to show that there exist unitary butterfly factors $\mathbf { L } , \mathbf { R }$ such that LHR has the form $\begin{array}{c} \begin{array} { r l } { \left[ { \displaystyle { \bf { I } } _ { n / 2 } - 2 { \bf { v } } _ { 0 } { \bf { v } } _ { 0 } ^ { * } } & { { \bf { 0 } } } \\ { { \bf { 0 } } } & { { \bf { I } } _ { n / 2 } - 2 { \bf { v } } _ { 1 } { \bf { v } } _ { 1 } ^ { * } } \end{array} \right] } & { { } } \end{array}$ for some unit vectors $\mathbf { v } _ { 0 } , \mathbf { v } _ { 1 } \in \mathbb { C } ^ { n / 2 }$ . + +Define + +$$ +\left( \mathbf { v } _ { 0 } [ i ] , \mathbf { v } _ { 1 } [ i ] \right) = \left\{ \begin{array} { l l } { \left( \frac { \mathbf { u } _ { 0 } [ i ] } { \sqrt { | \mathbf { u } _ { 0 } [ i ] | ^ { 2 } + | \mathbf { u } _ { 1 } [ i ] | ^ { 2 } } } , \frac { \mathbf { u } _ { 1 } [ i ] } { \sqrt { | \mathbf { u } _ { 0 } [ i ] | ^ { 2 } + | \mathbf { u } _ { 1 } [ i ] | ^ { 2 } } } \right) } & { \mathrm { ~ i f ~ } | \mathbf { u } _ { 0 } [ i ] | ^ { 2 } + | \mathbf { u } _ { 1 } [ i ] | ^ { 2 } \neq 0 } \\ { ( 1 , 0 ) } & { \mathrm { ~ o t h e r w i s e } } \end{array} \right. . +$$ + +It is easily checked that + +$$ +\begin{array} { r l } & { \mathbf { v } _ { 0 } [ i ] ^ { * } \mathbf { v } _ { 0 } [ i ] + \mathbf { v } _ { 1 } [ i ] ^ { * } \mathbf { v } _ { 1 } [ i ] = 1 } \\ & { \mathbf { v } _ { 0 } [ i ] ^ { * } \mathbf { u } _ { 0 } [ i ] + \mathbf { v } _ { 1 } [ i ] ^ { * } \mathbf { u } _ { 1 } [ i ] = \sqrt { | \mathbf { u } _ { 0 } [ i ] | ^ { 2 } + | \mathbf { u } _ { 1 } [ i ] | ^ { 2 } } . } \\ & { \mathbf { v } _ { 1 } [ i ] \mathbf { u } _ { 0 } [ i ] - \mathbf { v } _ { 0 } [ i ] ^ { * } \mathbf { u } _ { 1 } [ i ] = 0 } \end{array} +$$ + +We choose + +$$ +\mathbf { L } = \left[ \begin{array} { c c } { \mathrm { D i a g } ( \mathbf { v } _ { 0 } ^ { * } ) } & { \mathrm { D i a g } ( \mathbf { v } _ { 1 } ^ { * } ) } \\ { \mathrm { D i a g } ( \mathbf { v } _ { 1 } ) } & { \mathrm { D i a g } ( - \mathbf { v } _ { 0 } ) } \end{array} \right] +$$ + +and ${ \bf R } = { \bf L } ^ { * } . { \bf L } , { \bf R }$ are (permuted) direct sums of blocks of the form $\begin{array} { r l } { \left[ \mathbf { v } _ { 0 } [ i ] ^ { * } } & { { } \mathbf { v } _ { 1 } [ i ] ^ { * } \right] } \\ { \mathbf { v } _ { 1 } [ i ] } & { { } - \mathbf { v } _ { 0 } [ i ] } \end{array}$ , which are orthogonal by construction (via (5)). Hence, $\mathbf { L } \in \widetilde { \mathbf { B } } _ { n } ^ { ( n ) }$ and $\mathbf { R } \in ( \widetilde { \mathbf { B } } ^ { * } ) _ { n } ^ { ( n ) }$ . Further, + +$$ +\begin{array} { r l } & { \mathbf { L H R } = \left[ \mathrm { D i a g } ( \mathbf { v } _ { 0 } ^ { * } ) \quad \mathrm { D i a g } ( \mathbf { v } _ { 1 } ^ { * } ) \right] \left( \mathbf { I } - 2 \left[ \mathbf { u } _ { 0 } \right] \left[ \mathbf { u } _ { 0 } \right] ^ { * } \right) \left[ \mathrm { D i a g } ( \mathbf { v } _ { 0 } ^ { * } ) \quad \mathrm { D i a g } ( \mathbf { v } _ { 1 } ^ { * } ) \right] ^ { * } } \\ & { \qquad = \mathbf { I } - 2 \left[ \mathrm { D i a g } ( \mathbf { v } _ { 0 } ^ { * } ) \quad \mathrm { D i a g } ( - \mathbf { v } _ { 0 } ) \right] \left[ \mathbf { u } _ { 0 } \right] \left[ \mathbf { u } _ { 0 } \right] ^ { * } \left[ \mathrm { D i a g } ( \mathbf { v } _ { 0 } ^ { * } ) \quad \mathrm { D i a g } ( - \mathbf { v } _ { 0 } ) \right] ^ { * } } \\ & { \qquad = \mathbf { I } - 2 \left[ \mathrm { D i a g } ( \mathbf { v } _ { 1 } ) \quad \mathrm { D i a g } ( - \mathbf { v } _ { 0 } ) \right] \left[ \mathbf { u } _ { 1 } \right] \left[ \mathbf { u } _ { 1 } \right] ^ { * } \left[ \mathrm { D i a g } ( \mathbf { v } _ { 0 } ^ { * } ) \quad \mathrm { D i a g } ( \mathbf { v } _ { 1 } ^ { * } ) \right] ^ { * } } \\ & { \qquad = \mathbf { I } - 2 \underbrace { \left[ \mathbf { v } _ { 0 } ^ { * } \circ \mathbf { u } _ { 0 } + \mathbf { v } _ { 1 } ^ { * } \circ \mathbf { u } _ { 1 } \right] } _ { \mathbf { W } } \left[ \underbrace { \mathbf { v } _ { 0 } ^ { * } \circ \mathbf { u } _ { 0 } + \mathbf { v } _ { 1 } ^ { * } \circ \mathbf { u } _ { 1 } } _ { \mathbf { W } } \right] ^ { * } , } \end{array} +$$ + +where $\circ$ denotes the Hadamard product. From (6) + +$$ +\mathbf { w } [ i ] = \left\{ \begin{array} { l l } { \sqrt { | \mathbf { u } _ { 0 } [ i ] | ^ { 2 } + | \mathbf { u } _ { 1 } [ i ] | ^ { 2 } } } & { i \in [ n / 2 ] } \\ { 0 } & { i \in [ n / 2 : n ] } \end{array} \right. +$$ + +Denoting the first half of this vector by $\mathbf { w } _ { 0 } \in \mathbb { C } ^ { n / 2 }$ , we have + +$$ +\mathbf { L H R } = \left[ \begin{array} { c c } { \mathbf { I } - 2 \mathbf { w } _ { 0 } \mathbf { w } _ { 0 } ^ { * } } & { \mathbf { 0 } } \\ { \mathbf { 0 } } & { \mathbf { I } } \end{array} \right] , +$$ + +where $\| \mathbf { w } _ { 0 } \| _ { 2 } = \| \mathbf { u } \| _ { 2 } = 1$ . The result follows inductively. + +As an immediate corollary, we can use Singular Value Decomposition to obtain a factorization for an arbitrary $n \times n$ matrix. + +Corollary K.3. Let M be an arbitrary $n \times n$ matrix. Then, $\mathbf { M } \in ( { \mathcal { O } } B B ) ^ { 2 n - 1 }$ , where all but one matrix in the decomposition is orthogonal (unitary). + +Proof. By employing Singular Value Decomposition, we can decompose $\mathbf { M }$ as $\mathbf { M } = \mathbf { U } \pmb { \Sigma } \mathbf { V } ^ { * }$ , where $\mathbf { U } , \mathbf { V } ^ { * }$ are orthogonal and $\pmb { \Sigma }$ is diagonal. By Lemma $\mathbf { K } . 2 , \mathbf { U } , \bar { \mathbf { V } } ^ { * } \in ( \mathcal { O B } \mathcal { B } ) ^ { n - 1 }$ , and trivially $\pmb { \Sigma } \in \mathcal { O } B B$ . Hence, $\mathbf { M } \in ( \mathcal { O } B B ) ^ { 2 n - 1 }$ . Note that $\pmb { \Sigma }$ is the only matrix in the decomposition that is not orthogonal (unitary). □ + +# K.3 CONSTRUCTIONS + +We show that we can construct $s$ -sparse matrices in the OBB hierarchy with the same width as the $B B ^ { * }$ hierarchy. The proof follows a structure to that of Theorem 3. We begin by arguing about permutation and step matrices, then using the same factorization to argue that matrices with at most $n$ NNZ are contained in $( B B ^ { * } ) ^ { 4 }$ . Then, we will appeal to a modified sum closure lemma to extend the argument to matrices of general $s$ NNZ. Similar to Appendix F, we can use these results to place all matrices with low-depth circuits for matrix vector multiplication in the $\ O B B$ hierarchy. + +# K.3.1 PERMUTATIONS + +We begin by presenting the argument that permutations are included in OBB as a corollary to Theorem 2. + +Corollary K.4. Let P be a permutation matrix. Then $\mathbf { P } \in \widetilde { B } \widetilde { B } ^ { * }$ . + +Proof. We appeal to the decomposition from Theorem 2, noting that all butterfly factor matrices constructed in the proofs of Lemmas G.3 and G.1 are permutation matrices, and thus are orthogonal. Hence, $\mathbf { P } \in { \mathcal { O } } B B$ where the inner diagonal matrix is I. □ + +Similarly, the construction of Lemma G.6 also show that permutations are included in $\widetilde { B } ^ { * } \widetilde { B }$ . + +Corollary K.5. Let $\mathbf { P }$ be a permutation matrix. Then $\mathbf { P } \in \widetilde { B } ^ { * } \widetilde { B }$ . + +To prove the containment of sparse matrices within the $o s B$ hierarchy, we make use of the following lemma. + +Lemma K.6. Let $\mathbf { P }$ be a permutation matrix and $\mathbf { D }$ a diagonal matrix. Then there exist diagonal matrices $\mathbf { D ^ { \prime } }$ and $\mathbf { D } ^ { \prime \prime }$ such that: + +$$ +\mathbf { P D } = \mathbf { D } ^ { \prime } \mathbf { P } \qquad \mathbf { D P } = \mathbf { P D } ^ { \prime \prime } . +$$ + +Proof. Let $\sigma$ be the permutation such that $\mathbf { P } [ i , j ] = \delta _ { i , \sigma ( j ) }$ + +Define $\mathbf { D ^ { \prime } }$ such that $\mathbf { D } ^ { \prime } [ \sigma ( j ) , \sigma ( j ) ] = \mathbf { D } [ j , j ]$ . Then, if $i = \sigma ( j )$ : + +$$ +\begin{array} { r } { ( \mathbf { P D } ) [ i , j ] = \mathbf { P } [ i , j ] \mathbf { D } [ j , j ] = \mathbf { D } ^ { \prime } [ \sigma ( j ) , \sigma ( j ) ] \mathbf { P } [ \sigma ( j ) , j ] = ( \mathbf { D } ^ { \prime } \mathbf { P } ) [ \sigma ( j ) , j ] = ( \mathbf { D } ^ { \prime } \mathbf { P } ) [ i , j ] . } \end{array} +$$ + +Otherwise, if $i \neq \sigma ( j )$ , then $( { \bf P D } ) [ i , j ] = 0 = ( { \bf D ^ { \prime } P } ) [ i , j ]$ . Hence, $\mathbf { P D } = \mathbf { D } ^ { \prime } \mathbf { P }$ . + +Define $\mathbf { D } ^ { \prime \prime }$ such that ${ \bf D } ^ { \prime \prime } [ j , j ] = { \bf D } [ \sigma ( j ) , \sigma ( j ) ]$ . An analogous argument to above shows that $\mathbf { D P } = \mathbf { P D ^ { \prime \prime } }$ . + +# K.3.2 STEP MATRICES + +In the $B B ^ { * }$ hierarchy (Lemma I.2), we were able to show that horizontal step matrices are butterfly matrices. Here, we present a similar result for the $\ O B B$ hierarchy. + +Lemma K.7. Let $\mathbf { H }$ be an $n \times n$ horizontal step matrix. Then we can decompose ${ \bf H } = { \bf D } { \bf O }$ , where $\mathbf { D }$ is a diagonal matrix and $\mathbf { O } \in \widetilde { B }$ . + +Proof. Throughout the proof, we make reference to the original horizontal step matrix construction given in Lemma I.2 and its proof. + +To begin, we show that an arbitrary $2 ^ { k } \times 2 ^ { k }$ butterfly factor $\mathbf { H } _ { 2 ^ { k } }$ in the decomposition of $\mathbf { H }$ can be expressed as the product of a diagonal matrix and an orthogonal butterfly factor. Since a butterfly factor is direct sum of $2 \times 2$ matrices, there is a permutation matrix $\mathbf { P } _ { 2 ^ { k } }$ such that conjugation of $\mathbf { H } _ { 2 ^ { k } }$ by $\mathbf { P } _ { 2 ^ { k } }$ gives a block diagonal matrix $\mathbf { H } _ { 2 ^ { k } } ^ { \prime }$ of ${ \frac { n } { 2 } } \ 2 \times 2$ matrices, i.e. + +$$ +\mathbf { P } _ { 2 ^ { k } } \mathbf { H } _ { 2 ^ { k } } \mathbf { P } _ { 2 ^ { k } } ^ { * } = \mathbf { H } _ { 2 ^ { k } } ^ { \prime } . +$$ + +(See Figure 10 for an illustration.) Specifically, $\mathbf { P } _ { 2 ^ { k } }$ is the permutation where: + +$$ +{ \bf P } _ { s } [ 2 i , : ] = { \bf e } _ { i } ^ { T } { \bf \mathrm { ~ ~ { ~ \cal ~ ~ { ~ P ~ } ~ } ~ } } _ { s } [ 2 i + 1 , : ] = { \bf e } _ { i + \frac { n } { 2 } } ^ { T } . +$$ + +![](images/a4607d3febb414e4ea1a9f3500d9dc6ca268a2509f0b7cd81b00a8b90f87d6e6.jpg) +Figure 10: Block diagonalization of ${ \bf { H } } _ { 8 }$ + +We argue that each of these $2 \times 2$ blocks can be decomposed into a diagonal matrix times an orthogonal matrix. Note that the butterfly factor matrices constructed in the proof of Lemma I.2 each have at most one non-zero entry per column. Hence, there are 4 cases to consider. Note that matrices with at most one non-zero entry are exhausted by Cases 1 and 2. + +$$ +\mathfrak { e } \ 1 : \left[ 0 \atop 0 \right] = \underbrace { \left[ 0 \begin{array} { l l } { 0 } & { 0 } \\ { 0 } & { b } \end{array} \right] } _ { \textbf { D } } \underbrace { \left[ 0 \begin{array} { l l } { 1 } & { 0 } \\ { 0 } & { 1 } \end{array} \right] } _ { \textbf { O } } +$$ + +Case 2: $\left[ \begin{array} { l l } { 0 } & { a } \\ { b } & { 0 } \end{array} \right] = \underbrace { \left[ \begin{array} { l l } { a } & { 0 } \\ { 0 } & { b } \end{array} \right] } _ { \textbf { D } } \underbrace { \left[ \begin{array} { l l } { 0 } & { 1 } \\ { 1 } & { 0 } \end{array} \right] } _ { \textbf { O } }$ + +![](images/7f28d28d7daab240206962aaddca12847be0539a1b0d4e28308bd7d0b076eccb.jpg) + +In the last two cases, $\mathbf { O }$ is a $2 \times 2$ rotation matrix, which is commonly known to be orthogonal. Assume that we perform the above decomposition on all of the blocks of $\mathbf { H } _ { 2 ^ { k } } ^ { \prime }$ in parallel, therefore expressing $\mathbf { H } _ { 2 ^ { k } } ^ { \prime } \doteq \mathbf { D } ^ { \prime } \mathbf { O } ^ { \prime }$ . We now have + +$$ +{ \bf H } _ { 2 ^ { k } } = { \bf P } _ { 2 ^ { k } } ^ { * } { \bf D } ^ { \prime } { \bf O } ^ { \prime } { \bf P } _ { 2 ^ { k } } . +$$ + +By Lemma K.6, we can rewrite this as + +$$ +{ \bf H } _ { 2 ^ { k } } = { \bf D } ^ { \prime \prime } { \bf P } _ { 2 ^ { k } } ^ { * } { \bf O } ^ { \prime } { \bf P } _ { 2 ^ { k } } . +$$ + +Note that $\mathbf { P } _ { 2 ^ { k } } ^ { * } \mathbf { O } ^ { \prime } \mathbf { P } _ { 2 ^ { k } }$ is the product of three orthogonal matrices, and thus orthogonal. Additionally, the construction of $\mathbf { P } _ { 2 ^ { k } }$ ensures that $\mathbf { P } _ { 2 ^ { k } } ^ { * } \mathbf { O } ^ { \prime } \mathbf { P } _ { 2 ^ { k } }$ is butterfly factor.11 Hence, $\mathbf { H } _ { 2 ^ { k } }$ can be expressed as the product of a diagonal matrix and an orthogonal butterfly factor, as desired. + +Now, we show that this decomposition of butterfly factors implies Lemma K.7. By performing this decomposition in parallel on each butterfly factor, we conclude that any butterfly factor matrix $\mathbf { \bar { H } } _ { 2 ^ { k } } ^ { ( n ) }$ of $\mathbf { H }$ can be decomposed as ) = D2k O(n)2k . 12 + +We complete the argument by induction on $n$ . The base case $n = 2$ holds by the observation about butterfly factor matrices above. Assume that any horizontal step matrix of size ${ \frac { n } { 2 } } \times { \frac { n } { 2 } }$ can be expressed as a diagonal matrix times an orthogonal butterfly matrix. Now, consider the $n \times n$ horizontal step matrix $\mathbf { H }$ . From Lemma I.2, $\mathbf { H }$ can be expressed as + +$$ +\mathbf { H } = \mathbf { B } _ { n } ^ { ( n ) } \left[ \begin{array} { c c } { \mathbf { H } _ { 1 } } & { 0 } \\ { 0 } & { \mathbf { H } _ { 2 } } \end{array} \right] , +$$ + +where $\mathbf { H } _ { 1 } , \mathbf { H } _ { 2 }$ are ${ \frac { n } { 2 } } \times { \frac { n } { 2 } }$ horizontal step matrices. By our inductive hypothesis, + +$$ +\mathbf { H } = \mathbf { B } _ { n } ^ { ( n ) } \mathbf { D } _ { 1 } \left[ \begin{array} { c c } { \mathbf { O } _ { 1 } } & { 0 } \\ { 0 } & { \mathbf { O } _ { 2 } } \end{array} \right] , +$$ + +where $\mathbf { D } _ { 1 }$ is diagonal and $\mathbf { O } _ { 1 } , \mathbf { O } _ { 2 }$ are ${ \frac { n } { 2 } } \times { \frac { n } { 2 } }$ matrices in $\widetilde { B }$ . However, ${ \bf B } _ { n } ^ { ( n ) } { \bf D } _ { 1 }$ is a butterfly factor, and therefore can be expressed as $\mathbf { D } _ { n } \mathbf { O } _ { n } ^ { ( n ) }$ . Therefore, + +$$ +{ \bf H } = { \bf D } _ { n } { \bf O } _ { n } ^ { ( n ) } \left[ \begin{array} { c c } { { \bf O } _ { 1 } } & { 0 } \\ { 0 } & { { \bf O } _ { 2 } } \end{array} \right] = { \bf D } _ { n } { \bf O } , +$$ + +with $\mathbf { O } \in \widetilde { B }$ , as desired. + +Just as with the $B B ^ { * }$ hierarchy, the decomposition of vertical step matrices falls out as an immediate corollary to the horizontal step matrix proof. + +Corollary K.8. Let $\mathbf { V }$ be a vertical step matrix. Then we can decompose $\mathbf { V } = \mathbf { O } ^ { * } \mathbf { D }$ , where $\mathbf { D }$ is $a$ diagonal matrix and $\mathbf { O } ^ { * } \in \widetilde { B } ^ { * }$ . + +# K.3.3 SPARSE MATRICES + +Now that we have argued about the decomposition of permutation and step matrices in the OBB hierarchy, we can leverage the construction from Lemma I.1 to argue about matrices with at most $n$ NNZ. + +Corollary K.9. Let S be an $n \times n$ matrix with at most n NNZ. Then, $\mathbf { S } \in ( \mathcal { O } B B ) ^ { 4 }$ . + +Proof. We use the construction from Lemma I.1, along with Lemma K.7 and Corollary K.8, to express S as: + +$$ +\mathbf { S } = \underbrace { \mathbf { O } _ { 1 } \mathbf { O } _ { 1 } ^ { \prime } } _ { \mathbf { P } _ { 1 } } \underbrace { \mathbf { D } _ { 2 } \mathbf { O } _ { 2 } } _ { \mathbf { H } } \underbrace { \mathbf { O } _ { 3 } \mathbf { O } _ { 3 } ^ { \prime } } _ { \mathbf { P } _ { 2 } } \underbrace { \mathbf { O } _ { 4 } ^ { \prime } \mathbf { D } _ { 4 } } _ { \mathbf { V } } \underbrace { \mathbf { O } _ { 5 } \mathbf { O } _ { 5 } ^ { \prime } } _ { \mathbf { P } _ { 3 } } , +$$ + +with each $\mathbf { O } _ { i } \in \widetilde { B }$ , each $\mathbf { O } _ { j } ^ { \prime } \in \widetilde { B } ^ { * }$ , and each $\mathbf { D } _ { k }$ diagonal. Since $\mathbf { P } _ { 2 }$ is a permutation, by Corollary K.5, we can write it as $\tilde { \bf { O } } _ { 3 } ^ { ' } \tilde { \bf { O } } _ { 3 }$ for some $\tilde { \mathbf { O } _ { 3 } } ^ { \prime } \in \widetilde { B } ^ { * }$ and $\tilde { \mathbf { O } } _ { 3 } \in \tilde { B }$ . Moreover, noting that $\mathbf { O } _ { 1 } ^ { \prime }$ and $\mathbf { O } _ { 5 }$ are permutations, we make use of Lemma K.6 to re-express $\mathbf { S }$ as: + +$$ +\mathbf { S } = \underbrace { \mathbf { O } _ { 1 } \mathbf { D } _ { 2 } ^ { \prime } \mathbf { O } _ { 1 } ^ { \prime } } _ { \mathbf { M } _ { 1 } } \underbrace { \mathbf { O } _ { 2 } \tilde { \mathbf { O } _ { 3 } } } _ { \mathbf { M } _ { 2 } } \underbrace { \mathbf { \tilde { O } _ { 3 } O _ { 4 } ^ { \prime } } } _ { \mathbf { M } _ { 3 } } \underbrace { \mathbf { O } _ { 5 } \mathbf { D } _ { 4 } ^ { \prime } \mathbf { O } _ { 5 } ^ { \prime } } _ { \mathbf { M } _ { 4 } } . +$$ + +Note that each $\mathbf { M } _ { \ell } \in \mathcal { O B B }$ . Hence, $\mathbf { S } \in ( \mathcal { O } B B ) ^ { 4 }$ , as desired. + +Just as in Appendix I, we would like to extend this orthogonal-based construction to capture matrices of general sparsity. To accomplish this, we introduce an addition closure lemma analogous to Lemma K.10 for the $\ O B B$ hierarchy. + +Lemma K.10. Let $\mathbf { A } _ { 1 } , \ldots , \mathbf { A } _ { m }$ be $k \times k$ matrices in $( O B B ) _ { e } ^ { w }$ then $\begin{array} { r } { \sum _ { i = 1 } ^ { m } \mathbf { A } _ { i } \in ( \mathcal { O } B \mathcal { B } ) _ { 4 e } ^ { m w } . } \end{array}$ + +With Lemma K.10, we arrive at the following Corollary on general orthogonal sparsity. + +Corollary K.11. Let S be an $n \times n$ matrix with s NNZ. Then, $\mathbf { S } \in \left( \mathcal { O } B B \right) _ { 4 } ^ { 4 \left\lceil \frac { s } { n } \right\rceil }$ . + +Proof. Just as in the proof of Theorem 3, we accomplish this using a sum of $\textstyle { \left\lceil { \frac { s } { n } } \right\rceil }$ matrices of at most $n$ NNZ. For handling the sum of matrices, we need to appeal to Lemma K.10. □ + +To conclude the argument, we give the proof of Lemma K.10. + +Proof of Lemma $K . I O$ . For each $1 \leq i \leq m$ , let $\mathbf { E } _ { \mathbf { A } _ { i } } \in \mathbb { F } ^ { e k \times e k }$ be defined such that $\mathbf { A } _ { i } = \mathbf { S } \mathbf { E } _ { \mathbf { A } _ { i } } \mathbf { S } ^ { * }$ (with $\mathbf { S }$ as in Definition 2.4). Note that $\mathbf { E } _ { \mathbf { A } _ { i } } \in ( \mathcal { O } B B ) ^ { w }$ . Consider matrices of the form: + +$$ +\begin{array} { r l } & { \left[ \mathbf { I } _ { e k } \quad 0 \quad 0 \quad 0 \quad \mathbf { E } _ { \mathbf { A } _ { i } } \right] } \\ & { \left[ \begin{array} { l l l l } { \mathbf { 0 } } & { \mathbf { I } _ { e k } \quad 0 \quad } & { \mathbf { 0 } \quad } & { 0 } \\ { 0 \quad \mathbf { I } _ { e k } \quad 0 \quad } & { 0 \quad } & { \mathbf { 0 } } \\ { \mathbf { I } _ { e k } \quad 0 \quad } & { 0 \quad } & { \mathbf { 0 } } \end{array} \right] } \\ & { \underbrace { \left[ \begin{array} { l l l l } { \mathbf { I } _ { e k } \quad 0 \quad } & { 0 \quad } & { \mathbf { 0 } } \\ { 0 \quad } & { \mathbf { I } _ { e k } \quad 0 \quad } & { 0 } \end{array} \right] } _ { \mathbf { M } _ { i } \in \mathbb { F } ^ { 4 e k \times 4 e k } } = \sqrt { 2 } \underbrace { \left[ \frac { 1 } { \sqrt { 2 } } \mathbf { I } _ { 2 e k } \quad \frac { 1 } { \sqrt { 2 } } \mathbf { I } _ { 2 e k } \right] } _ { \mathbf { O } \in \widetilde { \mathbf { B } } _ { 4 e k } ^ { ( 4 e k ) } } \underbrace { \left[ \begin{array} { l l l l l } { \mathbf { I } _ { e k } \quad 0 \quad } & { 0 \quad } & { 0 } & { 0 } \\ { 0 \quad } & { \mathbf { I } _ { e k } \quad } & { 0 \quad } & { 0 } \\ { 0 \quad } & { 0 \quad } & { \mathbf { E } _ { \mathbf { A } _ { i } } \quad 0 } \\ { 0 \quad } & { 0 \quad } & { 0 } & { 0 } \end{array} \right] } _ { \mathbf { K } } \underbrace { \left[ \begin{array} { l l l l } { \mathbf { I } _ { e k } \quad 0 \quad } & { 0 \quad } & { 0 } & { 0 } \\ { 0 \quad } & { \mathbf { I } _ { e k } \quad } & { 0 \quad } & { 0 } \\ { 0 \quad } & { 0 \quad } & { \mathbf { I } _ { e k } \quad } \\ { 0 \quad } & { 0 \quad } & { \mathbf { I } _ { e k } \quad 0 } \end{array} \right] } _ { \mathbf { P } \in \widetilde { \mathbf { B } } _ { 2 e k } ^ { ( 4 e k ) } } } \end{array} +$$ + +Note that $\mathbf { K }$ , a block diagonal matrix composed of matrices in $( \mathcal { O } B B ) ^ { w }$ , is itself in $( \mathcal { O } B B ) ^ { w }$ since + +$$ +\mathbf { K } = \prod _ { j = 1 } ^ { w } \underbrace { \left[ \begin{array} { c c c c } { \mathbf { I } _ { e k } } & { 0 } & { 0 } & { 0 } \\ { 0 } & { \mathbf { I } _ { e k } } & { 0 } & { 0 } \\ { 0 } & { 0 } & { \mathbf { O } _ { j } } & { 0 } \\ { 0 } & { 0 } & { 0 } & { \mathbf { I } _ { e k } } \end{array} \right] } _ { \mathbf { L } _ { j } \in \widetilde { B } } \underbrace { \left[ \begin{array} { c c c c } { \mathbf { I } _ { e k } } & { 0 } & { 0 } & { 0 } \\ { 0 } & { \mathbf { I } _ { e k } } & { 0 } & { 0 } \\ { 0 } & { 0 } & { \mathbf { D } _ { j } } & { 0 } \\ { 0 } & { 0 } & { 0 } & { 0 } \end{array} \right] } _ { \mathrm { D i a g o n a l } } \underbrace { \left[ \begin{array} { c c c c } { \mathbf { I } _ { e k } } & { 0 } & { 0 } & { 0 } \\ { 0 } & { \mathbf { I } _ { e k } } & { 0 } & { 0 } \\ { 0 } & { 0 } & { \mathbf { O } _ { j } ^ { \prime } } & { 0 } \\ { 0 } & { 0 } & { 0 } & { \mathbf { I } _ { e k } } \end{array} \right] } _ { \mathbf { R } _ { j } \in \widetilde { B } ^ { * } } , +$$ + +where each $\mathbf { O } _ { j }$ is a $e k \times e k$ matrix in $\widetilde { B }$ , and each ${ \bf O } _ { j } ^ { \prime }$ is a $e k \times e k$ matrix in ${ \widetilde { B } } ^ { * }$ . $\mathbf { L } _ { w }$ (the leftmost factor) is a block diagonal matrix composed of $4 ~ e k \times e k$ matrices in $\widetilde { B }$ . Therefore, we can fold $\mathbf { O }$ into this factor (since a butterfly factor in Be (4ek)4ek was not yet used in $\mathbf { L } _ { w }$ ) to conclude that $\mathbf { O L } _ { w } \in \widetilde { B }$ . Similarly, since no btterfly factor from Be (4ek)2ek has been used in ${ \bf R } _ { 1 }$ , we may fold $\mathbf { P }$ into ${ \bf R } _ { 1 }$ to conclude that $\mathbf { R } _ { 1 } \mathbf { P } \in \widetilde { B } ^ { * }$ . Finally, we address the scalar multiple of $\sqrt { 2 }$ by multiplying all entries of any diagonal matrix in the decomposition of $\mathbf { K }$ by $\sqrt { 2 }$ . Hence, we may conclude that $\mathbf { M } _ { i } \in ( \mathcal { O } B B ) ^ { w }$ . + +Through repeated application $\textbar { m }$ times) of the identity + +$$ +\left[ \begin{array} { c c c c } { \mathbf { I } } & { \mathbf { A } _ { 1 } } & { 0 } & { \mathbf { B } _ { 1 } } \\ { 0 } & { \mathbf { I } } & { 0 } & { 0 } \\ { \mathbf { I } } & { \mathbf { A } _ { 2 } } & { 0 } & { \mathbf { B } _ { 2 } } \\ { 0 } & { \mathbf { I } } & { 0 } & { 0 } \end{array} \right] \left[ \begin{array} { c c c c } { \mathbf { I } } & { 0 } & { 0 } & { \mathbf { C } _ { 1 } } \\ { 0 } & { \mathbf { I } } & { 0 } & { 0 } \\ { \mathbf { I } } & { 0 } & { 0 } & { \mathbf { C } _ { 2 } } \\ { 0 } & { \mathbf { I } } & { 0 } & { 0 } \end{array} \right] = \left[ \begin{array} { c c c c } { \mathbf { I } } & { \mathbf { A } _ { 1 } + \mathbf { B } _ { 1 } } & { 0 } & { \mathbf { C } _ { 1 } } \\ { 0 } & { \mathbf { I } } & { 0 } & { 0 } \\ { \mathbf { I } } & { \mathbf { A } _ { 2 } + \mathbf { B } _ { 2 } } & { 0 } & { \mathbf { C } _ { 1 } } \\ { 0 } & { \mathbf { I } } & { 0 } & { 0 } \end{array} \right] , +$$ + +we see that + +$$ +\prod _ { i = 1 } ^ { m } \mathbf { M } _ { i } = \left[ \begin{array} { c c c c } { \mathbf { I } _ { e k } } & { \sum _ { i = 2 } ^ { m } \mathbf { E } _ { \mathbf { A } _ { i } } } & { 0 } & { \mathbf { E } _ { \mathbf { A } _ { 1 } } } \\ { 0 } & { \mathbf { I } _ { e k } } & { 0 } & { 0 } \\ { \mathbf { I } _ { e k } } & { \mathbf { - E } _ { \mathbf { A } _ { m } } + \sum _ { i = 2 } ^ { m - 1 } \mathbf { E } _ { \mathbf { A } _ { i } } } & { 0 } & { \mathbf { E } _ { \mathbf { A } _ { 1 } } } \\ { 0 } & { \mathbf { I } _ { e k } } & { 0 } & { 0 } \end{array} \right] . +$$ + +Therefore, $\mathbf { M } \in ( \mathcal { O } B B ) ^ { m w }$ . Next, we note that + +$$ +\sum _ { i = 1 } ^ { m } \mathbf { A } _ { i } = \mathbf { S } \mathbf { M } \underbrace { \left[ \begin{array} { c c c c } { 0 } & { \mathbf { I } _ { e k } } & { 0 } & { \mathbf { I } _ { e k } } \\ { \mathbf { I } _ { e k } } & { 0 } & { \mathbf { I } _ { e k } } & { 0 } \\ { 0 } & { \mathbf { I } _ { e k } } & { 0 } & { - \mathbf { I } _ { e k } } \\ { \mathbf { I } _ { e k } } & { 0 } & { - \mathbf { I } _ { e k } } & { 0 } \end{array} \right] } _ { \mathbf { Q } } \mathbf { S } ^ { T } . +$$ + +We would like to show that we can fold $\mathbf { Q }$ into the rightmost $\ O B B$ factor of M. The rightmost matrix in the decomposition of $\mathbf { M }$ is $\mathbf { P }$ . Note that + +$$ +\mathbf { P Q } = \left[ \begin{array} { c c c c } { 0 } & { \mathbf { I } _ { e k } } & { 0 } & { \mathbf { I } _ { e k } } \\ { \mathbf { I } _ { e k } } & { 0 } & { \mathbf { I } _ { e k } } & { 0 } \\ { \mathbf { I } _ { e k } } & { \mathbf { I } _ { e k } } & { \mathbf { I } _ { e k } } & { 0 } \\ { 0 } & { \mathbf { I } _ { e k } } & { 0 } & { \mathbf { - I } _ { e k } } \end{array} \right] = \sqrt { 2 } \underbrace { \left[ \begin{array} { c c c c } { 0 } & { \mathbf { I } _ { e k } } & { 0 } & { 0 } \\ { \mathbf { I } _ { e k } } & { 0 } & { 0 } & { 0 } \\ { 0 } & { 0 } & { \mathbf { I } _ { e k } } & { 0 } \\ { 0 } & { 0 } & { 0 } & { \mathbf { I } _ { e k } } \end{array} \right] } _ { \widetilde { \mathbf { B } } _ { 2 e k } ^ { ( 4 e k ) } } \underbrace { \left[ \frac { 1 } { \sqrt { 2 } } \mathbf { I } _ { 2 e k } \quad \frac { 1 } { \sqrt { 2 } } \mathbf { I } _ { 2 e k } \right] } _ { \widetilde { \mathbf { B } } _ { 4 e k } ^ { ( 4 e k ) } } . +$$ + +Just as earlier, the factor of $\sqrt { 2 }$ can be multiplied through any diagonal matrix. Also, these two orthogonal butterfly factor matrices can be folded into the the rightmost $\mathbf { R }$ matrix (the decomposition of $\mathbf { K }$ above does not use these two, rightmost butterfly factors). Hence, $\begin{array} { r } { \sum _ { i = 1 } ^ { m } \mathbf { A } _ { i } \in ( \mathcal { O B } \dot { B } ) _ { 4 e } ^ { m w } } \end{array}$ )mw4e , as desired. □ + +# K.3.4 ARITHMETIC CIRCUITS + +Just as in Theorem 1, we can use the sparsity result in Lemma K.10 to place matrices with low-depth (linear) arithmetic circuits for matrix vector multiplication in the $\ O B B$ hierarchy. + +Corollary K.12. Let M be an $n \times n$ matrix such that matrix-vector multiplication of M times an +arbitrary vector v can be represented as a be a linear arithmetic circuit $C$ comprised of s gates +(including inputs) and having depth $d .$ . Then, $\mathbf { M } \in ( \mathcal { O } B B ) _ { O ( \frac { s } { n } ) } ^ { O ( d ) }$ . + +Proof. We use the construction given in the proof of Theorem 1. Corollaries K.9 and K.4 allow us to recover the same width and expansion factor with the $o s B$ hierarchy. □ + +# L RELU NETWORK WITH STRUCTURED WEIGHT MATRICES + +We show that for any neural network with ReLU nonlinearities and whose weight matrices have arithmetic circuits with few gates, its linear network counterpart (obtained by removing all the ReLU’s) also has an arithmetic circuit with not too many more gates. This implies that in trying to find the smallest arithmetic circuit augmented with ReLU gates to represent a ReLU network, one might as well try to find the smallest arithmetic circuits that represent the matrix-vector multiplication of each weight matrix. + +Proposition 2. Consider a neural network architecture consisting of $L$ layers with weight matrices $\mathbf { W } _ { 1 } , \dots , \mathbf { W } _ { L } \in \mathbb { F } ^ { n \times n }$ and ReLU nonlinearity in between. + +Suppose that matrix-vector multiplication of $\mathbf { W } _ { i }$ times an arbitrary vector v can be represented as a linear arithmetic circuit with $s _ { i }$ gates (including inputs). Then there exists an arithmetic circuit augmented with ReLU gates with PLi=1 si + Ln total gates that computes the output ReLU $( \mathbf { W } _ { L } ( \dots \mathrm { R e L U } ( \mathbf { W } _ { 1 } \mathbf { v } ) ) )$ ) of the network for an arbitrary input vector v. + +Conversely, if there is an arithmetic circuit augmented with ReLU gates with s total gates that computes all the activations of the network ReL $\operatorname { \mathrm { . U } } ( \mathbf { W } _ { 1 } \mathbf { v } ) , \ldots , \operatorname { R e L U } ( \mathbf { W } _ { L } \ldots \operatorname { \mathrm { . R e L U } } ( \mathbf { W } _ { 1 } \mathbf { v } ) )$ for an arbitrary input v, then there exists an arithmetic circuit augmented with ReLU gates with $2 s + 2 L n$ total gates that computes the activations of the network without ReLU $\mathbf { W } _ { 1 } \mathbf { v } , \dots , \mathbf { W } _ { L } \dots \mathbf { W } _ { 1 } \mathbf { v }$ . + +Proof of Proposition 2. To compute the output of the network Re $\operatorname { \mathrm { . U } } ( \mathbf { W } _ { L } ( \dots \operatorname { R e L U } ( \mathbf { W } _ { 1 } \mathbf { v } ) ) )$ ), we first compute the matrix-vector product $\mathbf { W } _ { 1 } \mathbf { v }$ with an arithmetic circuit of $s _ { 1 }$ gates by assumption, and use $n$ other ReLU gates to compute the pointwise ReLU. Then we repeat the process for layer $2 , 3 , \ldots , L$ , using the arithmetic circuits of $\mathbf { W } _ { 1 } , \dots , \mathbf { W } _ { L }$ and $L n$ additional gates for ReLU. In total we obtain an arithmetic circuit augmented with ReLU gates with $\textstyle \sum _ { i = 1 } ^ { L } s _ { i } + L n$ total gates. + +Conversely, to build an arithmetic circuit augmented with ReLU gates to compute $\mathbf { W } _ { 1 } \mathbf { v } , \dots , \mathbf { W } _ { L } \dots \mathbf { W } _ { 1 } \mathbf { v }$ , we pass $\mathbf { v }$ and then $- \mathbf { v }$ through the circuit that computes $\mathrm { R e L U } ( \mathbf { W } _ { 1 } \mathbf { x } )$ for an arbitrary $\mathbf { x }$ to get ReLU $\left( \mathbf { W } _ { 1 } \mathbf { v } \right)$ and ReLU $\left( - \mathbf { W } _ { 1 } \mathbf { v } \right)$ . Noting that $x = \mathrm { R e L U } ( x ) - \mathrm { R e L U } ( - x )$ , we can use $n$ additional gates to compute $\mathbf { W } _ { 1 } \mathbf { v }$ from ReLU $\left( \mathbf { W } _ { 1 } \mathbf { v } \right)$ and $\operatorname { R e L U } ( - \mathbf { W } _ { 1 } \mathbf { v } )$ . + +Repeat the process for layer $2 , 3 , \ldots , L$ (for example, pass $\mathbf { W } _ { 1 } \mathbf { v }$ and $- \mathbf { W } _ { 1 } \mathbf { v }$ to the circuit that computes $\mathbf { W } _ { \mathrm { 2 } } \mathbf { x }$ for an arbitrary $\mathbf { x }$ on layer 2). Overall we need to double the circuits that computes all the activations of the network $\operatorname { R e L U } ( \mathbf { \dot { W } } _ { 1 } \mathbf { v } ) , \dots , \operatorname { R e L U } ( \mathbf { W } _ { L } \dots \operatorname { R e L U } ( \mathbf { W } _ { 1 } \mathbf { v } ) )$ , requiring $2 s$ gates. We also need $n$ additional gates per layer to compute the negation of the input to that layer (e.g. computing $- \mathbf { v }$ from v), and $n$ additional gates per layer to subtract the output of the ReLU circuit (e.g. computing $\mathbf { W } _ { 1 } \mathbf { v }$ from $\mathrm { R e L U } ( \mathbf { W } _ { 1 } \mathbf { v } )$ and $\operatorname { R e L U } ( - \mathbf { W } _ { 1 } \mathbf { v } )$ .) Therefore we can construct an arithmetic circuit augmented with ReLU gates with $2 s + 2 L$ total gates that computes the activations of the network without ReLU $\mathbf { W } _ { 1 } \mathbf { v } , \dots , \mathbf { W } _ { L } \dots \mathbf { W } _ { 1 } \mathbf { v }$ . + +We now prove an asymptotic bound on the VC dimension of a ReLU network whose weight matrices are kaleidoscope matrices with bounded width and expansion. + +Proposition 3. Let $\mathcal { F }$ be the class of ReLU neural networks consisting of $L$ layers, where each layer is a $K$ -matrix with width and expansion bounded by some constant $C$ . Suppose that the network has $W$ total parameters. Let sign $\mathcal { F }$ denote the corresponding classification functions: $\{ x \mapsto \operatorname { s i g n } f ( x ) : f \in { \mathcal { F } } \}$ . Then this class has VC dimension: + +$$ +\operatorname { V C d i m } ( \operatorname { s i g n } { \mathcal { F } } ) = O ( L W \log W ) . +$$ + +We leverage the result from Thomas et al. (2018) for the case where the entries of the weight matrices interact multiplicatively, but with polynomially bounded degrees. This proof is similar to the VC bound for ReLU networks whose weight matrices are butterfly matrices (Dao et al., 2019). + +Proof. To use Theorem 3 of Thomas et al. (2018), we simply need to check that the entries of the linear layer, as polynomials of the parameters, has degree at most $c _ { 1 } m _ { l } ^ { c _ { 2 } }$ for some universal constant $c _ { 1 } , c _ { 2 } > 0$ , where $m _ { l }$ is the size of output of the $l$ -th layer. If the network weight matrices are K-matrices with bounded width and expansion, each weight matrix is a product of at most $c _ { 3 } \log m _ { l }$ sparse factors, for some universal constant $c _ { 3 } > 0$ . This means that the degree is polynomially bounded, which satisfies the condition of the theorem. Therefore the VC dimension is bounded to be almost linear in the number of parameters: + +$$ +\operatorname { V C d i m } ( \operatorname { s i g n } { \mathcal { F } } ) = O ( L W \log W ) . +$$ + +# M ARITHMETIC CIRCUIT PRIMER + +We give a quick overview of arithmetic circuits. This is a model of computation that has been studied for numerous computational problems (and is the basic model for algebraic complexity theory). For our purposes, we will exclusively focus on arithmetic circuits for the matrix-vector multiplication problem. For a more detailed exposition, the reader is referred to the standard book on this topic (Bürgisser et al., 2013). + +Definition M.1 (Arithmetic Circuits). An arithmetic circuit that computes $\mathbf { y } = \mathbf { A x }$ (for $\mathbf { A } \in \mathbb { F } ^ { m \times n } ,$ ) has $n$ input gates (corresponding to $\mathbf { x } [ 0 ] , \ldots , \mathbf { x } [ n - 1 ] )$ and $m$ output gates (corresponding to $\mathbf { y } [ 0 ] , \ldots , \mathbf { y } [ m - 1 ] )$ . All the internal gates correspond to addition, subtraction, multiplication and division13 over the underlying field F. The circuit is also allowed to use constants from F for ‘free.’ The definition of the internal gates can depend on A (as well as x of course). In other words, one can ‘bake’ the knowledge about A into the circuit. + +The size s of a circuit is n plus the number of addition, multiplication, subtraction and division gates used in the circuit. The depth d of a circuit is the minimum number of layers such that all gates in a given layer take as its input gates from previous layers.14 + +One drawback of arithmetic circuits (especially for infinite fields e.g. $\mathbb { F } = \mathbb { R }$ , which is our preferred choice in this work) is that they assume operations over $\mathbb { F }$ can be performed exactly. In particular, it ignores precision issues involved with real arithmetic. Nonetheless, this model turns out to be a very useful model in reasoning about the complexity of doing matrix-vector multiplication for any family of matrices. + +Perhaps the strongest argument in support of arithmetic circuits is that a large (if not an overwhelming) majority of matrix-vector multiplication algorithm also imply an arithmetic circuit of size comparable to the runtime of the algorithm (and the depth of the circuit roughly correponds to the time taken to for each compute it by a parallel algorithm). For example consider the obvious algorithm to compute $i \in [ m ]$ , compute $\mathbf { y } [ i ]$ as the sum $\begin{array} { r } { \bar { \sum } _ { i = 0 } ^ { n - 1 } \mathbf { A } [ i , j ] \mathbf { x } [ j ] ) } \end{array}$ . It is easy to see that this algorithm $\mathbf { A x }$ (i.e. implies an arithmetic circuit of size $O ( n m )$ and depth $O ( \log n )$ .15 + +One thing to note about the arithmetic circuit above is that all the multiplications involve at least one input that is a constant from $\mathbb { F }$ (recall that we can assume that the entries of A are constants that can be used to build the circuit). This leads to the following important sub-class of arithmetic circuits: + +Definition M.2 (Linear Arithmetic Circuits). An arithmetic circuit is called a linear arithmetic circuit if it only uses addition, subtraction and multiplication. Further, every multiplcation has a fixed constant from $\mathbb { F }$ as at least one of its two inputs. In other words, all gates in the circuit are linear functions of their inputs (i.e. of the form $a x +$ by for fixed constants $a , b \in \mathbb { F }$ ). + +Intuitively for the matrix-vector multiplication, it makes sense to consider linear arithmetic circuits since the final function we want to compute $\mathbf { A x }$ is indeed a linear function of its inputs. For inifinite fields (e.g. $\mathbb { F } = \mathbb { R }$ or $\mathbb { F } = \mathbb { C }$ ), it turns out that this is essentially without loss of generality: + +Theorem 4 ((Bürgisser et al., 2013)). Let F be an infinite field. Any (general) arithmetic circuit to compute Ax over $\mathbb { F }$ of size s and depth d can be converted into a linear arithmetic circuit of size $O ( s )$ and depth $O ( d )$ . + +The above result implies that for asymptotic considerations, linear arithmetic circuits for matrix-vector multiplication are equivalent to general arithmetic circuits.16 + +One important property of linear arithmetic circuits of depth $d$ , which we will use in our arguments, is that such a circuit can be equivalently represented as product of $d$ sparse matrices (see the proof of Theorem 1 for the precise derivation17). + +As mentioned earlier, a vast majority of efficient matrix vector multiplication algorithms are equivalent to small (both in size and depth) linear arithmetic circuit. For example the FFT can be thought of as an efficient arithmetic circuit to compute the Discrete Fourier Transform (indeed when one converts the linear arithmetic circuit for FFT into a matrix decomposition,18 then each matrix in the decomposition is a butterfly factor, with each block matrix in each factor being the same). For an illustration of this consider the DFT with $n = 4$ as illustrated in Figure 11. + +![](images/a3cbb522b99a7e6b2265f68e5eb9f6e0cce9cdc88c2032108159d5ac677d7182.jpg) +Figure 11: DFT of order 4. + +Figure 12 represent the arithmetic circuit corresponding to FFT with $n = 4$ + +![](images/ed59117b4e076a126cd83314d57abe6b306ececc3fd8efefc8ff242d70ff5fe8.jpg) +Figure 12: Arithmetic circuit for 4-DFT from Figure 11. + +Finally, Figure 13 is representation of the arithmetic circuit of Figure 12 as a product of a butterfly matrix and (the bit-reversal) permutation. We note that our generic arithmetic circuit to decomposition into $B B ^ { * }$ is not as tight as in Figure 13. + +![](images/1cb545a085d2ee937c721349e630bd97a789faba060e50b28a39b329f3ede1fd.jpg) +Figure 13: Decomposition of DFT of Figure 11 via the arithmetic circuit of Figure 12. + +One reason for the vast majority of existing efficient matrix vector algorithms leading to (linear) arithmetic circuits is that they generally are divide and conquer algorithms that use polynomial operations such as polynomial multiplication or evaluation (both of which themselves are divide and conquer algorithms that use FFT as a blackbox) or polynomial addition. Each of these pieces are well known to have small (depth and size) linear arithmetic circuits (since FFT has these properties). Finally, the divide and conquer structure of the algorithms leads to the circuit being of low depth. See the book of Pan (Pan, 2001) for a more elaborate description of this connection. + +In fact, the recent work of De Sa et al. (De Sa et al., 2018) makes this fact explicit and presents the most general known structure on matrices that imply near-linear size linear arithmetic circuits for the corresponding matrix vector multiplication. Their work combines two separate classes of structures matrices– orthogonal polynomial transforms (Driscoll et al., 1997; Szegö, 1967) as well as matrices with low displacement rank (Kailath et al., 1979; Olshevsky & Shokrollahi, 2000)– and presents a linear class of linear arithmetic circuits to solve their matrix vector multiplication problem. We note that structured matrices with low displacement rank have been used to replace fully connected layers in some neural network architectures (Sainath et al., 2013; Thomas et al., 2018). \ No newline at end of file diff --git a/parse/train/BkgrBgSYDS/BkgrBgSYDS_content_list.json b/parse/train/BkgrBgSYDS/BkgrBgSYDS_content_list.json new file mode 100644 index 0000000000000000000000000000000000000000..38c6d06a35c53978920bdf21d425b8ceb47edbe3 --- /dev/null +++ b/parse/train/BkgrBgSYDS/BkgrBgSYDS_content_list.json @@ -0,0 +1,6873 @@ +[ + { + "type": "text", + "text": "KALEIDOSCOPE: AN EFFICIENT, LEARNABLE REPRESENTATION FOR ALL STRUCTURED LINEAR MAPS ", + "text_level": 1, + "bbox": [ + 176, + 98, + 828, + 146 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Tri Dao 1, Nimit Sharad Sohoni∗2, Albert $\\mathbf { G u } ^ { * 1 }$ , Matthew Eichhorn 3, Amit Blonder 4, Megan Leszczynski 1, Atri Rudra 4, Christopher Ré 1 ", + "bbox": [ + 183, + 169, + 784, + 199 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "1 Department of Computer Science, Stanford University \n2 Institute for Computational and Mathematical Engineering, Stanford University \n3 Center for Applied Mathematics, Cornell University \n4 Department of Computer Science and Engineering, University at Buffalo, The State University of New York \n{trid,nims,albertgu}@stanford.edu, mae226@cornell.edu, amitblon@buffalo.edu, \nmleszczy@stanford.edu, atri@buffalo.edu, chrismre@cs.stanford.edu ", + "bbox": [ + 184, + 200, + 903, + 284 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "ABSTRACT ", + "text_level": 1, + "bbox": [ + 452, + 320, + 544, + 335 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Modern neural network architectures use structured linear transformations, such as low-rank matrices, sparse matrices, permutations, and the Fourier transform, to improve inference speed and reduce memory usage compared to general linear maps. However, choosing which of the myriad structured transformations to use (and its associated parameterization) is a laborious task that requires trading off speed, space, and accuracy. We consider a different approach: we introduce a family of matrices called kaleidoscope matrices (K-matrices) that provably capture any structured matrix with near-optimal space (parameter) and time (arithmetic operation) complexity. We empirically validate that K-matrices can be automatically learned within end-to-end pipelines to replace hand-crafted procedures, in order to improve model quality. For example, replacing channel shuffles in ShuffleNet improves classification accuracy on ImageNet by up to $5 \\%$ . K-matrices can also simplify hand-engineered pipelines—we replace filter bank feature computation in speech data preprocessing with a learnable kaleidoscope layer, resulting in only $0 . 4 \\%$ loss in accuracy on the TIMIT speech recognition task. In addition, K-matrices can capture latent structure in models: for a challenging permuted image classification task, a K-matrix based representation of permutations is able to learn the right latent structure and improves accuracy of a downstream convolutional model by over $9 \\%$ . We provide a practically efficient implementation of our approach, and use K-matrices in a Transformer network to attain $36 \\%$ faster end-to-end inference speed on a language translation task. ", + "bbox": [ + 233, + 352, + 766, + 642 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "1 INTRODUCTION ", + "text_level": 1, + "bbox": [ + 176, + 669, + 334, + 684 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Structured linear maps are fundamental and ubiquitous in modern machine learning. Their efficiency in speed (fast algorithms) and space (few parameters) can reduce computation and memory usage. The class of structured linear maps includes fixed specialized transforms such as the discrete Fourier transform (DFT) and Hadamard transform used in signal processing (Cooley et al., 1969), convolutions for image, language, and speech modeling (Gu et al., 2018), and low-rank and sparse matrices for efficient storage and inference on edge devices (Yu et al., 2017). Forms of structure such as sparsity have been at the forefront of recent advances in ML (Frankle & Carbin, 2019), and are critical for on-device and energy-efficient models, two application areas of tremendous recent interest (Tsidulko, 2019; Schwartz et al., 2019). ", + "bbox": [ + 174, + 700, + 825, + 825 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "There are a plethora of classes of structured linear maps, each with a significantly different representation, algorithm, and implementation. They have different tradeoffs in terms of inference speed, training speed, and accuracy, and the conventional wisdom is that no one class works uniformly well across all applications. As a result, ML practitioners currently hand-pick specific classes of structured linear maps for each of their applications. This is a difficult and labor-intensive task. ", + "bbox": [ + 176, + 832, + 825, + 901 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Ideally, these problems should be addressed with a universal representation for structured linear maps: (i) Such a parameterization should be expressive enough to capture important classes of structure, with a nearly tight parameter count and runtime: the space required to represent the linear map should be close to optimal, and the resulting algorithm for matrix vector multiplication should be close to the fastest possible algorithm. (ii) The parameterization should be differentiable in order to be learned as a component of end-to-end ML pipelines, enabling it to easily be used as a drop-in replacement for manually engineered structured components. (iii) The parameterization should admit practically efficient algorithms for training and inference, in terms of both speed and memory. ", + "bbox": [ + 173, + 103, + 825, + 215 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "Currently, no class of structured linear maps satisfies all of these criteria. Most existing classes of structured matrices—such as the class of low-rank matrices—fail to tightly capture other important types of structure. For example, the DFT has an efficient structured representation of size $O ( n \\log n )$ , yet cannot be well-approximated by a low-rank transform of size $\\dot { \\ll } n ^ { 2 }$ . Another important type of structure is sparsity; lots of exciting recent work has focused on the design of sparse neural networks. For instance, sparse networks of comparable quality to their dense counterparts—yet an order of magnitude fewer parameters—may be created via pruning (Han et al., 2016) or by identifying “winning lottery tickets” (Frankle & Carbin, 2019). In parallel, recent theoretical results by De Sa et al. (2018) show that sparsity and the notion of structure in linear maps are fundamentally linked: any given matrix can be factored into a product of sparse matrices with total parameter count equal to the efficiency (i.e. minimum arithmetic circuit complexity) of the matrix. In other words, the representation of linear maps as products of sparse matrices tightly captures all forms of structure. Unfortunately, it is difficult to actually learn these sparse factorizations, because it requires finding the sparsity patterns of the factors—a discrete, nondifferentiable search problem. Thus, current methods for training sparse neural networks are either expensive (Frankle & Carbin, 2019) or rely on highly hand-tuned heuristics for evolving the sparsity patterns throughout training (Dettmers & Zettlemoyer, 2019). ", + "bbox": [ + 173, + 222, + 825, + 457 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "By contrast, we propose a representation of linear maps as products of sparse matrices with specific predefined sparsity patterns (Section 2), and show that it does satisfy our desiderata: it retains the expressiveness of unstructured sparsity, while being differentiably learnable and efficient like other structured representations. Concretely, our representation is based on products of a particular building block known as a butterfly matrix (Parker, 1995; Dao et al., 2019); we term such products kaleidoscope matrices (K-matrices for short).1 (i) Our main theoretical contribution (Section 2.3) concerns the expressiveness of this representation: we show that any structured linear map (i.e. one that can be applied using $s \\ll n ^ { 2 }$ arithmetic operations) can be represented as a K-matrix, with a nearly tight number of parameters and algorithmic complexity (both on the order of $s$ up to logarithmic factors). (ii) The kaleidoscope representation is fully differentiable; thus, all the parameters of a K-matrix can be learned using standard optimization algorithms such as SGD. (iii) Because of their simple, regular structure, K-matrices are practical and easy to use. We provide memory- and runtime-efficient implementations of K-matrix multiplication on CPU and GPU for training and inference, with a simple PyTorch interface. ", + "bbox": [ + 174, + 464, + 825, + 659 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "We empirically validate that, due to their expressiveness, learnability, and efficiency, we can use K-matrices as a drop-in replacement for linear components in deep learning models. In Section 3.1, we use K-matrices to replace hand-crafted structure in two different settings. We simplify the six steps of filter bank computation in speech preprocessing into a single learnable K-matrix step, with only an $0 . 4 \\%$ accuracy drop on the TIMIT speech recognition task. We use K-matrices to replace channel shuffles in ShuffleNet, improving ImageNet classification accuracy by up to $5 \\%$ . In Section 3.2, we show that K-matrices can successfully recover latent structure; a K-matrix is used to learn latent permutations in a permuted image dataset (Permuted CIFAR), resulting in 9 points higher accuracy in a downstream CNN model. In Section 3.3, we show that our efficient K-matrix multiplication implementation can be applied to speed up real-world tasks: we replace linear layers with K-matrices in a DynamicConv-Transformer network to attain $36 \\%$ faster end-to-end inference speed with a 1.0 drop in BLEU score on the IWSLT14 German English translation task. ", + "bbox": [ + 174, + 666, + 825, + 832 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "We first present some background on the characterization of all structured matrices (i.e. those with subquadratic multiplication algorithms) as products of sparse factors, along with the definition of butterfly matrices. We then propose a differentiable family of kaleidoscope matrices, composed of products of butterfly matrices, and prove their expressivity: all structured matrices can be represented in this form, with almost optimal parameter count and runtime. ", + "bbox": [ + 174, + 132, + 825, + 203 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "2.1 BACKGROUND: SPARSE FACTORIZATION, BUTTERFLY MATRICES ", + "text_level": 1, + "bbox": [ + 176, + 220, + 658, + 234 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Sparse factorization One method of constructing matrices with theoretically fast matrix-vector multiplication algorithms is as a product of sparse matrices, so that multiplication by an arbitrary vector has cost proportional to the total number of nonzeros (NNZ) of the matrices in the product. Surprisingly, the converse is also true. De Sa et al. (2018) introduce the concept of sparse product width (SPW), which roughly corresponds to the total NNZ in a factorization of a matrix, and show that it is an asymptotically optimal descriptor of the algorithmic complexity of matrix-vector multiplication (Bürgisser et al., 2013). We use a similar argument in the proof of our main theorem (Section 2.3). However, attempting to learn such a factorization of a given matrix is difficult, as the sparsity constraint is not continuous. Moreover, because of the possibly irregular sparsity patterns, it is difficult to realize the theoretical speedups in practice (Gray et al., 2017; Gahvari et al., 2007). ", + "bbox": [ + 174, + 244, + 825, + 386 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Butterfly matrices Butterfly matrices, encoding the recursive divide-and-conquer structure of the fast Fourier transform (FFT) algorithm, have long been used in numerical linear algebra (Parker, 1995; Li et al., 2015) and machine learning (Mathieu & LeCun, 2014; Jing et al., 2017; Munkhoeva et al., 2018; Dao et al., 2019; Choromanski et al., 2019). Here we define butterfly matrices, which we use as a building block for our hierarchy of kaleidoscope matrices. ", + "bbox": [ + 173, + 400, + 825, + 469 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Definition 2.1. A butterfly factor of size $k \\geq 2$ (denoted as $\\mathbf { B } _ { k }$ ) is a matrix of the form $\\mathbf { B } _ { k } \\ =$ $\\left[ \\begin{array} { c c } { \\mathbf { D } _ { 1 } } & { \\mathbf { D } _ { 2 } } \\\\ { \\mathbf { D } _ { 3 } } & { \\mathbf { D } _ { 4 } } \\end{array} \\right]$ where each $\\mathbf { D } _ { i }$ is a ${ \\begin{array} { l } { { \\frac { k } { 2 } } \\times { \\frac { k } { 2 } } } \\end{array} }$ diagonal matrix. We restrict $k$ to be a power of 2. ", + "bbox": [ + 173, + 472, + 825, + 520 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Definition 2.2. A butterfly factor matrix of size n with block size $k$ (denoted as $\\mathbf { B } _ { k } ^ { \\left( n \\right) }$ ) is a block diagonal matrix of $\\frac { n } { k }$ (possibly different) butterfly factors of size $k$ : ", + "bbox": [ + 174, + 523, + 825, + 555 + ], + "page_idx": 2 + }, + { + "type": "equation", + "img_path": "images/7e206429ba9471ddd994e66e0577cf3ee0242b5ee73bf922a44600792ee16eb1.jpg", + "text": "$$\n\\mathbf B _ { k } ^ { ( n ) } = \\operatorname { d i a g } \\left( \\left[ \\mathbf B _ { k } \\right] _ { 1 } , \\left[ \\mathbf B _ { k } \\right] _ { 2 } , \\ldots , \\left[ \\mathbf B _ { k } \\right] _ { \\frac n k } \\right)\n$$", + "text_format": "latex", + "bbox": [ + 359, + 561, + 637, + 588 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "product of butterfly factor matrices: B(n) = B(n)n B(n)n2 . Definition 2.3. A butterfly matrix of size $n$ (denoted as $\\mathbf { B } ^ { ( n ) }$ ) is a matrix that can be expressed as a . Equivalently, we may define $\\mathbf { B } ^ { ( n ) }$ recursively as a matrix that can be expressed in the following form: ", + "bbox": [ + 173, + 593, + 826, + 643 + ], + "page_idx": 2 + }, + { + "type": "equation", + "img_path": "images/cbc52fb80e487a19f7e27a87dd8dcdac4c78d5c32d90781bbe0b3c788ff03326.jpg", + "text": "$$\n\\mathbf { B } ^ { ( n ) } = \\mathbf { B } _ { n } ^ { ( n ) } \\left[ { \\big [ } { \\mathbf { B } ^ { ( \\frac { n } { 2 } ) } } { \\big ] } _ { 1 } \\right. \\qquad 0 \\qquad { \\big ] }\n$$", + "text_format": "latex", + "bbox": [ + 383, + 647, + 614, + 683 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "(Note that $\\big [ \\mathbf { B } ^ { ( \\frac { n } { 2 } ) } \\big ] _ { 1 }$ and $\\big [ \\mathbf { B } ^ { ( \\frac { n } { 2 } ) } \\big ] _ { 2 }$ may be different.) ", + "bbox": [ + 171, + 686, + 501, + 705 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "2.2 THE KALEIDOSCOPE HIERARCHY ", + "text_level": 1, + "bbox": [ + 176, + 719, + 444, + 734 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Using the building block of butterfly matrices, we formally define the kaleidoscope $( B B ^ { * } )$ hierarchy and prove its expressiveness. This class of matrices serves as a fully differentiable alternative to products of sparse matrices (Section 2.1), with similar expressivity. In Appendix J, we show where various common structured matrix classes are located within this hierarchy. ", + "bbox": [ + 174, + 746, + 825, + 803 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "The building block for this hierarchy is the product of a butterfly matrix and the (conjugate) transpose of another butterfly matrix (which is simply a product of butterfly factors taken in the opposite order). Figure 1 visualizes the sparsity patterns of the butterfly factors in $B B ^ { * }$ , where the red and blue dots represent the allowed locations of nonzero entries. ", + "bbox": [ + 174, + 809, + 825, + 866 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Definition 2.4 (Kaleidoscope hierarchy, kaleidoscope matrices). ", + "bbox": [ + 173, + 868, + 594, + 883 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "• Define $\\boldsymbol { B }$ as the set of all matrices that can be expressed in the form $\\mathbf { B } ^ { ( n ) }$ (for some $n$ ). • Define $B B ^ { * }$ as the set of matrices M of the form $\\mathbf { M } = \\mathbf { M } _ { 1 } \\mathbf { M } _ { 2 } ^ { * }$ for some $\\mathbf { M } _ { 1 } , \\mathbf { M } _ { 2 } \\in B$ ", + "bbox": [ + 173, + 891, + 759, + 925 + ], + "page_idx": 2 + }, + { + "type": "image", + "img_path": "images/f75bf8d4d0ee21f357fac50dfdf80482c8af5cd021491925696495d635d08d3a.jpg", + "image_caption": [ + "Figure 1: Visualization of the fixed sparsity pattern of the building blocks in $B B ^ { * }$ , in the case $n = 1 6$ The red and blue dots represent all the possible locations of the nonzero entries. " + ], + "image_footnote": [], + "bbox": [ + 181, + 99, + 820, + 178 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "• Define $( B B ^ { * } ) ^ { w }$ as the set of matrices M that can be expressed as $\\mathbf { M } = \\mathbf { M } _ { w } \\ldots \\mathbf { M } _ { 2 } \\mathbf { M } _ { 1 }$ , with each $\\mathbf { M } _ { i } \\in B B ^ { * }$ $1 \\leq i \\leq w )$ . (The notation w represents width.) \n• Define $( B B ^ { * } ) _ { e } ^ { w }$ as the set of $n \\times n$ matrices M that can be expressed as ${ \\bf M } = { \\bf S } { \\bf E } { \\bf S } ^ { T }$ for some $e n \\times e n$ matrix $\\mathbf { E } \\in ( B B ^ { \\ast } ) ^ { w }$ , where $\\mathbf { S } \\in \\mathbb { F } ^ { n \\times e n } = [ \\mathbf { I } _ { n } \\quad 0 \\quad \\ldots \\quad 0 ]$ (i.e. M is the upper-left corner of $\\mathbf { E }$ ). (The notation e represents expansion relative to $n$ .) \n• M is a kaleidoscope matrix, abbreviated as $\\pmb { K }$ -matrix, if $M \\in ( B B ^ { * } ) _ { e } ^ { w }$ for some w and e. ", + "bbox": [ + 173, + 242, + 826, + 335 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "The kaleidoscope hierarchy, or $( B B ^ { * } )$ hierarchy, refers to the families of matrices $( B B ^ { * } ) _ { e } ^ { 1 } \\subseteq$ $( B B ^ { * } ) _ { e } ^ { 2 } \\subseteq \\dots$ , for a fixed expansion factor $e$ . Each butterfly matrix can represent the identity matrix, so $( B B ^ { * } ) _ { e } ^ { w } ~ \\subseteq ~ ( B B ^ { * } ) _ { e } ^ { \\bar { w } + 1 }$ . We show that the inclusion is proper in Appendix E. This hierarchy generalizes the $B \\mathcal { P }$ hierarchy proposed by Dao et al. (2019), as shown in Appendix J. ", + "bbox": [ + 173, + 344, + 825, + 401 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Efficiency in space and speed Each matrix in $( B B ^ { * } ) _ { e } ^ { w }$ is a product of $2 w$ total butterfly matrices and transposes of butterfly matrices, each of which is in turn a product of $\\log ( n e )$ factors with $2 n e$ nonzeros (NNZ) each. Therefore, each matrix in $( B B ^ { * } ) _ { e } ^ { w }$ has $4 w n e \\log ( n e )$ parameters and a matrix-vector multiplication algorithm of complexity $O ( w n e \\log n e )$ (by multiplying the vector with each sparse factor sequentially). We prove this more formally in Appendix E. For the applications in Section 3, $w$ and $e$ are small constants (up to 2), so those $\\mathrm { K }$ -matrices have $O ( n \\log n )$ parameters and runtime. ", + "bbox": [ + 173, + 416, + 825, + 513 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "2.3 ALL LOW-DEPTH STRUCTURED MATRICES ARE IN THE KALEIDOSCOPE HIERARCHY ", + "text_level": 1, + "bbox": [ + 174, + 531, + 787, + 545 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "We now present our main theoretical result: the fact that general linear transformations, expressed as low-depth linear arithmetic circuits, are captured in the $B B ^ { * }$ hierarchy with low width. Arithmetic circuits are commonly used to formalize algebraic algorithmic complexity (Bürgisser et al., 2013); we include a primer on this in Appendix M. The quantities of interest are the total number of gates in the circuit, representing the total number of steps required to perform the algorithm for a serial processor, and the depth, representing the minimum number of steps required for a parallel processor. ", + "bbox": [ + 174, + 556, + 825, + 640 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Theorem 1. Let M be an $n \\times n$ matrix such that multiplication of M times an arbitrary vector v can be represented as a linear arithmetic circuit with s total gates and depth d. Then, M ∈ (BB∗)O(d)O( sn ). ", + "bbox": [ + 174, + 643, + 823, + 679 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "The representation of such a matrix $\\mathbf { M }$ in the $B B ^ { * }$ hierarchy has $O ( d s \\log s )$ parameters and yields a $O ( d s \\log s )$ multiplication algorithm, compared to the $O ( s )$ parameters and runtime of the circuit representation. To the best of our knowledge, the most general classes of efficient matrices that have been studied (De Sa et al., 2018) have depth $d$ on the order of $\\log n$ or poly $\\log n$ . In these cases, the representation with K-matrices matches the best known bounds up to polylogarithmic factors. ", + "bbox": [ + 174, + 690, + 825, + 761 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "The crux of the proof of Theorem 1 (shown in Appendix F) is the construction of an almost tight representation of any sparse matrix as a $\\mathbf { K }$ -matrix (i.e. a product of butterfly matrices): specifically, we show that any $n \\times n$ sparse matrix with $s$ nonzeros is in $( B B ^ { * } ) _ { O ( 1 ) } ^ { O ( \\left\\lceil \\frac { s } { n } \\right\\rceil ) }$ (Theorem 3, Appendix I). We then leverage the expressivity result of products of sparse matrices to represent all arithmetic circuits (similar to the sparse product width result of De Sa et al. (2018) referenced in Section 2.1) to complete the proof of Theorem 1. ", + "bbox": [ + 173, + 767, + 825, + 861 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "This intermediate result is also a novel characterization of sparse matrices. For a matrix with $s$ NNZ, the kaleidoscope representation has $O ( s \\log { n } )$ parameters and runtime, instead of the optimal $O ( s )$ parameters and runtime; so, we trade off an extra logarithmic factor in space and time for full differentiability (thanks to the fixed sparsity patterns in the representation). The intuition behind the result is as follows: a sparse matrix with $s$ NNZ can be written as a sum of $\\lceil s / n \\rceil$ matrices each with at most $n$ NNZ. Any $n \\times n$ matrix with at most $n$ NNZ, up to permuting the rows and columns, is a product of two butterfly matrices (Lemma I.1). Sorting networks (Knuth, 1997) imply that permutation matrices are in $( \\tilde { B B ^ { * } } ) ^ { O ( \\log n ) }$ , but we tighten the result to show that they are in fact in $B B ^ { * }$ (Theorem 2, Appendix $\\mathbf { G }$ ). We thus obtain a kaleidoscope representation for each summand matrix with $O ( n \\log n )$ parameters. By the addition closure property of the $B B ^ { * }$ hierarchy (Lemma H.5), each sparse matrix with $s$ NNZ then has a kaleidoscope representation with $O ( s \\log { n } )$ parameters. ", + "bbox": [ + 174, + 867, + 825, + 924 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "", + "bbox": [ + 173, + 103, + 825, + 217 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Tight representation for structured linear maps common in ML Even though Theorem 1 suggests that the kaleidoscope representation can be loose by logarithmic factors, many structured linear maps common in ML can be represented in this hierarchy with an optimal number of parameters and runtime compared to the best known parameterizations, up to constant factors. Appendix J includes several examples such as discrete transforms (the DFT, discrete cosine transform (DCT), discrete sine transform (DST), and Hadamard transform), convolution (i.e. circulant matrices), Toeplitz matrices (Gray, 2006), structured matrices for kernel approximation $( ( H D ) ^ { 3 }$ (Yu et al., 2016)) and compact neural network design (Fastfood (Le et al., 2013), ACDC (Moczulski et al., 2016)). There have been other large classes of structured matrices proposed in the machine learning literature, such as Toeplitz-like (Sindhwani et al., 2015) and low displacement rank (LDR) (Thomas et al., 2018), but they are not known to be able to capture these common structures as tightly as K-matrices can. More detailed discussions are in Appendix A. ", + "bbox": [ + 174, + 237, + 825, + 404 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "2.4 EXTENSIONS ", + "text_level": 1, + "bbox": [ + 176, + 426, + 305, + 440 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "ReLU networks with low-depth structured weight matrices In Appendix L, we prove that finding an efficient circuit for a ReLU network can be reduced to finding efficient circuits for each of its weight matrices, with at most a constant factor greater size and run-time (i.e. number of gates). We also show that ReLU networks with kaleidoscope weight matrices have near-linear VC dimension in the number of parameters, matching the bound for networks with unconstrained weight matrices (Bartlett et al., 1999; Harvey et al., 2017) and LDR (Thomas et al., 2018). This yields a corresponding sample complexity bound. ", + "bbox": [ + 173, + 454, + 825, + 553 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Orthogonal kaleidoscope hierarchy Orthogonal butterfly matrices are one commonly used variant due to their improved stability (Parker, 1995), where each butterfly factor is constrained to be orthogonal: $\\left[ \\begin{array} { l l } { \\mathbf { C } } & { \\mathbf { s } } \\\\ { - \\mathbf { S } } & { \\mathbf { C } } \\end{array} \\right]$ with $\\mathbf { C } , \\mathbf { S }$ being diagonal and $\\mathbf { C } ^ { 2 } + \\mathbf { S } ^ { 2 } = \\mathbf { I }$ . Similar to the $B B ^ { * }$ hierarchy, in Appendix K, we define the OBB hierarchy consisting of products of orthogonal butterfly matrices and diagonal matrices, and show that this hierarchy has the same expressiveness as the $B B ^ { * }$ hierarchy. ", + "bbox": [ + 173, + 571, + 825, + 660 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "3 EMPIRICAL EVALUATION ", + "text_level": 1, + "bbox": [ + 176, + 684, + 415, + 700 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "We validate three claims that suggest that kaleidoscopes are a promising technique to learn different types of structure in modern architectures. ", + "bbox": [ + 176, + 719, + 823, + 747 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "1. Section 3.1: for applications in speech and lightweight computer vision relying on highly hand-crafted structured transformations, we show that we can recover—and even improve—the quality of such architectures by simply replacing existing hand-structured components with K-matrices, with only a small overhead in memory and computation. \n2. In Section 3.2, for a challenging task with latent structure (Permuted CIFAR-10), a K-matrixbased relaxation of permutations is able to learn the right latent permutation, yielding 9 points better accuracy in a downstream CNN compared to standard RNN and CNN baselines used on such permuted image classification tasks. \n3. In Section 3.3, we show that, although not yet highly optimized, our current implementation of K-matrices can improve the inference throughput of DynamicConv Transformer, a state-ofthe-art fast machine translation model, by $36 \\%$ , with only a relatively small drop in translation quality. ", + "bbox": [ + 186, + 755, + 826, + 924 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "In all of the above applications, as K-matrices are fully differentiable, we simply train them jointly with the rest of the model using standard learning algorithms (such as SGD). Full details for all of the experiments (precise architectures, hyperparameters, etc.) are in Appendix $\\boldsymbol { \\mathrm { ~ B ~ } ^ { 2 } }$ . ", + "bbox": [ + 174, + 103, + 825, + 146 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "3.1 REPLACING HAND-CRAFTED STRUCTURES ", + "text_level": 1, + "bbox": [ + 174, + 164, + 509, + 178 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "We validate that kaleidoscope matrices can recover or improve on the performance of hand-crafted structure in ML models. For example, a single learnable kaleidoscope layer can be used to replace the hand-engineered filter bank speech preprocessing pipeline with only $0 . 4 \\%$ loss in accuracy on the TIMIT speech recognition task (Section 3.1.1). Replacing channel shuffles in ShuffleNet with learnable K-matrices improves classification accuracy on ImageNet by up to $5 . 0 \\%$ (Section 3.1.2). ", + "bbox": [ + 174, + 189, + 825, + 260 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "3.1.1 SPEECH PREPROCESSING ", + "text_level": 1, + "bbox": [ + 176, + 276, + 401, + 290 + ], + "page_idx": 5 + }, + { + "type": "image", + "img_path": "images/84b8117d4d1439b40cfd9f8f6af50403ab20bc6079dd9c46a5e034984993d116.jpg", + "image_caption": [ + "Figure 2: Comparison of the standard MFSC featurization pipeline with our “kaleidoscope” pipeline. " + ], + "image_footnote": [], + "bbox": [ + 259, + 304, + 740, + 414 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "We show that K-matrices can remove the need for hand-tuning by significantly simplifying speech recognition data preprocessing pipelines. In particular, we can entirely replace the complex handcrafted MFSC featurization commonly used in speech recognition tasks with a fully learnable kaleidoscope layer, with only $0 . 4 \\%$ drop in accuracy on the TIMIT speech recognition benchmark. Results are presented in Table 1. Our approach is competitive with the accuracy of standard models that use hand-crafted features, and significantly outperforms current approaches for learning from raw audio input. ", + "bbox": [ + 173, + 458, + 825, + 556 + ], + "page_idx": 5 + }, + { + "type": "table", + "img_path": "images/c70107dac1436f4755420324f7ec7319dd8f28d137b1072ba173b61895fcf588.jpg", + "table_caption": [ + "Table 1: TIMIT phoneme error rate $( \\mathrm { P E R } \\% )$ for different methods. Our kaleidoscope, raw-input version of the model (row 3) performs competitively with the original model trained on MFSC features (row 1), with only an $0 . 4 \\%$ drop in PER. It significantly outperforms existing approaches that learn from raw audio, i.e. without handcrafted featurization (e.g. SincNet [row 2], which to our knowledge attains the previous state-of-the-art for learning from raw audio), and is only $0 . 8 \\%$ less accurate than the overall state-of-the-art on TIMIT.3 Additional comparisons are given in Appendix B.1. " + ], + "table_footnote": [], + "table_body": "
MethodTest set PER%Raw audio input
MFSC features +LSTM14.2x//
SincNet (Ravanelli et al., 2019)17.2
Kaleidoscope + LSTM14.6
", + "bbox": [ + 259, + 662, + 736, + 734 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Modern speech recognition models currently rely on carefully hand-crafted features extracted from the audio, which are then fed into an acoustic model. By contrast, learning directly from the raw audio—i.e. end-to-end learning from the audio waveform without any manual featurization—obviates the need for this complicated and often expensive preprocessing step. There have been recent attempts to learn directly from raw audio, such as SincNet (Ravanelli & Bengio, 2018); however, they often rely on specialized architectures designed by domain experts. Instead, we use a standard RNN speech recognition architecture, but use a learnable kaleidoscope layer to replace the featurization steps. ", + "bbox": [ + 173, + 748, + 825, + 847 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "The baseline architecture takes as input filter bank (MFSC) features, which are a popular standard featurization for speech recognition (Paliwal, 1999) and involve several steps hand-crafted specifically for this domain. These features are extracted from the raw audio waveform, and fed as the input into a Bi-LSTM model. We significantly simplify this pipeline by replacing the featurization step with a trainable kaleidoscope layer that is trained end-to-end together with the Bi-LSTM. The original pipeline and our modified kaleidoscope version are depicted in Figure 2. ", + "bbox": [ + 174, + 103, + 823, + 188 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "The computation of MFSC features involves a series of painstakingly hand-designed steps (further described in Appendix B.1), each involving their own hyperparameters: (i) the waveform is framed (split into chunks), (ii) the waveform is dithered (noise is added), (iii) pre-emphasis is applied, (iv) the Hamming window is applied, (v) the FFT is applied and the power spectrum is computed, (vi) the result is mapped to the mel scale (which involves applying a particular linear transformation and then taking the logarithm of the result), (vii) cepstral mean and variance normalization is applied. We replace the last six steps (ii-vii) of this featurization process with a learnable kaleidoscope layer; specifically, after windowing, we multiply the input by a K-matrix, and then compute the logarithm of the power spectrum; the output is fed into the Bi-LSTM model. ", + "bbox": [ + 173, + 194, + 825, + 319 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "3.1.2 REPLACING CNN CHANNEL SHUFFLE ", + "text_level": 1, + "bbox": [ + 176, + 337, + 486, + 351 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "We evaluate how K-matrices can improve the quality of hand-crafted, lightweight architectures for computer vision tasks, without the need for hand-tuning. We select ShuffleNet (Zhang et al., 2018), which is a state-of-the-art lightweight CNN architecture that uses a manually designed “channel shuffle” permutation matrix to improve performance. By replacing this fixed permutation with a learnable K-matrix, we achieve up to $5 \\%$ further improvement in classification accuracy, without hand-tuned components and with a modest space penalty of up to $10 \\%$ . Results are given in Table 2. ", + "bbox": [ + 174, + 362, + 825, + 445 + ], + "page_idx": 6 + }, + { + "type": "table", + "img_path": "images/39938c1550e0d77d1e492aa1259916c5d21638d60855471c5ad907f5d5640f7a.jpg", + "table_caption": [ + "Table 2: Top-1 classification accuracy of ShuffleNet on ImageNet validation set (parameter counts in parentheses). We compare our approach (col. 3) with our reimplementation of ‘vanilla’ ShuffleNet (col. 1) and a recent approach based on the Hadamard transform (col. 2).4 We report results for different network width multipliers (# channels). The last column shows the differences in accuracy and parameter count between our approach and vanilla ShuffleNet; using a learnable K-matrix in place of each fixed permutation (shuffle) or Hadamard matrix improves accuracy by up to $5 \\%$ . " + ], + "table_footnote": [], + "table_body": "
ShuffleHadamardKaleidoscope (K.)K. vs. Shuffle
0.25 ShuffleNet g844.1% (0.46M)43.9% (0.46M)49.2% (0.51M)+5.0% (+0.05M)
0.5 ShuffleNet g857.1% (1.0M)56.2% (1.0M)59.5% (1.1M)+2.4% (+0.1M)
1.0 ShuffleNet g865.3% (2.5M)65.0% (2.5M)66.5% (2.8M)+1.2% (+0.2M)
", + "bbox": [ + 173, + 556, + 841, + 628 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "Grouped convolution (Krizhevsky et al., 2012) is often used to reduce parameter count and speed up inference compared to standard convolution, but, by default, channels in different groups cannot exchange information. To remedy this, ShuffleNet uses a permutation matrix to shuffle the channels after each grouped convolution. Zhao et al. (2019) propose to instead use the Hadamard transform before and after each grouped convolution to mix the channels. In place of these hand-engineered solutions, we use a K-matrix before and after each grouped convolution, and learn these end-to-end together with the rest of the network. As shown in Table 2, across a range of sizes, replacing the channel shuffles with K-matrices results in improved performance at comparable parameter counts. ", + "bbox": [ + 174, + 643, + 825, + 755 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "3.2 LEARNING A LATENT PERMUTATION ", + "text_level": 1, + "bbox": [ + 176, + 775, + 464, + 789 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "We show that K-matrices can be used in a challenging task for which existing classes of structured linear maps have not been found suitable. We investigate the problem of image classification on a permuted image dataset (Permuted CIFAR-10). This problem is challenging due to the discrete nature of learning the latent permutation of the dataset; we present a differentiable relaxation for this using a K-matrix as a key component. Results are presented in Table 3; compared to methods that do not have a permutation learning step, our approach gets 9 points higher accuracy $8 4 . 4 \\%$ to $9 3 . 6 \\%$ ), coming within 2 points of the accuracy on the un-permuted dataset $( 9 4 . 9 \\% )$ . ", + "bbox": [ + 174, + 801, + 825, + 871 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "", + "bbox": [ + 171, + 103, + 823, + 132 + ], + "page_idx": 7 + }, + { + "type": "table", + "img_path": "images/5f1101d48ba5865118e3c1608568b61d4f566e3cab2d79ef0d448fe765e42054.jpg", + "table_caption": [ + "Table 3: Permuted CIFAR-10 validation set classification accuracy $( \\% )$ . Our kaleidoscope layer is able to nearly perfectly recover the latent structure, allowing a downstream CNN to approach the accuracy of a standard ResNet18 on the unpermuted dataset (last column). " + ], + "table_footnote": [], + "table_body": "
ModelFCRNNCNNDense +CNNK +CNN Unpermuted
Accuracy61.257.873.784.493.694.9
", + "bbox": [ + 233, + 193, + 759, + 236 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "In this task, we use a permuted image classification dataset (Permuted CIFAR-10), wherein a fixed global permutation is applied to the pixels of every image in the original input set. Typically, only fully-connected (FC) and recurrent models are applied to such datasets (Le et al., 2015), because the permutation destroys locality in the image, presenting a difficulty for CNNs. However, CNNs are much better-suited for standard image tasks. We thus expect that learning the permutation and then applying a standard CNN should outperform these baselines. As mentioned in Section 2, the kaleidoscope hierarchy provides a nearly tight parameterization of permutations; this makes them a natural fit for the permutation learning step. ", + "bbox": [ + 173, + 248, + 825, + 361 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "Experimentally, we use a K-matrix to represent a distribution over permutations, which converges to a single permutation at the end of training. The correct latent structure is learned by applying samples from this distribution to the permuted training images, and minimizing an auxiliary smoothness-based loss that encourages the reconstructed images to be more “natural” (i.e. vary smoothly pixel-to-pixel). The learned permutation is evaluated by training a ResNet18 with the K-matrix permutation layer inserted at the beginning. Full details of our approach are provided in Appendix B.3. ", + "bbox": [ + 173, + 367, + 825, + 452 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "In Table 3, we compare our approach to a ResNet18 without this extra K-matrix layer, a ResNet18 with an extra dense matrix at the beginning instead of a K-matrix, and other baselines. As generic representations such as unstructured matrices do not have the requisite properties to fit in the pipeline, these baselines fail to effectively learn the latent permutation. We emphasize that a K-matrix provides this ability to recover latent structure despite not being specialized for permutations. Figure 3 describes the pipeline and displays examples of permuted and unpermuted images. ", + "bbox": [ + 173, + 458, + 825, + 542 + ], + "page_idx": 7 + }, + { + "type": "image", + "img_path": "images/413753fc1737fb267ca8776a147dae87b59ab479d8f3db8b527eccd5a0dbb78b.jpg", + "image_caption": [], + "image_footnote": [], + "bbox": [ + 616, + 554, + 816, + 662 + ], + "page_idx": 7 + }, + { + "type": "image", + "img_path": "images/e9a1d672dbf9a7bf87c5651a9c2850de7f5858227bf8566efc1af6180d95ea50.jpg", + "image_caption": [ + "Figure 3: (a) (Left) Schematic describing permutation learning approach. The inputs are multiplied by a K-matrix and then fed into a CNN, from which the classification loss is computed. Separately, the input is permuted by a permutation matrix sampled from the distribution described by the Kmatrix, and a “smoothness” loss (Rudin et al., 1992) is computed from the result, as described in Appendix B.3. (b) (Right) Left panel: original (unpermuted) example images. Center panel: the permuted versions. Right panel: these images after then applying the permutation recovered by the K-matrix. The K-matrix is able to nearly unscramble the images into their unpermuted versions. " + ], + "image_footnote": [], + "bbox": [ + 174, + 555, + 565, + 659 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "3.3 SPEEDING UP INFERENCE ", + "text_level": 1, + "bbox": [ + 174, + 792, + 392, + 806 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "We evaluate the inference speed benefit of using K-matrices on a real language translation model. We choose the state-of-the-art DynamicConv Transformer translation model (Wu et al., 2019), which offers $20 \\%$ inference speedup over the standard Transformer model, and replace dense matrices in the decoder’s linear layers with K-matrices, which leads to a further $36 \\%$ inference speedup (Table 4). ", + "bbox": [ + 174, + 819, + 825, + 875 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "As outlined in Section 2.3, K-matrices admit a simple and fast $O ( n \\log n )$ matrix-vector multiplication algorithm. We provide fast implementations of this algorithm in $\\mathrm { C } { + } { + }$ and CUDA, with an interface to PyTorch (Paszke et al., 2017), and use this implementation in our experiments. ", + "bbox": [ + 176, + 881, + 823, + 924 + ], + "page_idx": 7 + }, + { + "type": "table", + "img_path": "images/33d7090a3b401b7bd778984c4ecb8b8dabd9756dd68e21819c1398b5d10f9cf1.jpg", + "table_caption": [ + "Table 4: Inference speed on the IWSLT-14 German-English translation task (test set). Using Kmatrices instead of dense matrices in the DynamicConv decoder linear layers results in $36 \\%$ faster inference speed (measured on a single-threaded CPU with a batch size of 1 and beam size of 1). " + ], + "table_footnote": [], + "table_body": "
Model# paramsBLEUSentences/secTokens/sec
Transformer (Vaswani et al., 2017)43M34.43.066.4
DynamicConv Transformer (Wu et al., 2019)39M35.23.680.2
DynamicConv Transformer w/ K-matrices (ours)30M34.24.9103.4
", + "bbox": [ + 173, + 154, + 856, + 227 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "We use K-matrices to replace all the linear layers in the decoder of DynamicConv (since $90 \\%$ of inference time is spent in the decoder). As shown in Table 4, on the IWSLT-14 German-English translation task, this yields a $2 5 \\%$ smaller model with $36 \\%$ faster inference time on CPU, at the cost of 1.0 drop in BLEU score.5 (Our model also nearly matches the state-of-the-art BLEU performance of 2 years ago obtained by the Transformer model (Vaswani et al., 2017), despite being over $60 \\%$ faster for inference than the Transformer.) The majority $( 5 5 \\% )$ of inference time is spent in matrix-vector multiplication; our implementation of K-matrix-vector multiplication is about 2 times faster than the optimized implementation of dense matrix-vector multiplication in the Intel MKL library. Direct comparisons of K-matrix multiplication with this and other highly-optimized routines such as the FFT are further detailed in Appendix C. ", + "bbox": [ + 174, + 250, + 825, + 388 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "4 CONCLUSION ", + "text_level": 1, + "bbox": [ + 176, + 409, + 318, + 425 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "We address the problem of having to manually choose among the numerous classes of structured linear maps by proposing the universal (expressive, efficient, and learnable) family of kaleidoscope matrices. We prove that K-matrices can represent any structured linear maps with near-optimal space and time complexity. Empirical validations suggest that K-matrices are a promising and flexible way to employ structure in modern ML; they can be used to reduce the need for hand-engineering, capture challenging latent structure, and improve efficiency in models. We are excited about future work on further hardware-optimized implementations of K-matrices, to fully realize the size and speed benefits of structured matrices on a broad array of real-world applications. ", + "bbox": [ + 174, + 440, + 825, + 551 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "ACKNOWLEDGMENTS ", + "text_level": 1, + "bbox": [ + 176, + 568, + 326, + 580 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "We thank Avner May and Jian Zhang for their helpful feedback. ", + "bbox": [ + 174, + 590, + 591, + 604 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "We gratefully acknowledge the support of DARPA under Nos. FA87501720095 (D3M), FA86501827865 (SDH), and FA86501827882 (ASED); NIH under No. U54EB020405 (Mobilize), NSF under Nos. CCF1763315 (Beyond Sparsity), CCF1563078 (Volume to Velocity), and 1937301 (RTML); ONR under No. N000141712266 (Unifying Weak Supervision); the Moore Foundation, NXP, Xilinx, LETI-CEA, Intel, IBM, Microsoft, NEC, Toshiba, TSMC, ARM, Hitachi, BASF, Accenture, Ericsson, Qualcomm, Analog Devices, the Okawa Foundation, American Family Insurance, Google Cloud, Swiss Re, and members of the Stanford DAWN project: Teradata, Facebook, Google, Ant Financial, NEC, VMWare, and Infosys. The U.S. Government is authorized to reproduce and distribute reprints for Governmental purposes notwithstanding any copyright notation thereon. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views, policies, or endorsements, either expressed or implied, of DARPA, NIH, ONR, or the U.S. Government. Matthew Eichhorn and Atri Rudra’s research is supported by NSF grant CCF-1763481. 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Yu, Ananda T. Suresh, Krzysztof M. Choromanski, Daniel N. Holtmann-Rice, and Sanjiv Kumar. Orthogonal random features. In D. D. Lee, M. Sugiyama, U. V. Luxburg, I. Guyon, and R. Garnett (eds.), Advances in Neural Information Processing Systems 29, pp. 1975–1983. Curran Associates, Inc., 2016. ", + "bbox": [ + 173, + 347, + 825, + 405 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "Xiyu Yu, Tongliang Liu, Xinchao Wang, and Dacheng Tao. On compressing deep models by low rank and sparse decomposition. In IEEE Conference on Computer Vision and Pattern Recognition (CVPR), 2017. ", + "bbox": [ + 171, + 412, + 825, + 455 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "Neil Zeghidour, Nicolas Usunier, Iasonas Kokkinos, Thomas Schatz, Gabriel Synnaeve, and Emmanuel Dupoux. Learning filterbanks from raw speech for phone recognition. In IEEE International Conference on Acoustics, Speech, and Signal Processing (ICASSP), 2018. 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", + "bbox": [ + 173, + 619, + 821, + 648 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "A RELATED WORK ", + "text_level": 1, + "bbox": [ + 176, + 102, + 348, + 118 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "A.1 STRUCTURED MATRICES IN MACHINE LEARNING ", + "text_level": 1, + "bbox": [ + 176, + 135, + 555, + 148 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "Structured linear maps such as the DFT, the Hadamard transform and convolution are a workhorse of machine learning, with diverse applications including data preprocessing, random projection, featurization, and model compression. For example, the DFT is a crucial step in the standard filter bank speech preprocessing pipeline (Jurafsky & Martin, 2014), and is commonly used when dealing with time series data in general (Panaretos & Tavakoli, 2013). Fast random projection and kernel approximation methods rely on the fast Hadamard transform (Le et al., 2013; Yu et al., 2016) and convolution (Yu et al., 2015), and convolution is a critical component of modern image processing architectures (Krizhevsky et al., 2012) as well as being useful in speech recognition (Zeghidour et al., 2018) and natural language processing (Wu et al., 2019). Large learnable classes of structured matrices such as Toeplitz-like matrices (Sindhwani et al., 2015) and low-displacement rank (LDR) matrices (Thomas et al., 2018) have been used for model compression. However, despite their theoretical speedup, these structured matrix classes lack efficient implementations, especially on GPUs. Therefore, their use has largely been confined to small models (e.g. single hidden layer neural nets) and small datasets (e.g. CIFAR-10). ", + "bbox": [ + 174, + 160, + 825, + 354 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "Butterfly matrices encode the recursive divide-and-conquer structure of the fast Fourier transform (FFT) algorithm. They were first used in numerical linear algebra for fast preconditioning (Parker, 1995). The butterfly factorization is then generalized to encompass complementary low-rank matrices commonly encountered in solving differential and integral equations (Rokhlin & Tygert, 2006; Tygert, 2008; 2010b;a; Li et al., 2015; 2018). In machine learning, butterfly matrices have been use to approximate the Hessian for fast optimization (Mathieu & LeCun, 2014), and to perform fast random projection (Jing et al., 2017; Munkhoeva et al., 2018; Choromanski et al., 2019). Dao et al. (2019) show that butterfly matrices can be used to learn fast algorithms for discrete transforms such as the Fourier transform, cosine/sine transform, Hadamard transform, and convolution. ", + "bbox": [ + 174, + 362, + 825, + 487 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "A.2 SPARSE MATRICES ", + "text_level": 1, + "bbox": [ + 176, + 506, + 346, + 520 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "Several classes of structured linear transforms are ubiquitous in modern deep learning architectures; particularly widespread examples include convolution and multiheaded attention. Recently, attempts to impose sparsity on the neural network weights have been gaining traction. State-of-the art approaches of this type typically accomplish this by pruning small weights (either gradually during training (Zhu & Gupta, 2017), or post-training (Han et al., 2016)) or by training a dense network and then identifying “winning lottery tickets”—sparse subnetworks which may then be retrained from scratch with appropriate initialization (Frankle & Carbin, 2019). Importantly, these approaches start from a dense network, and therefore training is expensive. There is also a more nascent line of work that aims to train unstructured sparse neural networks directly (Mocanu et al., 2018; Mostafa & Wang, 2019; Dettmers & Zettlemoyer, 2019; Evci et al., 2019). These approaches maintain a constant network sparsity level throughout training, and use heuristics to evolve the sparsity pattern during training. One drawback is that the indices of the nonzero entries need to be stored in addition to the entry values themselves, which increases the memory required to store the sparse weight tensors. Another drawback is that these approaches to learn the sparsity pattern are based on intricate heuristics, which can be brittle. We note that these heuristic sparsification techniques could potentially be combined with our approach, to further sparsify the K-matrix factors. ", + "bbox": [ + 174, + 532, + 825, + 753 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "A.3 SPEECH RECOGNITION FROM RAW AUDIO ", + "text_level": 1, + "bbox": [ + 174, + 773, + 504, + 786 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "Numerous works focus on the problem of speech recognition from raw audio input, i.e. without manual featurization. SincNet (Ravanelli & Bengio, 2018) is a CNN-based architecture parameterized with sinc functions, designed so that the first convolutional layer imitates a band-pass filter. Zeghidour et al. (2018) formulate a learnable version of a filter bank featurization; their filters are initialized as an approximation of MFSC features and then fine-tuned jointly with the rest of the model. Sainath et al. (2015) proposed a powerful combined convolutional LSTM (CLDNN)-based model for learning from raw audio, using a large amount of training data. The WaveNet generative architecture (van den Oord et al., 2016), based on dilated convolutions, has been adapted to speech recognition and can be trained on raw audio. Other approaches that can learn from raw audio can be found in (Palaz et al., ", + "bbox": [ + 174, + 799, + 825, + 924 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "2013; Collobert et al., 2016; Ghahremani et al., 2016). To our knowledge, the $1 4 . 6 \\%$ PER achieved by our kaleidoscope $+ \\mathrm { L S T M }$ model on the TIMIT test set is the lowest error rate obtained by a model trained directly on the raw audio. ", + "bbox": [ + 174, + 103, + 823, + 145 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "A.4 LEARNING PERMUTATIONS ", + "text_level": 1, + "bbox": [ + 176, + 162, + 405, + 176 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "Permutation matrices find use in tasks such as matching and sorting (among many others). Techniques to obtain posterior distributions over permutations have been developed, such as the exponential weights algorithm (Helmbold & Warmuth, 2009) and the Gumbel-Sinkhorn network (Mena et al., 2018). ", + "bbox": [ + 174, + 189, + 825, + 243 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "Classifying images with permuted pixels is a standard task to benchmark the ability of RNNs to learn long range dependencies. Le et al. (2015) propose the Permuted MNIST task, in which the model has to classify digit images with all the pixels permuted. Many new RNN architectures, with unitary or orthogonal weight matrices to avoid gradient explosion or vanishing, have been proposed and tested on this task (Le et al., 2015; Arjovsky et al., 2016; Wisdom et al., 2016; Mhammedi et al., 2017; Trinh et al., 2018). Standard gated RNN architectures such as LSTM and GRU have also been found to be competitive with these new RNN architectures on this task (Bai et al., 2018). ", + "bbox": [ + 174, + 251, + 825, + 348 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "B ADDITIONAL EXPERIMENTAL DETAILS ", + "text_level": 1, + "bbox": [ + 176, + 369, + 534, + 385 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "B.1 SPEECH PREPROCESSING ", + "text_level": 1, + "bbox": [ + 176, + 400, + 390, + 414 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "In this section, we fully describe our settings and procedures for the speech preprocessing experiments in Section 3.1.1, and present additional auxiliary baselines and results. ", + "bbox": [ + 173, + 426, + 823, + 454 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "B.1.1 EXPERIMENTAL SETUP ", + "text_level": 1, + "bbox": [ + 176, + 469, + 390, + 483 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "We evaluate our speech recognition models on the TIMIT speech corpus (Garofolo et al., 1993), a standard benchmark for speech recognition. The input is audio (16-bit, $1 6 \\mathrm { k H z }$ .wav format), and the target is the transcription into a sequence of phonemes (units of spoken sound). Our evaluation metric is the phoneme error rate (PER) between the true phoneme sequence and the phoneme sequence predicted by our model. We use PyTorch (Paszke et al., 2017), the Kaldi speech recognition toolkit (Povey et al., 2011), and the PyTorch-Kaldi toolkit (Ravanelli et al., 2019) for developing PyTorch speech recognition models for all our experiments and evaluations. ", + "bbox": [ + 174, + 493, + 825, + 590 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "B.1.2 MODEL AND EVALUATION ", + "text_level": 1, + "bbox": [ + 176, + 606, + 413, + 621 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "Our baseline Bi-LSTM architecture is taken from the PyTorch-Kaldi repository.6 This is a strong baseline model that, to the best of our knowledge, matches state-of-the-art performance for models that use a single type of input featurization (Ravanelli et al., 2019). The original Bi-LSTM model takes as input filter bank features. These are computed as follows: (i) the waveform is framed (split into chunks of $2 5 ~ \\mathrm { m s }$ each that overlap by $1 0 \\mathrm { m s }$ each), (ii) the waveform is dithered (zero-mean Gaussian random noise is added), (iii) pre-emphasis is applied to amplify high frequencies, (iv) the Hamming window function (Harris, 1978) is applied, (v) the FFT is applied, and the power spectrum of the resulting (complex-valued) output is computed, (vi) the power spectrum (which has dimension 512) is mapped to the “mel scale” (which is a scale intended to mimic human auditory perception (Stevens et al., 1937)) by multiplication with a specific banded matrix of dimension $5 1 2 \\times 2 3$ , and the entrywise logarithm of the output is taken (the 23 outputs are called the filters), and (vii) cepstral mean and variance normalization (Liu et al., 1993) is applied. Numerical hyperparameters of this procedure include the dither noise scale, the pre-emphasis coefficient, the Hamming window size, the number of mel filters, and more; we kept all these the same as the Kaldi/PyTorch-Kaldi defaults. ", + "bbox": [ + 174, + 631, + 825, + 824 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "In contrast, our “K-matrix version” of the model takes as input the raw waveform, split into chunks the same way as before but with no normalization, dithering, or other preprocessing, which is then fed into a complex-valued kaleidoscope $[ ( B B ^ { * } ) ^ { 2 } ]$ matrix. Similarly to the nonlinear steps in computing filter bank features, the logarithm of the power spectrum of the output (which has dimension 512) ", + "bbox": [ + 174, + 832, + 825, + 887 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "is then computed. This output is fed into the Bi-LSTM; the Bi-LSTM and kaleidoscope layer are trained together in standard end-to-end fashion. The Bi-LSTM architecture is not modified aside from changing the input dimension from 23 to 512; this (along with the $\\approx 7 5 \\mathrm { K }$ parameters in the kaleidoscope layer itself) results in approximately a 1.1M increase in the total number of parameters compared to the model that takes in MFSC features (a modest $8 \\%$ relative increase). Total training time for our kaleidoscope-based architecture is $7 \\%$ greater than that required for the model that uses MFSC features, not counting the time required to precompute the MFSC features; the FLOPs for inference-time are approximately $15 \\%$ greater (mostly due to the larger dimension of the input to the Bi-LSTM; the kaleidoscope layer accounts for less than $0 . 5 \\%$ of the total FLOPs). ", + "bbox": [ + 174, + 103, + 825, + 229 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "As baselines, we also compare to inserting other types of linear transformations before the Bi-LSTM: fixed linear transformations (such as the fixed FFT, or no transform at all [i.e. the identity]), other trainable structured layers (low-rank, circulant, and sparse [using the sparse training algorithm of Dettmers & Zettlemoyer (2019)]), and a trainable unstructured (dense) linear layer. The kaleidoscope layer performs the best out of all such approaches. The fact that it outperforms even a dense linear layer with more parameters is particularly notable, as it suggests that the structural bias imposed by the K-matrix representation is beneficial for performance on this task. Full results are given in Table 5. ", + "bbox": [ + 174, + 236, + 825, + 333 + ], + "page_idx": 16 + }, + { + "type": "table", + "img_path": "images/339a0f9dd315c6fd892eac7a9f09786548540de0105e3cba345bef5d6336e941.jpg", + "table_caption": [ + "Table 5: TIMIT phoneme error rate $( \\mathrm { P E R } \\%$ , $\\pm$ standard deviation across 5 random seeds). " + ], + "table_footnote": [], + "table_body": "
ModelTest set PER%#Parameters
Low rank + LSTM23.6 ± 0.915.5M
Sparse +LSTM21.7 ± 0.915.5M
Circulant +LSTM23.9 ± 0.915.4M
Dense +LSTM15.4 ± 0.615.9M
FFT+LSTM15.7 ± 0.115.4M
Identity +LSTM20.7 ± 0.315.4M
Kaleidoscope + LSTM14.6 ± 0.315.4M
MFSC features + LSTM14.2 ± 0.214.3M
SincNet (Ravanelli et al., 2019)17.210.0M
LiGRU (Ravanelli et al., 2018)13.812.3M
", + "bbox": [ + 272, + 371, + 723, + 546 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "In our experiments, we grid search the initial learning rate for the “preprocessing layer” (if applicable) in {5e-5, 1e-4, 2e-4, 4e-4, 8e-4, 1.6e-3}, and fix all other hyperparameters (including the initial learning rates for the other parts of the network) to their default values in the PyTorch-Kaldi repository. The model and any preprocessing layers are trained end-to-end with the RMSProp optimizer for 24 epochs (as per the defaults in PyTorch-Kaldi). For each model, we use the validation set to select the best preprocessing learning rate, while the final error rates are reported on the separate held-out test set. For all structured matrix baselines except circulant (which always has $n$ parameters for an $n \\times n$ matrix), the number of parameters in the structured matrices is set to equal the number of parameters in the butterfly layer, while the unconstrained matrix is simply a standard dense complexvalued square matrix. For all experiments with a trainable “preprocessing layer,” we initialize the preprocessing matrix to represent the FFT (or approximate it as closely as possible [i.e. minimize the Frobenius error to the true FFT matrix], in the case of low-rank, sparse, and circulant), which we found to outperform random initialization. ", + "bbox": [ + 173, + 561, + 825, + 742 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "B.1.3 EXTENSION: COMBINING MFSC AND KALEIDOSCOPE ", + "text_level": 1, + "bbox": [ + 174, + 758, + 606, + 773 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "As an additional experiment, we sought to investigate whether combining the hand-engineered MFSC featurization pipeline and a learnable kaleidoscope layer (instead of replacing the former with the latter) could lead to accuracy gains. Specifically, in this experiment we first used the standard filter bank featurization pipeline described above, and trained end-to-end as usual. Then, we replaced the FFT step with a K-matrix initialized to the FFT, and made the weights of the Hamming window function and the mel filter bank matrix learnable as well (similarly to (Zeghidour et al., 2018)). We fine-tuned the resulting architecture for an additional 10 epochs. The final test $\\mathrm { P E R } \\%$ attained by this “hybrid” model is ${ \\bf 1 4 . 0 \\pm 0 . 3 }$ ; the model has 14.4M parameters—a negligible increase over the $1 4 . 3 \\mathbf { M }$ in the original architecture. Thus, by combining the manually encoded domain knowledge in the filter bank featurization and allowing this structure to be learnable rather than fixed, we are able to nearly match the state-of-the-art $1 3 . 8 \\%$ accuracy on TIMIT. While this “hybrid” model certainly involves some hand-engineering, the state-of-the-art results use a concatenation of three different speech audio featurizations—MFSC, MFCC, and fMLLR—as the neural network input, along with a customized RNN architecture (LiGRU) specifically designed for speech recognition, and thus require a more complicated pipeline that is arguably even more hand-crafted. ", + "bbox": [ + 174, + 785, + 825, + 924 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "", + "bbox": [ + 174, + 103, + 825, + 174 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "B.2 REPLACING CNN CHANNEL SHUFFLE ", + "text_level": 1, + "bbox": [ + 178, + 194, + 477, + 208 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "B.2.1 MODEL ARCHITECTURES ", + "text_level": 1, + "bbox": [ + 176, + 222, + 405, + 236 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "ShuffleNet is a convolutional neural network with residual (skip) connections that uses a permutation matrix to shuffle the channels after each grouped 1x1 convolution, sending the $i \\cdot$ -th channel to the $( i \\bmod g )$ -th group, where $g$ is the total number of groups. The architecture for each residual block in ShuffleNet is: 1x1 group conv Batch norm, ReLU Permutation $\\to 3 { \\bf x } 3$ depthwise conv Batch norm $ 1 \\mathrm { x } 1$ group conv. The permutation is fixed. ", + "bbox": [ + 174, + 247, + 825, + 316 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "Zhao et al. (2019) propose to instead use the Hadamard transform before and after each grouped 1x1 convolution to mix the channels. Note that the Hadamard transforms are placed before the batch normalization and ReLU layer (unlike the permutation matrix in the original ShuffleNet design). In particular, the architecture for each block is: Hadamard $\\to 1 \\mathbf { x } 1$ group conv Hadamard Batch norm, $\\mathrm { R e L U } \\to 3 \\mathrm { x } 3$ depthwise conv Batch norm $ 1 \\mathrm { x } 1$ group conv. The Hadamard transform is fixed. ", + "bbox": [ + 173, + 324, + 825, + 407 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "In our architecture, we use a kaleidoscope matrix in OBB (product of an orthogonal butterfly matrix, a diagonal matrix, and the transpose of another butterfly matrix) before and after each grouped 1x1 convolution. We place the second K-matrix after the batch norm and ReLU, to more closely mimic the original ShuffleNet design. The structure for each block is: K-matrix $ 1 \\mathrm { x } 1$ group conv Batch norm, $\\mathrm { R e L U } \\mathrm { K } \\mathrm { - m a t r i x } 3 \\mathrm { x } 3$ depthwise conv Batch norm $ 1 \\mathrm { x } 1$ group conv. The K-matrices are trained along with the rest of the network, rather than being fixed. ", + "bbox": [ + 174, + 414, + 825, + 498 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "B.2.2 EXPERIMENTAL SETUP ", + "text_level": 1, + "bbox": [ + 174, + 517, + 390, + 531 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "We evaluate the CNN architectures on the image classification task of the standard ImageNet dataset (Russakovsky et al., 2015). We use the standard data augmentation, training, and evaluation pipeline as in (Xie et al., 2017). We train with SGD on 8 GPUs for 90 epochs, with a total batch size of 2048 and initial learning rate 0.8. For the 1.0 ShuffleNet g8 architecture, we reduce the total batch size to 1792 to fit into GPU memory, and correspondingly linearly scale the initial learning rate to 0.7. Other hyperparameters (e.g. learning rate schedule, weight decay, etc.) are kept the same as in the ShuffleNet paper (Zhang et al., 2018). We use the training script from NVIDIA’s deep learning examples repository.7 ", + "bbox": [ + 173, + 542, + 825, + 655 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "B.2.3 ADDITIONAL RESULTS ", + "text_level": 1, + "bbox": [ + 176, + 674, + 387, + 688 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "In Table 6, we report top-5 classification accuracy on ImageNet, to complement the top-1 accuracies in Table 2. ", + "bbox": [ + 174, + 699, + 823, + 727 + ], + "page_idx": 17 + }, + { + "type": "table", + "img_path": "images/35b74b64554d9ef8eaeae336a08b6b161700721eca30043b2e3c192053b54d0e.jpg", + "table_caption": [ + "Table 6: Top-5 classification accuracy of ShuffleNet on ImageNet validation set. We report results for different network width multipliers (number of channels), and for different kinds of matrices used for channel mixing. Using a learnable K-matrix in place of each fixed permutation (shuffle) or Hadamard matrix improves top-5 accuracy by up to $4 . 8 \\%$ . Parameter counts are the same as in Table 2. " + ], + "table_footnote": [ + "7https://github.com/NVIDIA/DeepLearningExamples/tree/master/PyTorch/ Classification/RN50v1.5 " + ], + "table_body": "
ShuffleHadamardKaleidoscope (K.)|K. vs. Shuffle
0.25 ShuffleNet g868.6%68.4%73.4%+4.8%
0.5 ShuffleNet g879.9%79.2%81.7%+1.8%
1.0 ShuffleNet g886.0%85.8%86.8%+0.8%
", + "bbox": [ + 223, + 809, + 774, + 882 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "In each setting, the total training time of our K-matrix approach is within $20 \\%$ of the total training time of vanilla ShuffleNet. ", + "bbox": [ + 173, + 103, + 823, + 132 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "In Figure 4, we plot the loss and accuracy on the training set and validation set when we train 1.0 ShuffleNet $^ { \\mathrm { g 8 } }$ , with either a fixed permutation (Shuffle) or a K-matrix for channel shuffling. Even though each K-matrix is a product of multiple (sparse) matrices, the model with K-matrices takes about the same number of training steps to converge as the baseline model does. One possible reason is that we constrain the K-matrices to be orthogonal (Section 2.4), thus avoiding vanishing or exploding gradients. ", + "bbox": [ + 173, + 138, + 825, + 223 + ], + "page_idx": 18 + }, + { + "type": "image", + "img_path": "images/667c482d9c16deb50edde5ee298ef0038ca2d217e93dcabc91f95f10b2abadc8.jpg", + "image_caption": [ + "Figure 4: Loss and top-1 accuracy of 1.0 ShuffleNet $\\mathrm { g } 8$ with either a fixed permutation (Shuffle) or a K-matrix for channel shuffling. The K-matrix model takes about the same number of training steps to converge as does the baseline model. " + ], + "image_footnote": [], + "bbox": [ + 178, + 239, + 815, + 448 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "B.3 LEARNING PERMUTATIONS ", + "text_level": 1, + "bbox": [ + 176, + 537, + 403, + 553 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "B.3.1 DATASET ", + "text_level": 1, + "bbox": [ + 174, + 564, + 295, + 579 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "The permuted CIFAR-10 dataset is constructed by applying a fixed permutation to every input. We choose to use the 2-D bit-reversal permutation,8 i.e., the bit reversal permutation on 32 elements is applied to the rows and to the columns. This permutation was chosen because it is locality-destroying: if two indices $i , j$ are close, they must differ in a lower-order bit, so that the bit-reversed indices $i ^ { \\prime } , j ^ { \\prime }$ are far. This makes it a particularly challenging test case for architectures that rely on spatial locality such as “vanilla” CNNs. ", + "bbox": [ + 174, + 588, + 825, + 672 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "B.3.2 MODEL AND TRAINING ", + "text_level": 1, + "bbox": [ + 176, + 689, + 397, + 704 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "We describe the model architectures used in Section 3.1 (those reported in Table 3). ", + "bbox": [ + 176, + 713, + 717, + 728 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "Our model $\\mathbf { ( K + C N N ) }$ The model represents a fixed permutation $P$ , parametrized as a K-matrix, to learn to recover the true permutation, followed by a standard ResNet18 architecture (He et al., 2016). Because of the simple decomposable nature of the butterfly factors (Section 2.1), our parameterization is easily extensible with additional techniques: ", + "bbox": [ + 176, + 744, + 826, + 801 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "(i) We constrain each butterfly factor matrix in the K-matrix to be doubly-stochastic. For example, each $2 \\times 2$ block in the butterfly factor matrix of block size 2 has the form $\\left[ { \\begin{array} { c c } { a } & { 1 - a } \\\\ { 1 - a } & { a } \\end{array} } \\right]$ , where $a \\in [ 0 , 1 ]$ . We treat this block as a distribution over permutations, generating the identity $\\left[ { \\begin{array} { l l } { 1 } & { 0 } \\\\ { 0 } & { 1 } \\end{array} } \\right]$ with probability $a$ and the swap $\\left[ { \\begin{array} { c c } { 0 } & { 1 } \\\\ { 1 } & { 0 } \\end{array} } \\right]$ with probability $1 - a$ . Butterfly factor matrices with larger block sizes are constrained to be doubly-stochastic in a similar manner. In this way, a permutation is sampled for each butterfly factor matrix, and these permutations are composed to get the final permutation that is applied to the image. ", + "bbox": [ + 210, + 813, + 825, + 876 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "", + "bbox": [ + 230, + 99, + 825, + 188 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "(ii) For each minibatch, the examples $P x$ by applying permutation samples on the (permuted) inputs are fed into an additional unsupervised reconstruction loss ", + "bbox": [ + 200, + 190, + 821, + 218 + ], + "page_idx": 19 + }, + { + "type": "equation", + "img_path": "images/9ba2827e11ef54db7514cfe2810019e8bb35ba2a26ac46ed08c7328d515d5621.jpg", + "text": "$$\n\\sum _ { 0 \\leq i , j < n } \\left\\| \\left[ { \\binom { P x } { ( P x ) [ i , j + 1 ] - ( P x ) [ i , j ] } } \\right] \\right\\| _ { 2 }\n$$", + "text_format": "latex", + "bbox": [ + 390, + 219, + 665, + 260 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "measuring total variation smoothness of the de-noised inputs. Such loss functions are often used in image denoising (Rudin et al., 1992). A final regularization loss was placed on the entropy of $P$ , which was annealed over time to encourage $P$ to converge toward a sharper doubly-stochastic matrix (in other words, a permutation). ", + "bbox": [ + 232, + 260, + 825, + 315 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "The model is trained with just the reconstruction loss to convergence before the standard ResNet is trained on top. ", + "bbox": [ + 232, + 318, + 823, + 345 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "These techniques are applicable to the K-matrix as well as specialized methods for representing permutations such as Gumbel-Sinkhorn (Mena et al., 2018) and are important for recovering the true permutation. However, they are not applicable to a general linear layer, which showcases the flexibility of K-matrices for representing generic structure despite not being specially tailored for this task. We also remark that other classes of structured linear maps such as low-rank, circulant, and so on, are even less suited to this task than dense matrices, as they are incapable of representing all permutations. ", + "bbox": [ + 173, + 354, + 825, + 452 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "Baseline architectures ", + "text_level": 1, + "bbox": [ + 176, + 467, + 328, + 481 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "1. Fully connected (FC): This is a 3-layer MLP, with hidden size 1024 and ReLU nonlinearity in-between the fully connected layers. \n2. Recurrent neural network (RNN): We use a gated recurrent unit (GRU) model (Cho et al., 2014), with hidden size 1024. Many RNN architectures have been proposed to capture long-range dependency on permuted image dataset such as Permuted MNIST (Arjovsky et al., 2016). Standard gated architectures such as LSTM and GRU have shown competitive performance on the Permuted MNIST dataset, and we choose GRU as a baseline since it has been reported to slightly outperform LSTM (Bai et al., 2018). \n3. CNN: We use the standard ResNet18 architecture, adapted to smaller image size of the CIFAR-10 dataset (changing stride from 2 to 1 of the first convolutional layer, and removing max-pooling layer that follows). \n4. Dense $^ +$ CNN: We add an additional linear layer (i.e. a dense matrix) of size $1 0 2 4 \\times 1 0 2 4$ before the ResNet18 architecture. This dense layer can in theory represent a permutation, but cannot benefit from the additional techniques described above. \n5. Baseline CNN (unpermuted): We use the standard ResNet18 architecture applied to the unpermuted CIFAR-10 dataset. ", + "bbox": [ + 210, + 489, + 825, + 727 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "All models are trained for 200 total epochs, with the Adam optimizer. We use the standard learning rate schedule and weight decay from Mostafa & Wang (2019). We use Hyperband (Li et al., 2017) to tune other hyperparameters such as the initial learning rate and annealing temperature. ", + "bbox": [ + 176, + 738, + 825, + 780 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "B.4 SPEEDING UP DYNAMICCONV’S INFERENCE ", + "text_level": 1, + "bbox": [ + 174, + 795, + 524, + 810 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "B.4.1 MODEL ARCHITECTURE ", + "text_level": 1, + "bbox": [ + 176, + 821, + 398, + 837 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "We start with the DynamicConv Transformer architecture (Wu et al., 2019), which is a variant of the Transformer architecture (Vaswani et al., 2017) where the self-attention in each layer is replaced with a light-weight DynamicConv module. We use the implementation from the Fairseq library(Ott et al., 2019),9 with PyTorch version 1.2. ", + "bbox": [ + 174, + 847, + 826, + 902 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "The architecture of each layer of the decoder is: Linear DynamicConv Linear LayerNorm Encoder-decoder attention LayerNorm Linear ReLU Linear $ \\mathrm { R e L U } $ LayerNorm. In every layer of the decoder, we replace the dense weight matrix in each of the four Linear layers with a $\\mathrm { K }$ -matrix from the $\\boldsymbol { B }$ class (i.e. a butterfly matrix). ", + "bbox": [ + 174, + 103, + 825, + 159 + ], + "page_idx": 20 + }, + { + "type": "text", + "text": "B.4.2 TRAINING AND EVALUATION ", + "text_level": 1, + "bbox": [ + 176, + 175, + 429, + 189 + ], + "page_idx": 20 + }, + { + "type": "text", + "text": "The models are trained from scratch using the training script from the Fairseq repository, with the same hyperparameters (optimizer, learning rate, number of updates, etc.) used in the DynamicConv paper (Wu et al., 2019). We note that the DynamicConv model with K-matrices in the decoder trains slightly faster than the default DynamicConv model (both models are trained for 50,000 updates, which requires approximately $7 \\%$ less time for the K-matrix model than for the default model). ", + "bbox": [ + 174, + 200, + 825, + 268 + ], + "page_idx": 20 + }, + { + "type": "text", + "text": "To evaluate inference speed, we run the decoding script on the IWSLT-14 De-En test set in singlethreaded mode on a server Intel Xeon CPU E5-2690 v4 at 2.60GHz, and measure wall-clock time. The test set contains 6750 sentences, with 149241 tokens. Following Wu et al. (2019), we set the batch size to 1 and beam size to 1 for this evaluation. ", + "bbox": [ + 174, + 276, + 825, + 332 + ], + "page_idx": 20 + }, + { + "type": "text", + "text": "B.4.3 ADDITIONAL COMPARISON WITH OTHER STRUCTURED MATRICES ", + "text_level": 1, + "bbox": [ + 176, + 348, + 683, + 362 + ], + "page_idx": 20 + }, + { + "type": "text", + "text": "We additionally compare the speed-quality tradeoff of K-matrices with other classes of structured matrices, when used to replace the fully-connected layers of DynamicConv’s decoder. We consider the following additional classes of structured matrices: low-rank, circulant, Toeplitz-like (Sindhwani et al., 2015), ACDC (Moczulski et al., 2016), Fastfood (Le et al., 2013), and sparse. For classes with a variable number of parameters (e.g. low-rank, sparse), we set the number of parameters to match that of K-matrices. For sparse matrices, besides the result for an ensemble of 10 models (the default setting in the Fairseq repository), we also report the result for a single model, as that could have faster inference time (since ensembling/averaging sparse matrices produces a less sparse matrix). ", + "bbox": [ + 174, + 372, + 825, + 484 + ], + "page_idx": 20 + }, + { + "type": "text", + "text": "In Figure 5, we plot the tradeoff between translation quality (measured by BLEU score) and inference speed (sentences per second). Most classes of structured matrices produce similar translation quality (between 34.1 and 34.4 BLEU score). K-matrices have the second fastest inference time, only $7 \\%$ slower than low-rank matrices. We note that low-rank matrices benefit from very well-tuned BLAS routines (matrix-matrix multiplication). Even though our implementation of K-matrix multiplication is not yet highly optimized, it is already quite close to the speed of low-rank matrix multiplication at an equivalent parameter count. ", + "bbox": [ + 174, + 491, + 825, + 588 + ], + "page_idx": 20 + }, + { + "type": "text", + "text": "C SPEED BENCHMARK AND IMPLEMENTATION DETAILS ", + "text_level": 1, + "bbox": [ + 174, + 611, + 650, + 626 + ], + "page_idx": 20 + }, + { + "type": "text", + "text": "Each K-matrix (for fixed width and expansion), has an $O ( n \\log n )$ matrix-vector multiplication algorithm: sequentially multiply the input vector with each of the sparse factors. Our implementation of this simple algorithm is surprisingly competitive with optimized subroutines, both on GPU (e.g. for training) and on CPU (e.g. for inference). In Figure 6, we compare the speed of multiplying by a K-matrix in class $\\boldsymbol { B }$ (i.e. a butterfly matrix) against a specialized implementation of the FFT. We normalize the speed by the speed of dense matrix-matrix multiply (on GPU) or dense matrix-vector multiply (on CPU). On GPU, with input sizes $n = 1 0 2 4$ and batch size 2048, the training time (forward and backward) of K-matrices matrix is about $3 \\mathbf { x }$ faster than dense matrix multiply (GEMM from cuBLAS). For inference on CPU, the kaleidoscope fast multiplication can be one or two orders of magnitude faster than GEMV. Over a range of matrix sizes, our implementation is within a factor of $2 { - } 4 \\mathbf { x }$ of specialized implementations of the FFT, a highly optimized kernel. ", + "bbox": [ + 174, + 640, + 825, + 794 + ], + "page_idx": 20 + }, + { + "type": "text", + "text": "Our implementation is also memory-efficient. In the forward pass through the $O ( \\log n )$ sparse factors, we do not store the intermediate results, but recompute them during the backward pass. Therefore the activation memory required is $O ( b n )$ for an input batch size of $b$ . ", + "bbox": [ + 174, + 801, + 825, + 843 + ], + "page_idx": 20 + }, + { + "type": "text", + "text": "D SYNTHETIC MATRIX RECOVERY ", + "text_level": 1, + "bbox": [ + 176, + 863, + 472, + 880 + ], + "page_idx": 20 + }, + { + "type": "text", + "text": "We directly validate Theorem 1 on well-known types of structured matrices used in machine learning. Given a structured matrix $\\mathbf { M }$ , we attempt to represent $\\mathbf { M }$ as closely as possible using $\\mathbf { K }$ -matrices as well as the standard classes of structured matrices: sparse and low-rank. In Table 7, we quantify the expressivity of each of these three methods, as measured by their ability to approximate a range of different structures. Results for “global minimum” of kaleidoscope matrices are obtained from the theoretical expressiveness results in Section I and Section J. Low-rank and sparse approximation have closed form solutions: truncating the SVD and keeping the largest-magnitude entries, respectively. We also report the results using SGD for kaleidoscope matrices to validate that good approximation with K-matrices can be obtained even from standard first-order optimization algorithms. Even with imperfect optimization, kaleidoscope matrices can still capture out-of-class target matrices better than low-rank and sparse matrices. ", + "bbox": [ + 174, + 895, + 825, + 924 + ], + "page_idx": 20 + }, + { + "type": "image", + "img_path": "images/60e8527bd3e997ff76c45727293b2592a1bb95f4efe3920dcade7144a5d1b7fb.jpg", + "image_caption": [ + "Figure 5: Tradeoff between translation quality (measured by BLEU score) and inference speed (sentences per second). K-matrices have the second fastest inference speed, only $7 \\%$ slower than low-rank matrices. " + ], + "image_footnote": [], + "bbox": [ + 245, + 106, + 751, + 363 + ], + "page_idx": 21 + }, + { + "type": "image", + "img_path": "images/dab94d90fe5fadd003eb3561b1387c255889d8db809628de7ac84a68752cb799.jpg", + "image_caption": [ + "Figure 6: Speedup of FFT and Kaleidoscope against dense matrix-matrix multiply (GEMM) for training, and against dense matrix-vector multiply (GEMV) for inference. " + ], + "image_footnote": [], + "bbox": [ + 174, + 444, + 818, + 680 + ], + "page_idx": 21 + }, + { + "type": "text", + "text": "", + "bbox": [ + 173, + 750, + 825, + 875 + ], + "page_idx": 21 + }, + { + "type": "text", + "text": "The target matrices are kaleidoscope, low-rank, sparse, convolution (i.e. circulant matrices), Fastfood (Le et al., 2013), and entrywise random IID Gaussian matrix (to show the typical magnitude of the error). All target matrices M were randomly initialized such that $\\mathbb { E } [ \\mathbf { M } ^ { T } \\mathbf { M } ] ^ { - \\sum } \\mathbf { I }$ . ", + "bbox": [ + 176, + 882, + 825, + 924 + ], + "page_idx": 21 + }, + { + "type": "text", + "text": "Table 7: Expressiveness of different classes of structured matrices: Frobenius error of representing common structured matrices (columns) of dimension 256 using three structured representations of matrices with adjustable numbers of parameters. (Left group: Target matrices in the same class as the methods. Middle group: Target matrices with fixed number of parameters. Right: Random matrix to show typical scale of error.) Each method is allotted the same number of parameters, equal to a $\\log n$ factor more than that of the target matrix. Low-rank and sparse matrices are unable to capture any structure outside their own class, while the minima for kaleidoscope matrices found via optimization better capture the actual structure for out-of-class targets better than the baselines. ", + "bbox": [ + 173, + 101, + 825, + 213 + ], + "page_idx": 22 + }, + { + "type": "table", + "img_path": "images/1a8d699f6a22c163d448e6bc79eb72c0fe5d223837976d590783c4f9438ae20a.jpg", + "table_caption": [], + "table_footnote": [], + "table_body": "
Target MethodKaleidoscopeLow-rankSparseConvolutionFastfoodRandom
Global Min.Kaleidoscope0.00.00.00.00.0
Low-rank14.90.010.814.611.615.5
Sparse11.712.20.013.17.114.1
With SGDKaleidoscope0.00.018.00.05.114.5
", + "bbox": [ + 173, + 224, + 885, + 324 + ], + "page_idx": 22 + }, + { + "type": "text", + "text": "To find a kaleidoscope approximation with SGD, we used Hyperband to tune its learning rate (between 0.001 and 0.5). ", + "bbox": [ + 176, + 351, + 825, + 378 + ], + "page_idx": 22 + }, + { + "type": "text", + "text": "E PROPERTIES OF THE $B B ^ { * }$ HIERARCHY", + "text_level": 1, + "bbox": [ + 174, + 398, + 522, + 415 + ], + "page_idx": 22 + }, + { + "type": "text", + "text": "Here, we justify why the definitions in Section 2.2 give rise to a hierarchy. We first make some basic observations about the parameterization. ", + "bbox": [ + 173, + 430, + 823, + 458 + ], + "page_idx": 22 + }, + { + "type": "text", + "text": "Observation E.1. An $n \\times n$ matrix $\\mathbf { M } \\in B B ^ { * }$ has $4 n \\log n$ parameters. ", + "bbox": [ + 176, + 463, + 648, + 478 + ], + "page_idx": 22 + }, + { + "type": "text", + "text": "Proof. M can be expressed as a product of $2 \\log n$ butterfly factor matrices of size $n \\times n$ . Each of these factor matrices has 2 parameters per row, for a total of $2 n$ parameters each. Hence, the total number of parameters is $4 n \\log n$ . □ ", + "bbox": [ + 174, + 493, + 825, + 536 + ], + "page_idx": 22 + }, + { + "type": "text", + "text": "Observation E.2. Let M be an $n \\times n$ matrix in $( B B ^ { * } ) _ { e } ^ { w }$ . Then, given an arbitrary vector v of length $n$ , we can compute Mv with $O ( w n e \\log ( n e ) )$ field operations. ", + "bbox": [ + 176, + 544, + 823, + 574 + ], + "page_idx": 22 + }, + { + "type": "text", + "text": "Proof. Since $\\mathbf { M } \\in ( B B ^ { * } ) _ { e } ^ { w }$ , we can decompose it as $\\mathbf { S } \\mathbf { E } _ { 1 } \\mathbf { E } _ { 2 } \\ldots \\mathbf { E } _ { w } \\mathbf { S } ^ { T }$ , where $\\mathbf { S }$ is as given in Definition 2.4, and each $\\mathbf { E } _ { i }$ is an $e n \\times e n$ matrix in $B B ^ { * }$ . Therefore, to compute Mv, we can use associativity of matrix multiplication to multiply the vector by one of these matrices at a time. ", + "bbox": [ + 174, + 588, + 825, + 631 + ], + "page_idx": 22 + }, + { + "type": "text", + "text": "Since all of these factors are sparse, we use the naïve sparse matrix-vector multiplication algorithm (begin with a 0-vector and perform the corresponding multiplication and addition for each nonzero matrix entry). S (and thus $\\mathbf { \\vec { S } } ^ { T }$ ) have $n$ NNZ. Therefore, matrix-vector multiplication by S or $\\mathbf { S } ^ { T }$ requires $O ( n )$ operations, which is dominated by the butterfly matrix-vector multiplication. Each $\\mathbf { E } _ { i }$ can be further decomposed into $2 \\log ( n e )$ matrices with at most $2 n e$ non-zero entries each (by Observation E.1). Therefore, matrix vector multiplication by each $\\mathbf { E } _ { i }$ requires $O ( n e \\log ( n e ) )$ . Since there are $w$ such $\\mathbf { E } _ { i }$ , we require a total of $O ( w n \\bar { e } \\log ( n e ) )$ operations. □ ", + "bbox": [ + 174, + 637, + 825, + 736 + ], + "page_idx": 22 + }, + { + "type": "text", + "text": "Now, we are ready to show that our definition of classes $( B B ^ { * } ) _ { e } ^ { w }$ forms a natural hierarchy. ", + "bbox": [ + 173, + 751, + 767, + 766 + ], + "page_idx": 22 + }, + { + "type": "text", + "text": "First, we must argue that all matrices are contained within the hierarchy. ", + "bbox": [ + 176, + 772, + 647, + 786 + ], + "page_idx": 22 + }, + { + "type": "text", + "text": "Lemma E.3. Let M be an arbitrary $n \\times n$ matrix. Then $\\mathbf { M } \\in ( B B ^ { * } ) ^ { ( 2 n - 2 ) }$ ", + "bbox": [ + 174, + 792, + 676, + 808 + ], + "page_idx": 22 + }, + { + "type": "text", + "text": "Proof. Corollary E.3 in Appendix $\\mathbf { K }$ shows that any $n \\times n$ matrix can be written in the form $\\mathbf { M _ { 1 } } \\mathbf { \\bar { M } ^ { \\prime } } _ { 1 } ^ { * } \\dots \\mathbf { M } _ { n - 1 } \\mathbf { M ^ { \\prime } } _ { n - 1 } ^ { * } \\mathbf { \\bar { M } M _ { n } } \\mathbf { M ^ { \\prime } } _ { n } ^ { * } \\dots \\mathbf { M } _ { 2 n - 2 } \\mathbf { M ^ { \\prime } } _ { n - 2 } ^ { * } \\mathbf { \\bar { M } M ^ { \\prime } } _ { n } ^ { * } \\dots \\mathbf { M } _ { n } ^ { * } \\dots \\mathbf { M ^ { \\prime } } _ { n } ^ { * }$ , where $\\mathbf { M } _ { i } , \\mathbf { M } _ { i } ^ { \\prime }$ are orthogonal butterfly matrices and M is a diagonal matrix. We can combine $D$ with $M _ { n }$ to form another (possibly not orthogonal) butterfly matrix. This yields a decomposition of $\\mathbf { M }$ as products of (possibly not orthogonal) butterfly matrices and their (conjugate) transposes, completing the proof. □ ", + "bbox": [ + 174, + 824, + 825, + 893 + ], + "page_idx": 22 + }, + { + "type": "text", + "text": "Next, we argue that, up to a certain point, this hierarchy is strict. ", + "bbox": [ + 174, + 909, + 594, + 924 + ], + "page_idx": 22 + }, + { + "type": "text", + "text": "Lemma E.4. For every fixed $c \\geq 1$ , there is an $n \\times n$ matrix ${ { \\bf { M } } _ { n } }$ (with n sufficiently large) such that $\\mathbf { M } _ { n } \\in ( B B ^ { * } ) ^ { c + 1 }$ but $\\mathbf { M } _ { n } \\notin ( B B ^ { * } ) ^ { c }$ . ", + "bbox": [ + 171, + 103, + 823, + 133 + ], + "page_idx": 23 + }, + { + "type": "text", + "text": "Proof. Given $c$ , fix $n$ to be a power of 2 such that $\\begin{array} { r } { c < \\frac { n } { 4 \\log _ { 2 } n } } \\end{array}$ . For sake of contradiction, assume that every $n \\times n$ matrix in $( B B ^ { * } ) ^ { c + 1 }$ is also in $( B B ^ { * } ) ^ { c }$ . Let A be an arbitrary $n \\times n$ matrix. From Lemma E.3, $\\mathbf { A } \\in ( B B ^ { \\ast } ) ^ { ( 2 n - 2 ) }$ . From our assumption, we can replace the first $c + 1 B B ^ { * }$ factors of $\\mathbf { A }$ with $c$ (potentially different) $B B ^ { * }$ factors and still recover A. We can repeat this process until we are left with $c B B ^ { * }$ factors, implying that $\\mathbf { A } \\in ( B B ^ { * } ) ^ { c }$ . From Observation E.1, we require $4 c n \\log n < n ^ { 2 }$ (by our choice of $n$ ) parameters to completely describe A. This is a contradiction since $\\mathbf { A }$ is an arbitrary $n \\times n$ matrix, and therefore has $n ^ { 2 }$ arbitrary parameters. Hence, there must be some $n \\times n$ matrix in $( B B ^ { * } ) ^ { c + 1 }$ that is not in $( B B ^ { * } ) ^ { c }$ . □ ", + "bbox": [ + 173, + 146, + 825, + 266 + ], + "page_idx": 23 + }, + { + "type": "text", + "text": "F ARITHMETIC CIRCUITS IN $B B ^ { * }$ HIERARCHY ", + "text_level": 1, + "bbox": [ + 174, + 284, + 573, + 301 + ], + "page_idx": 23 + }, + { + "type": "text", + "text": "In this appendix, we prove our main theoretical result, namely, our ability to capture general transformations, expressed as low-depth linear arithmetic circuits, in the $B B ^ { * }$ hierarchy. This result is recorded in Theorem 1. ", + "bbox": [ + 176, + 315, + 825, + 358 + ], + "page_idx": 23 + }, + { + "type": "text", + "text": "Theorem 1. Let M be an $n \\times n$ matrix such that matrix-vector multiplication of M times an arbitrary vector v can be represented as a be a linear arithmetic circuit $C$ comprised of $s$ gates (including inputs) and having depth $d .$ . Then, $\\mathbf { M } \\in \\left( B B ^ { * } \\right) _ { O ( \\frac { s } { n } ) } ^ { O ( d ) }$ . ", + "bbox": [ + 173, + 362, + 825, + 412 + ], + "page_idx": 23 + }, + { + "type": "text", + "text": "To prove Theorem 1, we make use of the following two theorems. ", + "bbox": [ + 173, + 422, + 609, + 438 + ], + "page_idx": 23 + }, + { + "type": "text", + "text": "Theorem 2. Let P be an $n \\times n$ permutation matrix (with n a power of 2). Then $\\mathbf { P } \\in B B ^ { * }$ . ", + "bbox": [ + 171, + 440, + 771, + 457 + ], + "page_idx": 23 + }, + { + "type": "text", + "text": "Theorem 3. Let S be an $n \\times n$ matrix of s NNZ. Then $\\mathbf { S } \\in \\left( B B ^ { * } \\right) _ { 4 } ^ { 4 \\left\\lceil \\frac { s } { n } \\right\\rceil }$ . ", + "bbox": [ + 173, + 459, + 651, + 482 + ], + "page_idx": 23 + }, + { + "type": "text", + "text": "Theorem 2 is proven in Appendix $\\mathbf { G }$ , and Theorem 3 is proven in Appendix I. ", + "bbox": [ + 174, + 492, + 679, + 508 + ], + "page_idx": 23 + }, + { + "type": "text", + "text": "Proof of Theorem 1. We will represent $C$ as a product of $d$ matrices, each of size $s ^ { \\prime } \\times s ^ { \\prime }$ , where $s ^ { \\prime }$ is the smallest power of 2 that is greater than or equal to $s$ . ", + "bbox": [ + 174, + 522, + 825, + 551 + ], + "page_idx": 23 + }, + { + "type": "text", + "text": "To introduce some notation, define $w _ { 1 } , \\dots , w _ { d }$ such that $w _ { k }$ represents the number of gates in the $k$ ’th layer of $C$ (note that $\\begin{array} { r } { s = n + \\sum _ { k = 1 } ^ { d } w _ { k } ) } \\end{array}$ . Also, define $z _ { 1 } , \\dotsc z _ { d }$ such that $z _ { 1 } ~ = ~ n$ and $z _ { k } = w _ { k - 1 } + z _ { k - 1 }$ ( $z _ { k }$ is the number of gates that have already been used by the time we get to layer $k$ ). ", + "bbox": [ + 174, + 558, + 825, + 616 + ], + "page_idx": 23 + }, + { + "type": "text", + "text": "Let $g _ { i }$ denote the $i ^ { \\because }$ ’th gate (and its output) of $C$ $0 \\leq i < s$ ), defined such that: ", + "bbox": [ + 173, + 622, + 686, + 638 + ], + "page_idx": 23 + }, + { + "type": "equation", + "img_path": "images/eec6b7fd7be114295460a7139b1ed8896c1c09143d2dd04d91974e0a104d77dd.jpg", + "text": "$$\ng _ { i } = \\left\\{ { \\begin{array} { l l } { v _ { i } } & { 0 \\leq i < n } \\\\ { \\alpha _ { j } g _ { i _ { 1 } } + \\beta _ { i } g _ { i _ { 2 } } } & { n \\leq i < s } \\end{array} } \\right.\n$$", + "text_format": "latex", + "bbox": [ + 385, + 645, + 612, + 680 + ], + "page_idx": 23 + }, + { + "type": "text", + "text": "where $i _ { 1 } , i _ { 2 }$ are indices of gates in earlier layers. ", + "bbox": [ + 173, + 685, + 488, + 700 + ], + "page_idx": 23 + }, + { + "type": "text", + "text": "For the $k$ ’th layer of $C$ , we define the $s ^ { \\prime } \\times s ^ { \\prime }$ matrix ${ { \\bf { M } } _ { k } }$ such that it performs the computations of the gates in that layer. Define the $\\Ddot { \\iota }$ ’th row of ${ { \\bf { M } } _ { k } }$ to be: ", + "bbox": [ + 174, + 707, + 825, + 736 + ], + "page_idx": 23 + }, + { + "type": "equation", + "img_path": "images/1d36c6fbf850ea3b073447d78f37845aab1f383b391150bdf20216cfa903eef3.jpg", + "text": "$$\n\\begin{array} { r } { \\mathbf { M } _ { k } [ i : ] = \\left\\{ \\begin{array} { l l } { \\mathbf { e } _ { i } ^ { T } } & { 0 \\leq i < z _ { k } } \\\\ { \\alpha _ { i } \\mathbf { e } _ { i _ { 1 } } ^ { T } + \\beta _ { i } \\mathbf { e } _ { i _ { 2 } } ^ { T } } & { z _ { k } \\leq i < z _ { k } + w _ { k } } \\\\ { 0 } & { i \\geq z _ { k } + w _ { k } } \\end{array} \\right. } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 338, + 741, + 656, + 795 + ], + "page_idx": 23 + }, + { + "type": "text", + "text": "For any $0 \\leq k \\leq d$ , let $\\mathbf { v _ { k } }$ be vector ", + "bbox": [ + 174, + 808, + 411, + 823 + ], + "page_idx": 23 + }, + { + "type": "equation", + "img_path": "images/fc664909d16bd767110d29323b314398086c0df144f6fe112d49d2a94e66e913.jpg", + "text": "$$\n\\mathbf { v } _ { k } = \\mathbf { M } _ { k } \\ldots \\mathbf { M } _ { 2 } \\mathbf { M } _ { 1 } \\left[ \\mathbf { v } \\right] .\n$$", + "text_format": "latex", + "bbox": [ + 408, + 828, + 589, + 863 + ], + "page_idx": 23 + }, + { + "type": "text", + "text": "We’d like to argue that $\\mathbf { v } _ { d }$ contains the outputs of all gates in $C$ (i.e, the $n$ values that make up Mv). To do this we argue, by induction on $k$ , that $\\mathbf { v } _ { k }$ is the vector whose first $z _ { k + 1 }$ entries are $g _ { 0 } , g _ { 1 } , \\dotsc , g _ { \\left( z _ { k } - 1 \\right) } .$ , and whose remaining entries are 0. The base case, $k = 0$ is trivial. Assuming this holds for the case $k - 1$ , and consider multiplying $\\mathbf { v } _ { k - 1 }$ by ${ { \\bf { M } } _ { k } }$ . The first $z _ { k }$ rows of ${ { \\bf { M } } _ { k } }$ duplicate the first $z _ { k }$ entries of $\\mathbf { v } _ { k - 1 }$ The next $w _ { k }$ rows perform the computation of gates $g _ { z _ { k } } , \\ldots , g _ { ( z _ { k + 1 } - 1 ) }$ Finally, the remaining rows pad the output vector with zeros. Therefore, $\\mathbf { v } _ { k }$ is exactly as desired. ", + "bbox": [ + 173, + 867, + 825, + 924 + ], + "page_idx": 23 + }, + { + "type": "text", + "text": "", + "bbox": [ + 171, + 103, + 825, + 132 + ], + "page_idx": 24 + }, + { + "type": "text", + "text": "The final matrix product will contain all $n$ elements of the output. By left multiplying by some permutation matrix $\\mathbf { P }$ , we can reorder this vector such that the first $n$ entries are exactly Mv. Hence, we are left to argue the position of $\\mathbf { P M } _ { d } \\ldots \\mathbf { M } _ { 2 } \\mathbf { M } _ { 1 }$ within the $B B ^ { * }$ hierarchy. Each ${ { \\bf { M } } _ { k } }$ is a matrix with total $2 w _ { k } + z _ { k } < 2 s ^ { \\prime }$ NNZ. From Theorem 3, we can, therefore, represent ${ { \\bf { M } } _ { k } }$ as a product of $O ( 1 )$ matrices (of size $2 s ^ { \\prime }$ ) in $B B ^ { * }$ . From Theorem 2, $\\mathbf { P } \\in B B ^ { * }$ . Note that $s \\leq s ^ { \\prime } < 2 s$ , so $s ^ { \\prime } = \\Theta ( s )$ . ", + "bbox": [ + 173, + 138, + 825, + 223 + ], + "page_idx": 24 + }, + { + "type": "text", + "text": "Our final decomposition will have $O ( d ) B B ^ { * }$ factors, and requires an expansion from size $n$ to size $2 s ^ { \\prime }$ , or an expansion factor of $O ( { \\textstyle { \\frac { s } { n } } } )$ . Therefore, $\\mathbf { M } \\in \\left( B B ^ { * } \\right) _ { O ( \\frac { s } { n } ) } ^ { O ( d ) }$ , as desired. □ ", + "bbox": [ + 173, + 228, + 823, + 265 + ], + "page_idx": 24 + }, + { + "type": "text", + "text": "Remark F.1. By applying Observation $E . 2$ , we see that Theorem $I$ gives an $O ( s d \\log s )$ matrix vector multiplication algorithm for $\\mathbf { M }$ . ", + "bbox": [ + 173, + 276, + 825, + 305 + ], + "page_idx": 24 + }, + { + "type": "text", + "text": "G PERMUTATIONS IN $B B ^ { * }$ ", + "text_level": 1, + "bbox": [ + 176, + 325, + 403, + 343 + ], + "page_idx": 24 + }, + { + "type": "text", + "text": "In this appendix, we prove Theorem 2. In addition, we will also show that permutations are in $B ^ { * } B$ , where the set $B ^ { * } B$ is defined analogously to $B B ^ { * }$ (i.e. matrices of the form $\\mathbf { M } = \\mathbf { M } _ { 1 } ^ { * } \\mathbf { M } _ { 2 }$ for some $\\mathbf { M } _ { 1 } , \\mathbf { M } _ { 2 } \\in B )$ . ", + "bbox": [ + 174, + 358, + 826, + 401 + ], + "page_idx": 24 + }, + { + "type": "text", + "text": "To prove Theorem 2, we decompose permutation matrix $\\mathbf { P }$ into $\\mathbf { P } = \\mathbf { L R }$ , with $\\mathbf { L } \\in B$ and $\\mathbf { R } \\in B ^ { * }$ Throughout the proof, we make use of the following definition. ", + "bbox": [ + 174, + 407, + 821, + 436 + ], + "page_idx": 24 + }, + { + "type": "text", + "text": "Definition G.1. Let L be an $n \\times n$ permutation matrix (n a power of 2). We say that L meets the $2 ^ { j }$ balance condition if L can be divided into chunks of $2 ^ { j }$ (with each chunk having all columns $i$ such that $\\left\\lfloor { \\frac { i } { 2 ^ { j } } } \\right\\rfloor$ has the same value) such that for every $0 \\leq m < 2 ^ { j }$ , each chunk has exactly one $\\mathbf { L } [ : , k ] = \\mathbf { e } _ { \\pi _ { k } }$ with $\\pi _ { k } \\equiv m$ ( mod $2 ^ { j }$ ). We say that L is modular-balanced if it meets the $2 ^ { j }$ balance condition for each $2 \\leq 2 ^ { \\overset { . } { j } } \\leq n$ . ", + "bbox": [ + 173, + 440, + 825, + 515 + ], + "page_idx": 24 + }, + { + "type": "image", + "img_path": "images/e1a749d098c79becb5562ba1896bfb4d508efdd4f05072b76cda0be851a989ca.jpg", + "image_caption": [ + "Figure 7: First step of decomposition of modular-balanced matrix L. Here, the red entries must be permuted into the main diagonal blocks. " + ], + "image_footnote": [], + "bbox": [ + 225, + 535, + 772, + 696 + ], + "page_idx": 24 + }, + { + "type": "text", + "text": "Lemma G.1. Let L be an $n \\times n$ modular-balanced matrix. Then $\\mathbf { L } \\in B$ . ", + "bbox": [ + 173, + 761, + 651, + 776 + ], + "page_idx": 24 + }, + { + "type": "text", + "text": "Proof. We proceed by induction on $n$ . The base case $n = 2$ is trivial. As our inductive hypothesis, we assume that all modular-balanced matrices of size ${ \\frac { n } { 2 } } \\times { \\frac { n } { 2 } }$ are butterfly matrices of size $\\frac { n } { 2 }$ . From Definition 2.3, it is sufficient to show that can be decomposed as: ", + "bbox": [ + 174, + 795, + 826, + 838 + ], + "page_idx": 24 + }, + { + "type": "equation", + "img_path": "images/df1b27189f25d83f3c2706f4f55e1d3299d3ac12ed64ecc586c74c3ec0a96383.jpg", + "text": "$$\n\\begin{array} { r } { \\mathbf { L } = \\mathbf { B } _ { n } \\underbrace { \\left[ \\begin{array} { c c } { \\mathbf { L } _ { 1 } } & { 0 } \\\\ { 0 } & { \\mathbf { L } _ { 2 } } \\end{array} \\right] } _ { \\mathbf { L } ^ { \\prime } } , } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 429, + 844, + 566, + 902 + ], + "page_idx": 24 + }, + { + "type": "text", + "text": "where $\\mathbf { B } _ { n }$ is a butterfly factor of size $n$ and each $\\mathbf { L } _ { j }$ is an ${ \\frac { n } { 2 } } \\times { \\frac { n } { 2 } }$ modular-balanced matrix. ", + "bbox": [ + 166, + 909, + 767, + 926 + ], + "page_idx": 24 + }, + { + "type": "text", + "text": "Define $\\mathbf { L } _ { 1 }$ and $\\mathbf { L } _ { 2 }$ such that: ", + "bbox": [ + 174, + 103, + 359, + 117 + ], + "page_idx": 25 + }, + { + "type": "equation", + "img_path": "images/c0edfc17e42c571afe33c461cb103c2887e01cacdcb6cd0a0453244974c612af.jpg", + "text": "$$\n{ \\bf L } _ { 2 } [ i , j ] = { \\bf L } \\left[ i , j + \\frac { n } { 2 } \\right] + { \\bf L } \\left[ i + \\frac { n } { 2 } , j + \\frac { n } { 2 } \\right] .\n$$", + "text_format": "latex", + "bbox": [ + 486, + 117, + 787, + 145 + ], + "page_idx": 25 + }, + { + "type": "text", + "text": "Note that since $\\mathbf { L }$ is a permutation matrix (and thus has exactly one non-zero entry per column), at most one term of each of these sums can be non-zero. ", + "bbox": [ + 176, + 146, + 820, + 174 + ], + "page_idx": 25 + }, + { + "type": "text", + "text": "For sake of contradiction, assume $\\mathbf { L } _ { 1 }$ is not modular-balanced. Then, for some $2 ^ { j } \\leq { \\frac { n } { 2 } }$ , there are two columns $c _ { 1 } , c _ { 2 }$ such that $\\textstyle \\left\\lfloor { \\frac { c _ { 1 } } { 2 ^ { j } } } \\right\\rfloor = \\left\\lfloor { \\frac { c _ { 2 } } { 2 ^ { j } } } \\right\\rfloor$ and such that indices of the non-zero entries of $\\mathbf { L } _ { 1 }$ in columns $c _ { 1 }$ and $c _ { 2 }$ are the same modulo $2 ^ { j }$ . However, from the definition of $\\mathbf { L } _ { 1 }$ , this implies that the indices of the non-zero entries of $\\mathbf { L }$ in columns $c _ { 1 }$ and $c _ { 2 }$ are also the same modulo $2 ^ { j }$ , contradicting $\\mathbf { L }$ being modular-balanced. Hence, $\\mathbf { L } _ { 1 }$ is modular-balanced. An analogous argument (that instead considers columns $\\begin{array} { r } { c _ { 1 } + \\frac { n } { 2 } , c _ { 2 } + \\frac { n } { 2 } } \\end{array}$ of $\\mathbf { L }$ ) shows that $\\mathbf { L } _ { 2 }$ is also modular-balanced. ", + "bbox": [ + 173, + 179, + 825, + 270 + ], + "page_idx": 25 + }, + { + "type": "text", + "text": "To complete the proof, we must argue that $\\mathbf { B } _ { n }$ is a butterfly factor of size $n$ . Since each $\\mathbf { L } _ { i }$ is modular-balanced, it is a permutation matrix. Therefore, $\\mathbf { L } ^ { \\prime }$ has exactly 1 non-zero entry in each of the first $\\frac { n } { 2 }$ rows and columns from $\\mathbf { L } _ { 1 }$ and exactly 1 non-zero entry in each of the second $\\frac { n } { 2 }$ rows and columns from $\\mathbf { L } _ { 2 }$ . Hence, $\\mathbf { L } ^ { \\prime }$ is a permutation matrix. Since both $\\mathbf { L }$ and $\\mathbf { L } ^ { \\prime }$ are permutation matrices, $\\mathbf { B } = \\mathbf { L } \\left( \\mathbf { L } ^ { \\prime } \\right) ^ { - 1 }$ must also be a permutation matrix. Therefore, we can view $\\mathbf { B }$ as performing a permutation of the rows of $\\mathbf { L } ^ { \\prime }$ to get $\\mathbf { L }$ . ", + "bbox": [ + 173, + 275, + 825, + 363 + ], + "page_idx": 25 + }, + { + "type": "text", + "text": "Consider the $\\overrightarrow { \\imath } ^ { \\prime }$ ’th row of $\\mathbf { L } ^ { \\prime }$ , with $0 \\leq i < \\frac { n } { 2 }$ . There are two possible cases. ", + "bbox": [ + 176, + 367, + 663, + 385 + ], + "page_idx": 25 + }, + { + "type": "text", + "text": "Case $1 \\colon \\mathbf { L } ^ { \\prime } [ i , : ] = \\mathbf { L } [ i , : ]$ ", + "bbox": [ + 174, + 388, + 339, + 406 + ], + "page_idx": 25 + }, + { + "type": "text", + "text": "In this case, the column of $\\mathbf { L }$ with a non-zero entry in row $i$ is in the left $\\frac { n } { 2 }$ columns. The column of $\\mathbf { L }$ with a non-zero entry in row $\\begin{array} { r } { i + \\frac { n } { 2 } } \\end{array}$ must, therefore, be in the right $\\frac { n } { 2 }$ columns, otherwise $\\mathbf { L }$ would not satisfy the $\\frac { n } { 2 }$ balance condition. Therefore, $\\begin{array} { r } { \\mathbf { L } ^ { \\prime } \\left[ i + \\frac { n } { 2 } , : \\right] = \\bar { \\mathbf { L } } \\left[ i + \\frac { n } { 2 } , : \\right] } \\end{array}$ , so we set $\\begin{array} { r } { \\mathbf { B } [ i , i ] = \\mathbf { B } \\left[ i + \\frac { n } { 2 } , i + \\frac { n } { 2 } \\right] = 1 } \\end{array}$ . ", + "bbox": [ + 173, + 411, + 825, + 474 + ], + "page_idx": 25 + }, + { + "type": "equation", + "img_path": "images/dfb26de8f515e48be0c57737e3b622f24d777a25d59af0207281aa209773a162.jpg", + "text": "$$\n{ \\mathsf { C a s e 2 : L ^ { \\prime } } } [ i , : ] \\neq \\mathbf { L } [ i , : ]\n$$", + "text_format": "latex", + "bbox": [ + 173, + 478, + 341, + 497 + ], + "page_idx": 25 + }, + { + "type": "text", + "text": "By the definition of $\\mathbf { L } ^ { \\prime }$ , $\\begin{array} { r } { { \\bf L } ^ { \\prime } [ i , : ] = v L \\left[ i + \\frac { n } { 2 } , : \\right] } \\end{array}$ . In this case, the column of $\\mathbf { L }$ with a non-zero entry in row $\\begin{array} { r } { i + \\frac { n } { 2 } } \\end{array}$ must be in the left $\\frac { n } { 2 }$ columns. By the $\\frac { n } { 2 }$ balance condition of $\\mathbf { L }$ , the column of $\\mathbf { L }$ with a non-zero entry in row $i$ must be in the right $\\frac { n } { 2 }$ columns. Therefore, $\\begin{array} { r } { \\mathbf { L } ^ { \\prime } \\left[ i + \\frac { n } { 2 } , : \\right] = \\mathbf { L } \\left[ i , : \\right] } \\end{array}$ , so we set $\\begin{array} { r } { \\mathbf { B } \\left[ i , i + \\frac { n } { 2 } \\right] = \\mathbf { B } \\left[ i + \\frac { n } { 2 } , i \\right] = 1 . } \\end{array}$ ", + "bbox": [ + 173, + 501, + 825, + 566 + ], + "page_idx": 25 + }, + { + "type": "text", + "text": "In both cases, the non-zero entries of $\\mathbf { B }$ fall into the correct diagonal bands (the main diagonal, and the bands $\\frac { n } { 2 }$ away). Hence, $\\mathbf { B }$ is a butterfly factor of size $n$ . □ ", + "bbox": [ + 174, + 570, + 823, + 599 + ], + "page_idx": 25 + }, + { + "type": "text", + "text": "Now, we consider the process of transforming $\\mathbf { P }$ into a modular-balanced matrix. We make use of the following lemma. ", + "bbox": [ + 173, + 613, + 825, + 642 + ], + "page_idx": 25 + }, + { + "type": "image", + "img_path": "images/a11fbc418e35ae750d3922a0ae3f160b759972fa724560634f39157fd9066c93.jpg", + "image_caption": [ + "Figure 8: First step of balancing $8 \\times 8$ bit reversal permutation (a component of the $8 \\times 8$ DFT). Red signifies edges that must be flipped. " + ], + "image_footnote": [], + "bbox": [ + 241, + 656, + 754, + 815 + ], + "page_idx": 25 + }, + { + "type": "text", + "text": "Lemma G.2. Let M be a $k \\times k$ matrix with $I$ non-zero entry per column, such that for each $\\textstyle 0 \\leq m < { \\frac { k } { 2 } }$ , there are exactly 2 columns with non-zero entry in a row with index $\\equiv m$ \u0000mod $\\frac { k } { 2 }$ \u0001 . Then, there is a butterfly factor $\\mathbf { B } _ { k }$ such that $\\mathbf { M } \\mathbf { B } _ { k } = \\mathbf { M } ^ { \\prime }$ , where $\\mathbf { M } ^ { \\prime }$ meets the $\\frac { k } { 2 }$ balance condition. ", + "bbox": [ + 173, + 877, + 826, + 925 + ], + "page_idx": 25 + }, + { + "type": "text", + "text": "Proof. We construct a directed graph $G$ with nodes in $\\left[ \\frac { k } { 2 } \\right]$ . For each $\\begin{array} { r } { 0 \\leq i < \\frac { k } { 2 } } \\end{array}$ we add a directed edge from node $\\left( s \\mod \\frac { k } { 2 } \\right)$ \u0001 to node $\\left( t \\mod \\frac { k } { 2 } \\right)$ if $\\mathbf { M } [ : , i ] = \\mathbf { e } _ { s }$ and $\\begin{array} { r } { \\mathbf { M } \\left[ : , i + \\frac { k } { 2 } \\right] = \\mathbf { e } _ { t } } \\end{array}$ . Each node has (undirected) degree exactly 2 by the structure of M. Hence, $G$ is a union of disjoint (undirected) cycles. ", + "bbox": [ + 174, + 102, + 825, + 162 + ], + "page_idx": 26 + }, + { + "type": "text", + "text": "If M met the $\\frac { k } { 2 }$ balance condition, then each node would additionally have in-degree exactly 1 and out-degree exactly 1. By reversing edges of $G$ such that each (undirected) cycle becomes a directed cycle, we can achieve this. However, reversing edges corresponds to swapping columns of $\\mathbf { M }$ that are $\\frac { k } { 2 }$ apart. Let $\\mathbf { B } _ { k }$ be the permutation matrix that performs all such swaps. $\\mathbf { B } _ { k }$ has non-zero entries only along the main diagonal and the diagonal bands $\\frac { k } { 2 }$ away, and thus is a butterfly factor of size $k$ . □ ", + "bbox": [ + 173, + 169, + 825, + 256 + ], + "page_idx": 26 + }, + { + "type": "text", + "text": "We are ready to present the decomposition of $\\mathbf { P }$ ", + "bbox": [ + 174, + 273, + 490, + 287 + ], + "page_idx": 26 + }, + { + "type": "text", + "text": "Lemma G.3. Let $\\mathbf { P }$ be an $n \\times n$ permutation matrix. Then we can decompose $\\mathbf { P }$ into $\\mathbf { P } = \\mathbf { L R }$ , where L is modular-balanced and $\\mathbf { R } \\in B ^ { * }$ . ", + "bbox": [ + 173, + 292, + 825, + 320 + ], + "page_idx": 26 + }, + { + "type": "text", + "text": "Proof. We repeatedly apply Lemma G.2. First, we conclude that there is a butterfly factor $\\mathbf { B } _ { n }$ such that ", + "bbox": [ + 173, + 338, + 825, + 366 + ], + "page_idx": 26 + }, + { + "type": "equation", + "img_path": "images/a295912c9fdc3d8079021a4e4a58ef21f0cab90f50c2e152683860e89417f17c.jpg", + "text": "$$\n\\mathbf { P B } _ { n } = \\mathbf { P } ^ { \\prime } ,\n$$", + "text_format": "latex", + "bbox": [ + 457, + 366, + 539, + 383 + ], + "page_idx": 26 + }, + { + "type": "text", + "text": "where $\\mathbf { P ^ { \\prime } }$ meets the $\\frac { n } { 2 }$ balance condition. Now, we consider the first and last $\\frac { n } { 2 }$ columns of $\\mathbf { P ^ { \\prime } }$ independently. We can again apply Lemma G.2 (twice) to conclude that there are butterfly factors $\\left[ \\mathbf { B } _ { \\frac { n } { 2 } } \\right] _ { 1 } , \\left[ \\mathbf { B } _ { \\frac { n } { 2 } } \\right] _ { 2 }$ such that ", + "bbox": [ + 174, + 387, + 825, + 433 + ], + "page_idx": 26 + }, + { + "type": "equation", + "img_path": "images/f466ce619a581a644535e0693f9477d60a9d39de0b2dd7d8d8bfd4b730939922.jpg", + "text": "$$\n\\mathbf { P B } _ { n } \\left[ \\begin{array} { c c } { \\left[ \\mathbf { B } _ { \\frac { n } { 2 } } \\right] _ { 1 } } & { 0 } \\\\ { 0 } & { \\left[ \\mathbf { B } _ { \\frac { n } { 2 } } \\right] _ { 2 } } \\end{array} \\right] = \\mathbf { P B } _ { n } ^ { ( n ) } \\mathbf { B } _ { \\frac { n } { 2 } } ^ { ( n ) } = \\mathbf { P } ^ { \\prime \\prime } ,\n$$", + "text_format": "latex", + "bbox": [ + 339, + 439, + 655, + 477 + ], + "page_idx": 26 + }, + { + "type": "text", + "text": "where $\\mathbf { P } ^ { \\prime \\prime }$ meets the $\\frac { n } { 2 }$ and $\\frac { n } { 4 }$ balance conditions. ", + "bbox": [ + 173, + 483, + 496, + 501 + ], + "page_idx": 26 + }, + { + "type": "text", + "text": "We continue this process until we obtain a matrix that meets all of the balance conditions. Our final equation is of the form: ", + "bbox": [ + 176, + 506, + 823, + 534 + ], + "page_idx": 26 + }, + { + "type": "equation", + "img_path": "images/0c8bf0bb2c78363fdef2b9b2a33689d32ea3a5cba19e7d214bbf410262be555e.jpg", + "text": "$$\n\\mathbf { P } \\cdot \\mathbf { B } _ { n } ^ { \\left( n \\right) } \\mathbf { B } _ { \\frac { n } { 2 } } ^ { \\left( n \\right) } \\cdot \\ldots \\mathbf { B } _ { 2 } ^ { \\left( n \\right) } = \\mathbf { P B } = \\mathbf { L } ,\n$$", + "text_format": "latex", + "bbox": [ + 380, + 532, + 616, + 558 + ], + "page_idx": 26 + }, + { + "type": "text", + "text": "where $\\boldsymbol { B }$ is a butterfly matrix and $\\mathbf { L }$ is a modular-balanced matrix. Let $\\mathbf { R } = \\mathbf { B } ^ { - 1 } = \\mathbf { B } ^ { * }$ (since $\\mathbf { B }$ is a permutation matrix, and thus is orthogonal) and hence $\\mathbf { R } \\in B ^ { * }$ . Then $\\mathbf { P } = \\mathbf { L R }$ , as desired. □ ", + "bbox": [ + 173, + 561, + 828, + 590 + ], + "page_idx": 26 + }, + { + "type": "text", + "text": "Theorem 2 follows immediately from Lemmas G.3 and G.1. ", + "bbox": [ + 173, + 607, + 568, + 622 + ], + "page_idx": 26 + }, + { + "type": "text", + "text": "We now show that permutations are also in $B ^ { * } B$ . We start with the relationship between butterfly matrices and the bit-reversal permutation. ", + "bbox": [ + 176, + 628, + 823, + 659 + ], + "page_idx": 26 + }, + { + "type": "text", + "text": "Lemma G.4. Let $\\mathbf { P } _ { \\mathrm { b r } }$ be the $n \\times n$ bit-reversal permutation matrix where n is some power of 2, and let ${ \\bf M } _ { 1 } \\in { B }$ be an $n \\times n$ butterfly matrix. Then there is some butterfly matrix ${ \\bf M } _ { 2 } \\in { B }$ such that ", + "bbox": [ + 173, + 662, + 825, + 690 + ], + "page_idx": 26 + }, + { + "type": "equation", + "img_path": "images/38ebc511817bb20e47cd2363018c0b54183cacf0471e56cd3378fc2bf70e9b43.jpg", + "text": "$$\n\\mathbf { M } _ { 1 } ^ { * } = \\mathbf { P } _ { \\mathrm { b r } } \\mathbf { M } _ { 2 } \\mathbf { P } _ { \\mathrm { b r } } .\n$$", + "text_format": "latex", + "bbox": [ + 431, + 699, + 565, + 717 + ], + "page_idx": 26 + }, + { + "type": "text", + "text": "Proof sketch. For any input vector $\\mathbf { x }$ of length $n$ , to perform $\\mathbf { M } _ { 1 } ^ { * } \\mathbf { x }$ , we trace through $\\log _ { 2 } n$ steps of the multiplication algorithm. At each step, we perform $2 \\times 2$ matrix multiplication on elements of $\\mathbf { x }$ whose indices are $n / 2$ apart (e.g. indices 0 and $n / 2 , 1$ and $n / 2 + 1$ , etc.), then $n / 4$ apart, and so on, till indices are that 1 apart. If we apply the bit-reversal permutation on $\\mathbf { x }$ , then indices that are $n / 2$ apart will become 1 apart, indices that are $n / 4$ apart will become 2 apart, and so on. So the multiplication algorithm $\\mathbf { M } _ { 1 } ^ { * } \\mathbf { x }$ is equivalent to applying bit-reversal, then multiplying the permuted vector with another butterfly matrix (i.e. $2 \\times 2$ matrix multiplication on indices that are 1 apart, then 2 apart, and so on, till indices that are $n / 2$ apart). Finally we need to do another bit-reversal permutation to put all the indices back to the original order. If we call this other butterfly matrix $\\mathbf { M } _ { 2 }$ , then we have shown that $\\mathbf { M } _ { 1 } ^ { * } \\mathbf { x } = \\mathbf { P } _ { \\mathrm { b r } } \\mathbf { M } _ { 2 } \\mathbf { P } _ { \\mathrm { b r } } \\mathbf { x }$ . This holds for all $\\mathbf { x }$ (for the same matrix $\\mathbf { M } _ { 2 }$ ), so we have $\\mathbf { M } _ { 1 } ^ { * } = \\mathbf { P } _ { \\mathrm { b r } } \\mathbf { M } _ { 2 } \\mathbf { P } _ { \\mathrm { b r } }$ . □ ", + "bbox": [ + 173, + 731, + 826, + 886 + ], + "page_idx": 26 + }, + { + "type": "text", + "text": "Remark G.5. Lemma G.4 explains the connection between the two most common fast Fourier transform algorithm, decimation in time and decimation in frequency. Using the decimation-in-time ", + "bbox": [ + 173, + 895, + 823, + 924 + ], + "page_idx": 26 + }, + { + "type": "text", + "text": "FFT, we can write the DFT matrix $\\mathbf { F }$ as product of a butterfly matrix ${ { \\bf { M } } _ { 1 } }$ and the bit-reversal permutation (see Section J): ", + "bbox": [ + 169, + 102, + 825, + 132 + ], + "page_idx": 27 + }, + { + "type": "equation", + "img_path": "images/57737368eaabd7699e4a2c05595918e0b33408dadc98eb87c72f1ea99064baae.jpg", + "text": "$$\n\\mathbf { F } = \\mathbf { M } _ { 1 } \\mathbf { P } _ { \\mathrm { b r } } .\n$$", + "text_format": "latex", + "bbox": [ + 452, + 132, + 544, + 148 + ], + "page_idx": 27 + }, + { + "type": "text", + "text": "Taking conjugate transpose, we obtain $\\mathbf { F } ^ { * } = \\mathbf { P } _ { \\mathrm { b r } } \\mathbf { M } _ { 1 } ^ { * }$ (recall that $\\mathbf { P } _ { \\mathrm { b r } }$ is its own transpose/inverse). On the other hand, $\\mathbf { F } ^ { * }$ is just a scaled version of the inverse DFT matrix, so apply decimation-in-time FFT to the inverse DFT, we can write $\\mathbf { F } ^ { * } = \\mathbf { M } _ { 2 } \\mathbf { P } _ { \\mathrm { b r } }$ for some other butterfly matrix $\\mathbf { M } _ { 2 }$ . Hence $\\mathbf { P } _ { \\mathrm { b r } } \\mathbf { M } _ { 1 } ^ { * } = \\mathbf { M } _ { 2 } \\mathbf { P } _ { \\mathrm { b r } } ,$ , and thus $\\mathbf { P } _ { \\mathrm { b r } } \\mathbf { M } _ { 1 } ^ { * } \\mathbf { P } _ { \\mathrm { b r } } = \\mathbf { M } _ { 2 }$ (for these particular butterfly matrices ${ { \\bf { M } } _ { 1 } }$ and ${ { \\bf { M } } _ { 2 } }$ ). Note that this yields another decomposition of the DFT matrix, $\\mathbf { F } = \\mathbf { P } _ { \\mathrm { b r } } \\mathbf { M } _ { 2 } ^ { * }$ , which is exactly the decimation-in-frequency FFT algorithm. ", + "bbox": [ + 173, + 152, + 825, + 238 + ], + "page_idx": 27 + }, + { + "type": "text", + "text": "We are ready to show that permutations are in $B ^ { * } B$ . ", + "bbox": [ + 173, + 250, + 513, + 265 + ], + "page_idx": 27 + }, + { + "type": "text", + "text": "Lemma G.6. Let $\\mathbf { P }$ be an $n \\times n$ permutation matrix (with n a power of 2). Then there are butterfly matrices $\\mathbf { M } _ { 1 } , \\mathbf { M } _ { 2 } \\in B$ such that $\\mathbf { P } = \\mathbf { M } _ { 1 } ^ { * } \\mathbf { M } _ { 2 }$ . ", + "bbox": [ + 174, + 268, + 823, + 299 + ], + "page_idx": 27 + }, + { + "type": "text", + "text": "Proof. Consider the permutation $\\tilde { \\mathbf { P } } = \\mathbf { P } _ { \\mathrm { b r } } \\mathbf { P } \\mathbf { P } _ { \\mathrm { b r } }$ . By Theorem 2, there are some butterfly matrices $\\tilde { \\mathbf { M } } _ { 1 } , \\tilde { \\mathbf { M } } _ { 2 } \\in B$ such that $\\tilde { \\mathbf { P } } = \\tilde { \\mathbf { M } } _ { 1 } \\tilde { \\mathbf { M } } _ { 2 } ^ { * }$ . Applying Lemma G.4, we can replace ${ { \\tilde { \\mathbf { M } } } _ { 2 } } ^ { * }$ with $\\mathbf { P } _ { \\mathrm { b r } } \\mathbf { M } _ { 2 } \\mathbf { P } _ { \\mathrm { b r } }$ for some butterfly matrix ${ \\bf M } _ { 2 } \\in { B }$ . We thus have: ", + "bbox": [ + 173, + 318, + 825, + 364 + ], + "page_idx": 27 + }, + { + "type": "equation", + "img_path": "images/e1d5e6180591a144282515a08a0ca3670cb09d6428bd006e8d56f6dce90801d8.jpg", + "text": "$$\n\\mathbf { P } _ { \\mathrm { b r } } \\mathbf { P } \\mathbf { P } _ { \\mathrm { b r } } = \\tilde { \\mathbf { M } } _ { 1 } \\mathbf { P } _ { \\mathrm { b r } } \\mathbf { M } _ { 2 } \\mathbf { P } _ { \\mathrm { b r } } .\n$$", + "text_format": "latex", + "bbox": [ + 400, + 371, + 598, + 391 + ], + "page_idx": 27 + }, + { + "type": "text", + "text": "Pre- and post-multiply both sides by $\\mathbf { P } _ { \\mathrm { b r } }$ (which is its own inverse): ", + "bbox": [ + 173, + 398, + 619, + 415 + ], + "page_idx": 27 + }, + { + "type": "equation", + "img_path": "images/d2342287854341664e25674d5d9a093021eccd317b4596e1e4ab86275967cc29.jpg", + "text": "$$\n\\mathbf { P } = \\mathbf { P } _ { \\mathrm { b r } } \\tilde { \\mathbf { M } } _ { 1 } \\mathbf { P } _ { \\mathrm { b r } } \\mathbf { M } _ { 2 } .\n$$", + "text_format": "latex", + "bbox": [ + 426, + 421, + 571, + 440 + ], + "page_idx": 27 + }, + { + "type": "text", + "text": "Applying Lemma G.4 again, we can replace $\\mathbf { P } _ { \\mathrm { b r } } \\mathbf { \\tilde { M } } _ { 1 } \\mathbf { P } _ { \\mathrm { b r } }$ with $\\mathbf { M } _ { 1 } ^ { * }$ for some butterfly matrix ${ \\bf M } _ { 1 } \\in { B }$ . Thus: ", + "bbox": [ + 171, + 448, + 828, + 477 + ], + "page_idx": 27 + }, + { + "type": "equation", + "img_path": "images/e6a6be9d6c2fb0debc91d029fa4f36fdb17253558d89dd8a2f6a78ebf0c52fac.jpg", + "text": "$$\n\\mathbf { P } = \\mathbf { M } _ { 1 } ^ { * } \\mathbf { M } _ { 2 } .\n$$", + "text_format": "latex", + "bbox": [ + 450, + 477, + 545, + 494 + ], + "page_idx": 27 + }, + { + "type": "text", + "text": "H $B B ^ { * }$ CLOSURE LEMMAS ", + "text_level": 1, + "bbox": [ + 174, + 535, + 411, + 551 + ], + "page_idx": 27 + }, + { + "type": "text", + "text": "Here, we present some basic facts of the $B B ^ { * }$ hierarchy that will be useful for later constructions. For simplicity, we assume (WLOG via 0-padding) that all matrices are square matrices with size that is a power of 2. ", + "bbox": [ + 174, + 566, + 823, + 609 + ], + "page_idx": 27 + }, + { + "type": "text", + "text": "Lemma H.1. If $\\mathbf { M } \\in B$ (or $\\mathbf { M } \\in B ^ { * }$ ), then DM, $\\mathbf { M D } \\in B$ ( $B ^ { * }$ resp.) for any diagonal matrix D. ", + "bbox": [ + 174, + 613, + 816, + 630 + ], + "page_idx": 27 + }, + { + "type": "text", + "text": "Proof. Left multiplication by a diagonal matrix scales the rows of M by the corresponding diagonal entries. The same can be achieved by scaling all entries the leftmost butterfly factor matrix. Similarly, right multiplication by a diagonal matrix scales the columns of $\\mathbf { M }$ , which can be achieved by scaling all entries in the columns of the rightmost butterfly factor matrix. ", + "bbox": [ + 173, + 647, + 825, + 705 + ], + "page_idx": 27 + }, + { + "type": "text", + "text": "Lemma H.2. Let A, $\\mathbf { B } \\in \\mathbb { F } ^ { n \\times n }$ . If $\\mathbf { \\dot { A } } \\in ( B B ^ { * } ) _ { e } ^ { w _ { 1 } }$ and $\\mathbf { B } \\in ( B B ^ { * } ) _ { e } ^ { w _ { 2 } }$ then $\\mathbf { A B } \\in ( B B ^ { * } ) _ { e } ^ { w _ { 1 } + w _ { 2 } } .$ ", + "bbox": [ + 171, + 715, + 802, + 733 + ], + "page_idx": 27 + }, + { + "type": "text", + "text": "Proof. Let $\\mathbf { E _ { A } } , \\mathbf { E _ { B } } \\in \\mathbb { F } ^ { e n \\times e n }$ be defined such that $\\mathbf { A } = \\mathbf { S } \\mathbf { E } _ { \\mathbf { A } } \\mathbf { S } ^ { T }$ , ${ \\bf B } = { \\bf S } { \\bf E } _ { \\bf B } { \\bf S } ^ { T }$ (with S as in Definition 2.4). Then ", + "bbox": [ + 174, + 750, + 820, + 779 + ], + "page_idx": 27 + }, + { + "type": "equation", + "img_path": "images/163a4b39e4da26f297e8d3db683a356062efc3cb8eaa5dafffac0f17c76a7784.jpg", + "text": "$$\n\\mathbf { A } \\mathbf { B } = \\mathbf { S } \\underbrace { \\left[ \\mathbf { I } _ { n } \\mathbf { \\Gamma } _ { 0 } \\right] } _ { e n \\times e n } \\mathbf { E } _ { \\mathbf { A } } \\underbrace { \\left[ \\mathbf { I } _ { n } \\mathbf { \\Gamma } _ { 0 } \\right] } _ { e n \\times e n } \\mathbf { E } _ { \\mathbf { B } } \\mathbf { S } ^ { T }\n$$", + "text_format": "latex", + "bbox": [ + 367, + 777, + 632, + 833 + ], + "page_idx": 27 + }, + { + "type": "text", + "text": "$\\begin{array} { r l } { \\bigg [ \\mathbf { I } _ { n } } & { { } 0 \\bigg ] \\mathbf { E _ { A } } \\in ( B B ^ { * } ) ^ { w _ { 1 } } , \\bigg [ \\mathbf { I } _ { n } \\quad 0 \\bigg ] \\mathbf { E _ { B } } \\in ( B B ^ { * } ) ^ { w _ { 2 } } } \\end{array}$ by Lemma H.1. Hence, $\\mathbf { A B } \\in ( B B ^ { * } ) _ { e } ^ { w _ { 1 } + w _ { 2 } }$ by Definition 2.4. ", + "bbox": [ + 174, + 838, + 825, + 883 + ], + "page_idx": 27 + }, + { + "type": "text", + "text": "Lemma H.3. Let $\\mathbf { A } _ { 1 } , \\ldots , \\mathbf { A } _ { m } \\in \\mathbb { F } ^ { k \\times k }$ . If $\\mathbf { A } _ { 1 } , \\ldots , \\mathbf { A } _ { m } \\in ( B B ^ { * } ) _ { e } ^ { w }$ then $\\mathrm { D i a g } ( \\mathbf { A } _ { 1 } , \\ldots , \\mathbf { A } _ { m } ) \\in$ $( B B ^ { * } ) _ { e } ^ { w + 2 }$ . ", + "bbox": [ + 171, + 893, + 825, + 925 + ], + "page_idx": 27 + }, + { + "type": "text", + "text": "Proof. For each $1 \\leq i \\leq m$ , let $\\mathbf { E } _ { \\mathbf { A } _ { i } } \\in \\mathbb { F } ^ { e k \\times e k }$ be defined such that $\\mathbf { A } _ { i } = \\mathbf { S } \\mathbf { E } _ { \\mathbf { A } _ { i } } \\mathbf { S } ^ { T }$ (with $\\mathbf { S }$ as in Definition 2.4). Then ", + "bbox": [ + 171, + 102, + 825, + 132 + ], + "page_idx": 28 + }, + { + "type": "equation", + "img_path": "images/0a2908f8997014b205e28b77a1099c2cbba42d731002233b0d2602c885aeab37.jpg", + "text": "$$\n\\left[ \\begin{array} { c c c c } { \\mathbf { A } _ { 1 } } & { 0 } & { . . . } & { 0 } \\\\ { 0 } & { \\mathbf { A } _ { 2 } } & { . . . } & { 0 } \\\\ { \\vdots } & { \\vdots } & { \\ddots } & { 0 } \\\\ { 0 } & { 0 } & { . . . } & { \\mathbf { A } _ { m } } \\end{array} \\right] = \\mathbf { S } \\mathbf { P } \\left[ \\begin{array} { c c c c } { \\mathbf { E } _ { \\mathbf { A } _ { 1 } } } & { 0 } & { . . . } & { 0 } \\\\ { 0 } & { \\mathbf { E } _ { \\mathbf { A } _ { 2 } } } & { . . . } & { 0 } \\\\ { \\vdots } & { \\vdots } & { \\ddots } & { 0 } \\\\ { 0 } & { 0 } & { . . . } & { \\mathbf { E } _ { \\mathbf { A } _ { m } } } \\end{array} \\right] \\mathbf { P } ^ { T } \\mathbf { S } ^ { T }\n$$", + "text_format": "latex", + "bbox": [ + 277, + 138, + 723, + 207 + ], + "page_idx": 28 + }, + { + "type": "text", + "text": "where $\\mathbf { P }$ is a permutation that that moves the first $k$ rows of each $\\mathbf { E } _ { \\mathbf { A } _ { i } }$ (in order) into the top mk rows. From Theorem 2, $\\mathbf { P } \\in B B ^ { * }$ , (and so is $\\mathbf { P } ^ { T }$ , also a permutation). Within the RHS block matrix, the decompositions of each $\\mathbf { E } _ { \\mathbf { A } _ { i } }$ can be done in parallel, requiring total width $w$ . Hence, $\\mathrm { D i a g } ( \\mathbf { A } _ { 1 } , \\ldots , \\mathbf { A } _ { m } ) \\in ( B B ^ { * } ) _ { e } ^ { w + 2 }$ , as desired. □ ", + "bbox": [ + 173, + 213, + 826, + 271 + ], + "page_idx": 28 + }, + { + "type": "text", + "text": "Remark H.4. If $\\textit { e } = \\ 1$ in Lemma H.3, then $\\mathbf { P }$ is unnecessary. Hence, $\\mathrm { D i a g } ( \\mathbf { A } _ { 1 } , \\ldots , \\mathbf { A } _ { m } ) \\in$ $( B B ^ { * } ) ^ { w }$ . ", + "bbox": [ + 171, + 277, + 825, + 308 + ], + "page_idx": 28 + }, + { + "type": "text", + "text": "Lemma H.5. Let $\\mathbf { A } _ { 1 } , \\ldots , \\mathbf { A } _ { m }$ be $k \\times k$ matrices in $( B B ^ { * } ) _ { e } ^ { w }$ then $\\begin{array} { r } { \\sum _ { i = 1 } ^ { m } { \\bf A } _ { i } \\in ( B B ^ { * } ) _ { 4 e } ^ { m w } . } \\end{array}$ ", + "bbox": [ + 176, + 310, + 764, + 328 + ], + "page_idx": 28 + }, + { + "type": "text", + "text": "Proof. For each $1 \\leq i \\leq m$ , let $\\mathbf { E } _ { \\mathbf { A } _ { i } } \\in \\mathbb { F } ^ { e k \\times e k }$ be defined such that $\\mathbf { A } _ { i } = \\mathbf { S } \\mathbf { E } _ { \\mathbf { A } _ { i } } \\mathbf { S } ^ { T }$ (with $\\mathbf { S }$ as in Definition 2.4). Note that $\\mathbf { E } _ { \\mathbf { A } _ { i } } \\in ( B B ^ { * } ) ^ { w }$ . Consider matrices of the form: ", + "bbox": [ + 174, + 343, + 823, + 375 + ], + "page_idx": 28 + }, + { + "type": "equation", + "img_path": "images/0dce04dcda7d5ed24cc4bf04e4a43c10fa6f159677a6c0a89e55f4b870cc05de.jpg", + "text": "$$\n\\underbrace { \\left[ \\begin{array} { c c c c } { \\mathbf { I } _ { e k } } & { \\mathbf { E } _ { \\mathbf { A } _ { i } } } & { 0 } & { 0 } \\\\ { 0 } & { \\mathbf { I } _ { e k } } & { 0 } & { 0 } \\\\ { 0 } & { 0 } & { 0 } & { 0 } \\\\ { \\mathbf { M } _ { i } \\in \\mathbb { R } ^ { 4 e k \\times 4 e k } } & { \\mathbf { L } } \\end{array} \\right] } _ { \\mathbf { L } } = \\underbrace { \\left[ \\mathbf { I } _ { 2 e k } \\right. } _ { \\mathbf { L } } , \\mathbf { I } _ { 2 e k } \\underbrace { } _ { 0 } \\underbrace { \\left[ \\begin{array} { c c c c } { \\mathbf { I } _ { e k } } & { 0 } & { 0 } & { 0 } \\\\ { 0 } & { \\mathbf { I } _ { e k } } & { 0 } & { 0 } \\\\ { 0 } & { 0 } & { \\mathbf { E } _ { \\mathbf { A } _ { i } } } & { 0 } \\\\ { 0 } & { 0 } & { 0 } & { 0 } \\end{array} \\right] } _ { \\mathbf { S } } \\underbrace { \\left[ \\begin{array} { c c c c } { \\mathbf { I } _ { e k } } & { 0 } & { 0 } & { 0 } \\\\ { 0 } & { \\mathbf { I } _ { e k } } & { 0 } & { 0 } \\\\ { 0 } & { 0 } & { 0 } & { \\mathbf { I } _ { e k } } \\\\ { 0 } & { 0 } & { \\mathbf { I } _ { e k } } & { 0 } \\end{array} \\right] } _ { \\mathbf { P } _ { 1 } } \\underbrace { \\left[ \\mathbf { I } _ { 2 e k } \\right]} _ { \\mathbf { R } } , \\mathbf { I } _ { \\mathbf { R } } .\n$$", + "text_format": "latex", + "bbox": [ + 186, + 380, + 810, + 469 + ], + "page_idx": 28 + }, + { + "type": "text", + "text": "Here, $\\mathbf { L }$ and $\\mathbf { R }$ compute the sum of the $2 e k \\times 2 e k$ matrices on the diagonal of $\\mathbf { S P } _ { 1 }$ , where ${ \\bf P } _ { 1 }$ is a permutation swapping $\\mathbf { E } _ { \\mathbf { A } _ { i } }$ to the $4 ^ { t h } ~ e k$ -block column. Note that S is the diagonalization of four matrices in $( B B ^ { * } ) ^ { w }$ , so $\\mathbf { S } \\in ( B B ^ { * } ) ^ { w }$ by Remark H.4. In addition, since each block in $\\mathbf { S }$ is a butterfly matrix of size $e k$ , S only uses butterfly factors up to size $e k$ , so the outer factor matrices of sizes $4 e k$ and and $2 e k$ in S are unused. Also note that is a butterfly factor matrix of s $\\mathbf { L }$ e $\\mathbf { R }$ ar(or fly factor matrices of size ). This allows us to fold $4 e k$ (or urr (4ek)), g ${ \\bf P } _ { 1 }$ $2 e k$ $\\mathbf { B } _ { 2 e k } ^ { ( 4 e k ) } .$ \nmatrices $\\mathbf { L } , \\mathbf { P } _ { 1 } , \\mathbf { R }$ into S, so $\\mathbf { M } _ { i } \\in ( B B ^ { \\ast } ) ^ { w }$ . ", + "bbox": [ + 173, + 481, + 826, + 589 + ], + "page_idx": 28 + }, + { + "type": "text", + "text": "Through repeated application $\\textbar { m }$ times) of the identity ", + "bbox": [ + 174, + 594, + 532, + 609 + ], + "page_idx": 28 + }, + { + "type": "equation", + "img_path": "images/4047c18f0181a905709324bd8af59da63647dda2dcb591f27b60777ee4a9821b.jpg", + "text": "$$\n\\left[ { \\begin{array} { c c } { \\mathbf { I } } & { \\mathbf { A } } \\\\ { 0 } & { \\mathbf { I } } \\end{array} } \\right] \\left[ { \\begin{array} { c c } { \\mathbf { I } } & { \\mathbf { B } } \\\\ { 0 } & { \\mathbf { I } } \\end{array} } \\right] = \\left[ { \\begin{array} { c c } { \\mathbf { I } } & { \\mathbf { A } + \\mathbf { B } } \\\\ { 0 } & { \\mathbf { I } } \\end{array} } \\right] ,\n$$", + "text_format": "latex", + "bbox": [ + 380, + 614, + 616, + 651 + ], + "page_idx": 28 + }, + { + "type": "text", + "text": "we see that ", + "bbox": [ + 173, + 656, + 250, + 671 + ], + "page_idx": 28 + }, + { + "type": "equation", + "img_path": "images/de473cad6b527deca5dbeff498cfa88e80d02636f20c0a5ba57150883cec33c0.jpg", + "text": "$$\n\\underbrace { \\left[ \\begin{array} { c c c c } { \\mathbf { I } _ { e k } } & { \\sum _ { i = 1 } ^ { m } \\mathbf { E } _ { \\mathbf { A } _ { i } } } & { 0 } & { 0 } \\\\ { 0 } & { \\mathbf { I } _ { e k } } & { 0 } & { 0 } \\\\ { 0 } & { 0 } & { 0 } & { 0 } \\\\ { 0 } & { 0 } & { 0 } & { 0 } \\end{array} \\right] } _ { \\mathbf { M } \\in \\mathbb { R } ^ { 4 e n \\times 4 e n } } = \\prod _ { i = 1 } ^ { m } \\mathbf { M } _ { i } .\n$$", + "text_format": "latex", + "bbox": [ + 367, + 666, + 629, + 752 + ], + "page_idx": 28 + }, + { + "type": "text", + "text": "From Lemma H.2, $\\mathbf { M } \\in \\mathsf { \\Gamma } ( B B ^ { * } ) ^ { m w }$ . Finally, note that $\\begin{array} { r } { \\sum _ { i = 1 } ^ { m } \\mathbf { A } _ { i } = \\mathbf { S } \\mathbf { M } \\mathbf { P } _ { 2 } \\mathbf { S } ^ { T } } \\end{array}$ , where $\\mathbf { P } _ { 2 }$ is a permutation that moves the first $k$ columns of the second block-column of M to the left. $\\mathbf { P } _ { 2 }$ can be folded into the final summation factor $M _ { m }$ as follows: ", + "bbox": [ + 174, + 753, + 826, + 797 + ], + "page_idx": 28 + }, + { + "type": "equation", + "img_path": "images/342929852fdaea4b80c280f12ec2473f9213e0a974c11eeae3518f01525b3f35.jpg", + "text": "$$\n\\begin{array}{c} \\begin{array}{c} \\underbrace { \\left[ \\begin{array} { c c c c } { \\mathbf { I } _ { e k } } & { \\mathbf { 0 } } & { \\mathbf { 0 } } & { \\mathbf { 0 } } \\\\ { \\mathbf { 0 } } & { \\mathbf { I } _ { e k } } & { \\mathbf { 0 } } & { \\mathbf { 0 } } \\\\ { \\mathbf { 0 } } & { \\mathbf { 0 } } & { \\mathbf { I } _ { e k } } \\\\ { \\mathbf { 0 } } & { \\mathbf { D } } & { \\mathbf { I } _ { e k } } & { \\mathbf { 0 } } \\end{array} \\right] } _ { \\mathbf { P } _ { 1 } } \\underbrace { \\left[ \\mathbf { I } _ { 2 e k } \\right]} & { \\mathbf { 0 } } \\\\ & & { \\mathbf { R } } \\end{array} _ { \\mathbf { R } } \\underbrace { \\left[ \\begin{array} { c c c c } { \\mathbf { 0 } } & { \\mathbf { I } _ { e k } } & { \\mathbf { 0 } } & { \\mathbf { 0 } } \\\\ { \\mathbf { I } _ { e k } } & { \\mathbf { 0 } } & { \\mathbf { 0 } } & { \\mathbf { 0 } } \\\\ { \\mathbf { 0 } } & { \\mathbf { 0 } } & { \\mathbf { I } _ { e k } } & { \\mathbf { 0 } } \\\\ { \\mathbf { 0 } } & { \\mathbf { 0 } } & { \\mathbf { 0 } } & { \\mathbf { I } _ { e k } } \\end{array} \\right] } _ { \\mathbf { P } _ { 2 } } = \\underbrace { \\left[ \\begin{array} { c c c c } { \\mathbf { 0 } } & { \\mathbf { I } _ { e k } } & { \\mathbf { 0 } } & { \\mathbf { 0 } } \\\\ { \\mathbf { I } _ { e k } } & { \\mathbf { 0 } } & { \\mathbf { 0 } } & { \\mathbf { 0 } } \\\\ { \\mathbf { 0 } } & { \\mathbf { 0 } } & { \\mathbf { I } _ { e k } } & { \\mathbf { 0 } } \\\\ { \\mathbf { 0 } } & { \\mathbf { 0 } } & { \\mathbf { 0 } } & { \\mathbf { I } _ { e k } } \\end{array} \\right] } _ { \\mathbf { P } _ { 1 } ^ { \\prime } } \\underbrace { \\left[ \\mathbf { I } _ { 2 e k } & { \\mathbf { 0 } } \\\\ & & { \\mathbf { R } } \\end{array} \\right] } _ { \\mathbf { R } } \\underbrace { \\left[ \\begin{array} { c c c c } { \\mathbf { I } _ { e k } } & { \\mathbf { 0 } } & { \\mathbf { I } _ { e k } } & { \\mathbf { 0 } } \\\\ & & { \\mathbf { I } _ { e k } } & { \\mathbf { 0 } } \\\\ & & { \\mathbf { P } _ { 2 } ^ { \\prime } } & { \\mathbf { R } } \\end{array} \\right] } _ \\mathbf \n$$", + "text_format": "latex", + "bbox": [ + 187, + 803, + 790, + 895 + ], + "page_idx": 28 + }, + { + "type": "text", + "text": "Hence, $\\begin{array} { r } { \\sum _ { i = 1 } ^ { m } \\mathbf { A } _ { i } \\in ( B B ^ { * } ) _ { 4 e } ^ { m w } } \\end{array}$ , as desired. ", + "bbox": [ + 174, + 907, + 449, + 926 + ], + "page_idx": 28 + }, + { + "type": "text", + "text": "Lemma H.6. Let M be an invertible $n \\times n$ matrix such that $M \\in B$ . Then $\\mathbf { M } ^ { - 1 } \\in { \\mathfrak { B } } ^ { * }$ ", + "bbox": [ + 173, + 102, + 746, + 118 + ], + "page_idx": 29 + }, + { + "type": "text", + "text": "Proof. We prove this in a series of steps. ", + "bbox": [ + 174, + 137, + 441, + 154 + ], + "page_idx": 29 + }, + { + "type": "text", + "text": "First, let $\\mathbf { B } _ { k }$ be an invertible butterfly factor of size $k$ . Consider the method of computing $\\mathbf { B } _ { k } ^ { - 1 }$ by performing Gaussian elimination on the matrix $[ \\mathbf { B } _ { k } | \\mathbf { I } _ { k } ]$ to obtain the matrix $\\left[ \\mathbf { I } _ { k } | \\mathbf { B } _ { k } ^ { - 1 } \\right]$ . By the form of B, non-zero entries within a row or column are always exactly $\\frac { k } { 2 }$ positions apart. Therefore, the only row operations needed for this Gaussian elimination are: ", + "bbox": [ + 174, + 159, + 825, + 220 + ], + "page_idx": 29 + }, + { + "type": "text", + "text": "• Scaling a row by a constant factor $c \\neq 0$ • Addition of a row to another row exactly $\\frac { k } { 2 }$ rows apart ", + "bbox": [ + 214, + 238, + 589, + 280 + ], + "page_idx": 29 + }, + { + "type": "text", + "text": "Performing these operations on $\\mathbf { I } _ { k }$ will only allow non-zeros on the main diagonal and $\\frac { k } { 2 }$ diagonals away from the main diagonal. Hence, $\\mathbf { B } _ { k } ^ { - 1 }$ is also a butterfly factor of size $k$ . ", + "bbox": [ + 173, + 299, + 825, + 332 + ], + "page_idx": 29 + }, + { + "type": "text", + "text": "Next, let $\\mathbf { B } _ { k } ^ { \\left( n \\right) }$ be an invertible butterfly factor matrix of size $n$ and block size $k$ . Its inverse is the block diagonal matrix formed by the inverses of each of its constituent butterfly factors. From above, $\\left( \\mathbf { B } _ { k } ^ { \\left( n \\right) } \\right) ^ { - 1 }$ is also a butterfly factor matrix of size $n$ and block size $k$ . ", + "bbox": [ + 173, + 340, + 826, + 397 + ], + "page_idx": 29 + }, + { + "type": "text", + "text": "Finally, consider $\\mathbf { M } \\in B$ . ", + "bbox": [ + 174, + 401, + 341, + 417 + ], + "page_idx": 29 + }, + { + "type": "equation", + "img_path": "images/7e97d956a109d454e63c54b713f6d44ce6988d1399ea49191fff34622a6596bb.jpg", + "text": "$$\n\\mathbf { { \\cal { M } } } ^ { - 1 } = \\left( \\mathbf { B } _ { n } ^ { ( n ) } \\mathbf { B } _ { \\frac { n } { 2 } } ^ { ( n ) } \\dots \\mathbf { B } _ { 2 } ^ { ( n ) } \\right) ^ { - 1 } = \\left( \\mathbf { B } _ { 2 } ^ { ( n ) } \\right) ^ { - 1 } \\left( \\mathbf { B } _ { 4 } ^ { ( n ) } \\right) ^ { - 1 } \\dots \\left( \\mathbf { B } _ { n } ^ { ( n ) } \\right) ^ { - 1 } = \\mathbf { B } _ { 2 } ^ { \\prime { } ^ { ( n ) } } \\mathbf { B } _ { 4 } ^ { \\prime { } ^ { ( n ) } } \\dots \\mathbf { B } _ { n } ^ { \\prime { } ^ { ( n ) } } \\in \\mathcal { B } _ { n } ^ { ( n ) } .\n$$", + "text_format": "latex", + "bbox": [ + 181, + 424, + 839, + 455 + ], + "page_idx": 29 + }, + { + "type": "text", + "text": "Finally, we include a closure result for the Kronecker product, another common matrix composition operation. Although Lemma H.7 is not directly used in the subsequent proofs, it allows for examples the results for the DFT to be lifted to higher-dimensional Fourier transforms. We also note that the closure bound in Lemma H.7 can be tightened in such cases (cfRemark H.4). ˙ ", + "bbox": [ + 173, + 494, + 825, + 553 + ], + "page_idx": 29 + }, + { + "type": "text", + "text": "Lemma H.7. Let A, $\\mathbf { B } \\in \\mathbb { F } ^ { n \\times n }$ . I $^ { c } \\mathbf { A } \\in ( B B ^ { * } ) _ { e } ^ { w _ { 1 } }$ and $\\mathbf { B } \\in ( B B ^ { * } ) _ { e } ^ { w _ { 2 } }$ then $\\mathbf { A } \\otimes \\mathbf { B } \\in ( B B ^ { * } ) _ { e } ^ { w _ { 1 } + w _ { 2 } + 6 }$ ", + "bbox": [ + 171, + 556, + 823, + 574 + ], + "page_idx": 29 + }, + { + "type": "text", + "text": "Proof. Note that ", + "bbox": [ + 174, + 592, + 285, + 606 + ], + "page_idx": 29 + }, + { + "type": "equation", + "img_path": "images/0c1a34063ecb12b25aafe14a72e3df94b6b00236c59aab6a836d70d94dd574d8.jpg", + "text": "$$\n\\mathbf { A } \\otimes \\mathbf { B } = ( \\mathbf { A } \\otimes \\mathbf { I } ) ( \\mathbf { I } \\otimes \\mathbf { B } ) = \\mathbf { P } ^ { - 1 } ( \\mathbf { I } \\otimes \\mathbf { A } ) \\mathbf { P } ( \\mathbf { I } \\otimes \\mathbf { B } ) ,\n$$", + "text_format": "latex", + "bbox": [ + 320, + 604, + 674, + 623 + ], + "page_idx": 29 + }, + { + "type": "text", + "text": "for some permutation $\\mathbf { P }$ . By Lemma H.3, $\\mathbf { I } \\otimes \\mathbf { A }$ and $\\mathbf { I } \\otimes \\mathbf { B }$ are in $( B B ^ { * } ) _ { e } ^ { w _ { 1 } + 2 } , ( B B ^ { * } ) _ { e } ^ { w _ { 2 } + 2 }$ respectively. The result follows from combining with $\\mathbf { P } \\in B B ^ { * }$ and Lemma H.2. □ ", + "bbox": [ + 173, + 627, + 823, + 657 + ], + "page_idx": 29 + }, + { + "type": "text", + "text": "I SPARSE MATRICES IN $B B ^ { * }$ HIERARCHY ", + "text_level": 1, + "bbox": [ + 174, + 678, + 531, + 695 + ], + "page_idx": 29 + }, + { + "type": "text", + "text": "In this appendix, we prove Theorem 3. First, we consider matrices with at most $n$ NNZ. ", + "bbox": [ + 173, + 710, + 746, + 726 + ], + "page_idx": 29 + }, + { + "type": "text", + "text": "Lemma I.1. let S be an $n \\times n$ matrix with at most n NNZ. Then, $\\mathbf { S } \\in ( B B ^ { * } ) ^ { 4 }$ . ", + "bbox": [ + 174, + 729, + 692, + 746 + ], + "page_idx": 29 + }, + { + "type": "text", + "text": "We use this lemma and the addition closure lemma to prove Theorem 3. ", + "bbox": [ + 173, + 757, + 643, + 772 + ], + "page_idx": 29 + }, + { + "type": "text", + "text": "Proof of Theorem 3. We note that any $s$ sparse matrix is the sum of $\\textstyle { \\left\\lceil { \\frac { s } { n } } \\right\\rceil }$ matrices of at most $n$ NNZ, and we appeal to Lemma H.5. □ ", + "bbox": [ + 173, + 790, + 825, + 820 + ], + "page_idx": 29 + }, + { + "type": "text", + "text": "In the rest of the section we will prove Lemma I.1. We begin by defining two classes of matrices that will be used in our decomposition. ", + "bbox": [ + 173, + 839, + 823, + 869 + ], + "page_idx": 29 + }, + { + "type": "text", + "text": "Definition I.1. An $n \\times n$ matrix $\\mathbf { H }$ is a horizontal step matrix if for every $0 \\leq i , i ^ { \\prime } < n$ and $0 \\leq j \\leq j ^ { \\prime } < n ,$ , $i f \\mathbf { H } [ i , j ] \\neq 0$ and $\\mathbf { H } [ i ^ { \\prime } , j ^ { \\prime } ] \\neq 0$ , then $j ^ { \\prime } - j \\geq ( i ^ { \\prime } - i )$ mod $n$ . ", + "bbox": [ + 171, + 873, + 823, + 904 + ], + "page_idx": 29 + }, + { + "type": "text", + "text": "An $n \\times n$ matrix $\\mathbf { V }$ is a vertical step matrix if $\\mathbf { V } ^ { * }$ is a horizontal step matrix. ", + "bbox": [ + 173, + 909, + 679, + 924 + ], + "page_idx": 29 + }, + { + "type": "text", + "text": "With this definition, the horizontal step matrix obeys a “Lipschitz-like\" condition. Each column of a horizontal step matrix can have at most one non-zero entry, and given two non-zero columns $k$ apart, the non-zero entry in the right column must be between 0 and $k$ rows below the non-zero entry in the left column. Note that to show that a matrix is a horizontal step matrix, it is sufficient to argue that this condition holds for each pair of neighboring non-zero columns. ", + "bbox": [ + 173, + 103, + 825, + 174 + ], + "page_idx": 30 + }, + { + "type": "text", + "text": "Similarly, each row of a vertical step matrix can have at most one non-zero entry, and given two non-zero rows $k$ apart, the non-zero entry in the lower row must be between 0 and $k$ columns to the right of the non-zero entry in the upper row. ", + "bbox": [ + 176, + 180, + 821, + 223 + ], + "page_idx": 30 + }, + { + "type": "text", + "text": "Lemma I.2. Let $\\mathbf { H }$ be an $n \\times n$ horizontal step matrix. Then $\\mathbf { H } \\in B$ . ", + "bbox": [ + 173, + 227, + 630, + 242 + ], + "page_idx": 30 + }, + { + "type": "text", + "text": "Proof. We proceed by induction on $n$ . The base case $n = 2$ is trivial. As our inductive hypothesis, we assume that all horizontal step matrices of size ${ \\begin{array} { l } { { \\frac { n } { 2 } } } \\end{array} } \\times { \\begin{array} { l } { { \\frac { n } { 2 } } } \\end{array} }$ are butterfly matrices of size $\\frac { n } { 2 }$ . From Definition 2.3, it is sufficient to show that $\\mathbf { H }$ can be decomposed as: ", + "bbox": [ + 174, + 258, + 826, + 301 + ], + "page_idx": 30 + }, + { + "type": "equation", + "img_path": "images/41b679b92a8648bde29875077bb4aadc567c0ebffdcbcfa75f99146bf2323c56.jpg", + "text": "$$\n\\mathbf { H } = \\left[ \\mathbf { D } _ { 1 } \\quad \\mathbf { D } _ { 2 } \\right] \\left[ \\mathbf { H } _ { 1 } \\quad \\mathbf { \\Lambda } _ { \\mathbf { H } _ { 2 } } ^ { 0 } \\right] = \\left[ \\mathbf { D } _ { 1 } \\mathbf { H } _ { 1 } \\quad \\mathbf { D } _ { 2 } \\mathbf { H } _ { 2 } \\right] ,\n$$", + "text_format": "latex", + "bbox": [ + 323, + 308, + 671, + 344 + ], + "page_idx": 30 + }, + { + "type": "text", + "text": "where $\\mathbf { H } _ { 1 } , \\mathbf { H } _ { 2 }$ are ${ \\frac { n } { 2 } } \\times { \\frac { n } { 2 } }$ horizontal step matrices and each $\\mathbf { D } _ { k }$ is a ${ \\begin{array} { l } { { \\frac { n } { 2 } } \\times { \\frac { n } { 2 } } } \\end{array} }$ diagonal matrix. Denote the four, ${ \\frac { n } { 2 } } \\times { \\frac { n } { 2 } }$ 2 2corner submatrices of $\\mathbf { H }$ by: ", + "bbox": [ + 173, + 349, + 825, + 380 + ], + "page_idx": 30 + }, + { + "type": "equation", + "img_path": "images/2799457ad1bb24d7d3b9c7331cf3acfd156f715492a161f9c38b10a93519b2fb.jpg", + "text": "$$\n\\mathbf { H } = \\left[ \\begin{array} { c c } { \\mathbf { H } _ { 1 1 } } & { \\mathbf { H } _ { 1 2 } } \\\\ { \\mathbf { H } _ { 2 1 } } & { \\mathbf { H } _ { 2 2 } } \\end{array} \\right] .\n$$", + "text_format": "latex", + "bbox": [ + 429, + 387, + 568, + 422 + ], + "page_idx": 30 + }, + { + "type": "text", + "text": "Then, define $\\mathbf { H } _ { 1 }$ and $\\mathbf { H } _ { 2 }$ by: ", + "bbox": [ + 174, + 428, + 362, + 443 + ], + "page_idx": 30 + }, + { + "type": "equation", + "img_path": "images/014c86d8c88f60d7ad251673a7611e1e0bfe27696a07b76b977f91c2e29e1d9d.jpg", + "text": "$$\n\\mathbf { H } _ { 1 } = \\mathbf { H } _ { 1 1 } + \\mathbf { H } _ { 2 1 } \\qquad \\mathbf { H } _ { 2 } = \\mathbf { H } _ { 1 2 } + \\mathbf { H } _ { 2 2 }\n$$", + "text_format": "latex", + "bbox": [ + 351, + 450, + 647, + 468 + ], + "page_idx": 30 + }, + { + "type": "text", + "text": "For sake of contradiction, assume that $\\mathbf { H } _ { 1 }$ is not a horizontal step matrix. Then, there are $\\textstyle { 0 \\leq i , i ^ { \\prime } < { \\frac { n } { 2 } } }$ , $\\begin{array} { r } { 0 \\leq j \\leq j ^ { \\prime } < \\frac { n } { 2 } } \\end{array}$ such that $\\mathbf { H } _ { 1 } [ i , j ] \\neq 0$ , $\\mathbf { H } _ { 1 } [ i ^ { \\prime } , j ^ { \\prime } ] \\neq 0$ , and $j ^ { \\prime } - j < ( i ^ { \\prime } - i )$ mod $\\frac { n } { 2 }$ . From our definition of $\\mathbf { H } _ { 1 }$ , the non-zero entries in columns $j$ and $j ^ { \\prime }$ of $\\mathbf { H }$ are either $\\left( \\left( i ^ { \\prime } - i \\right) \\mod { \\frac { n } { 2 } } \\right)$ or $\\begin{array} { r } { \\left( \\frac { n } { 2 } + ( i ^ { \\prime } - i ) \\right. } \\end{array}$ mod ${ \\frac { n } { 2 } } .$ \u0001, both of which are greater than $j ^ { \\prime } - j$ , rows apart. This contradicts $\\mathbf { H }$ being a horizontal step matrix. Hence, $\\mathbf { H } _ { 1 }$ must be a horizontal step matrix, as must $\\mathbf { H } _ { 2 }$ from an analogous argument. ", + "bbox": [ + 171, + 481, + 826, + 571 + ], + "page_idx": 30 + }, + { + "type": "text", + "text": "Next, we define $\\mathbf { D } _ { 1 } , \\mathbf { D } _ { 2 } , \\mathbf { D } _ { 3 } , \\mathbf { D } _ { 4 }$ by: ", + "bbox": [ + 174, + 577, + 416, + 593 + ], + "page_idx": 30 + }, + { + "type": "equation", + "img_path": "images/2c99e29e47b32aab9acdbf1b8bdf0f32c95db2c94933646bad236e9aec4c80cd.jpg", + "text": "$$\n\\begin{array} { r l r l } & { \\mathbf { D } _ { 1 } [ k , k ] = \\left\\{ \\begin{array} { l l } { 1 } & { \\mathbf { H } _ { 2 1 } [ k , : ] = \\mathbf { 0 } } \\\\ { 0 } & { \\mathrm { o t h e r w i s e } } \\end{array} \\right. } & & { \\mathbf { D } _ { 2 } [ k , k ] = \\left\\{ \\begin{array} { l l } { 1 } & { \\mathbf { H } _ { 2 2 } [ k , : ] = \\mathbf { 0 } } \\\\ { 0 } & { \\mathrm { o t h e r w i s e } } \\end{array} \\right. } \\\\ & { \\mathbf { D } _ { 3 } [ k , k ] = \\left\\{ \\begin{array} { l l } { 1 } & { \\mathbf { H } _ { 1 1 } [ k , : ] = \\mathbf { 0 } } \\\\ { 0 } & { \\mathrm { o t h e r w i s e } . } \\end{array} \\right. } & & { \\mathbf { D } _ { 4 } [ k , k ] = \\left\\{ \\begin{array} { l l } { 1 } & { \\mathbf { H } _ { 1 2 } [ k , : ] = \\mathbf { 0 } } \\\\ { 0 } & { \\mathrm { o t h e r w i s e } . } \\end{array} \\right. } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 263, + 601, + 733, + 683 + ], + "page_idx": 30 + }, + { + "type": "text", + "text": "To finish the proof, we argue the correctness of the decomposition by equating arbitrary entries of each of the 4 corner submatrices. We begin with the upper left submatrix. ", + "bbox": [ + 171, + 693, + 825, + 722 + ], + "page_idx": 30 + }, + { + "type": "equation", + "img_path": "images/009086f621b094da4ef14bfe8e5b883b0e60b84b6a2ca6fadf45b409a63056ab.jpg", + "text": "$$\n\\begin{array} { r l r } { \\mathbf { D } _ { 1 } \\mathbf { H } _ { 1 } \\big [ i , j \\big ] = \\displaystyle \\sum _ { k = 0 } ^ { \\frac { n } { 2 } } \\mathbf { D } _ { 1 } [ i , k ] \\cdot \\mathbf { H } _ { 1 } \\big [ k , j \\big ] \\quad } & { \\mathrm { b y ~ d e f i n i t i o n ~ o f ~ m a t r i x ~ m u l t i p l i c a t i o n } } \\\\ { = \\mathbf { D } _ { 1 } \\big [ i , i \\big ] \\cdot \\mathbf { H } _ { 1 } \\big [ i , j \\big ] \\quad } & { \\mathbf { D } _ { 1 } \\mathrm { ~ i s ~ a d i a g o n a l ~ m a t r i x } } \\\\ { = \\mathbb { 1 } _ { ( \\mathbf { H } _ { 2 1 } [ i , : ] = 0 ) } \\cdot \\left( \\mathbf { H } _ { 1 1 } [ i , j ] + \\mathbf { H } _ { 2 1 } [ i , j ] \\right) \\quad } & { \\mathrm { b y ~ d e f i n i t i o n ~ o f ~ \\mathbf { D } _ 1 ~ a n d ~ \\mathbf { H } _ 1 ~ } } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 194, + 728, + 815, + 814 + ], + "page_idx": 30 + }, + { + "type": "text", + "text": "Here, we consider two cases: ", + "bbox": [ + 174, + 818, + 366, + 832 + ], + "page_idx": 30 + }, + { + "type": "text", + "text": "Case $1 : \\mathbf { H } _ { 2 1 } [ i , j ] \\neq 0$ ", + "bbox": [ + 173, + 837, + 326, + 854 + ], + "page_idx": 30 + }, + { + "type": "text", + "text": "Since $\\mathbf { H }$ is a horizontal step matrix (and hence may have at most one non-zero entry per column), it follows that $\\mathbf { H } _ { 1 1 } [ i , j ] = 0$ . In this case, the indicator function evaluates to 0, so $\\mathbf { D } _ { 1 } \\mathbf { H } _ { 1 } [ i , j ] = 0 =$ $\\mathbf { H } _ { 1 1 } [ i , j ]$ , as desired. ", + "bbox": [ + 174, + 861, + 825, + 902 + ], + "page_idx": 30 + }, + { + "type": "text", + "text": "Case $2 \\colon \\mathbf { H } _ { 2 1 } [ i , j ] = 0$ ", + "bbox": [ + 174, + 907, + 326, + 925 + ], + "page_idx": 30 + }, + { + "type": "text", + "text": "If $\\mathbf { H } _ { 1 1 } [ i , j ] = 0$ , then ${ \\bf D } _ { 1 } { \\bf H } _ { 1 } [ i , j ] = 0 = { \\bf H } _ { 1 1 } [ i , j ]$ . Otherwise, for sake of contradiction, suppose that $\\mathbf { H } _ { 2 1 } [ i , : ] \\neq \\mathbf { 0 }$ . Then, two of the first $\\frac { n } { 2 }$ columns of $\\mathbf { H }$ would have non-zero entries $\\frac { n } { 2 }$ rows apart, contradicting $\\mathbf { H }$ being a horizontal step matrix. Hence, $\\mathbf { H } _ { 2 1 } [ i , : ] = \\mathbf { 0 }$ , so $\\mathbf D _ { 1 } \\mathbf H _ { 1 } [ i , j ] = \\mathbf { \\bar { H } } _ { 1 1 } [ i , j ]$ , as desired. ", + "bbox": [ + 173, + 102, + 825, + 160 + ], + "page_idx": 31 + }, + { + "type": "text", + "text": "In all cases, $\\mathbf { D } _ { 1 } \\mathbf { H } _ { 1 } [ i , j ] = \\mathbf { H } _ { 1 1 } [ i , j ]$ , so our decomposition correctly recovers the upper left corner of $\\mathbf { H }$ . Analogous arguments show that the other three corners are also correctly recovered. Hence, our decomposition is correct, and by induction, $\\mathbf { H } \\in B$ . □ ", + "bbox": [ + 173, + 165, + 826, + 208 + ], + "page_idx": 31 + }, + { + "type": "text", + "text": "Corollary I.3. Let $\\mathbf { V }$ be a vertical step matrix. Then $\\mathbf { V } \\in B ^ { * }$ . ", + "bbox": [ + 173, + 217, + 583, + 232 + ], + "page_idx": 31 + }, + { + "type": "image", + "img_path": "images/d4daf5b9090c82cf9774fe7121e9c63dac335d55f5330daff1bf763ab1367e88.jpg", + "image_caption": [ + "Figure 9: Decomposition of $4 \\times 4$ sparse matrix S into $\\mathbf { P } _ { 1 } \\mathbf { H } \\mathbf { P } _ { 2 } \\mathbf { V } \\mathbf { P } _ { 3 }$ " + ], + "image_footnote": [], + "bbox": [ + 279, + 252, + 717, + 617 + ], + "page_idx": 31 + }, + { + "type": "text", + "text": "Now, we use step matrices to prove Lemma I.1. ", + "bbox": [ + 174, + 676, + 485, + 691 + ], + "page_idx": 31 + }, + { + "type": "text", + "text": "Proof of Lemma I.1. Given S, we decompose it as $\\mathbf { S } = \\mathbf { P } _ { 1 } \\mathbf { H } \\mathbf { P } _ { 2 } \\mathbf { V } \\mathbf { P } _ { 3 }$ , where each $P _ { \\ell }$ is a permutation matrix, $\\mathbf { H }$ is a horizontal step matrix, and $\\mathbf { V }$ is a vertical step matrix. For an example of this, see Figure 9. ", + "bbox": [ + 174, + 705, + 825, + 748 + ], + "page_idx": 31 + }, + { + "type": "text", + "text": "We first decompose S as $\\mathbf { S } = \\mathbf { P } _ { 1 } \\mathbf { S } ^ { \\prime } \\mathbf { P } _ { 3 }$ , where ${ \\bf P } _ { 1 }$ is the permutation that moves all 0 rows of S to the bottom and $\\mathbf { P } _ { 3 }$ is the permutation that moves all 0 columns of S to the right. ", + "bbox": [ + 171, + 755, + 823, + 785 + ], + "page_idx": 31 + }, + { + "type": "text", + "text": "Next, we further decompose $\\mathbf { S } ^ { \\prime }$ into $\\mathbf { S } ^ { \\prime } = \\mathbf { H } \\mathbf { V } ^ { \\prime }$ as follows. Since $\\mathbf { S } ^ { \\prime }$ has $s \\leq n$ NNZ, we can parameterize $\\mathbf { S } ^ { \\prime }$ by $\\theta = \\{ ( c _ { k } , i _ { k } , j _ { k } ) : 0 \\leq k < s \\}$ such that $\\mathbf { S } ^ { \\prime } [ i _ { k } , j _ { k } ] = c _ { k }$ , with the non-zero entries indexed in row-major order. Define matrix $\\mathbf { H }$ by: ", + "bbox": [ + 174, + 790, + 823, + 833 + ], + "page_idx": 31 + }, + { + "type": "equation", + "img_path": "images/4a1ff679031ec8990603c7ed3b5b69f99e487f6368838e8c33aec83fdf86f09c.jpg", + "text": "$$\n\\mathbf { H } [ : , k ] = { \\left\\{ \\begin{array} { l l } { c _ { k } \\cdot \\mathbf { e } _ { i _ { k } } } & { 0 \\leq k < s } \\\\ { \\mathbf { 0 } } & { { \\mathrm { o t h e r w i s e } } . } \\end{array} \\right. }\n$$", + "text_format": "latex", + "bbox": [ + 390, + 838, + 606, + 873 + ], + "page_idx": 31 + }, + { + "type": "text", + "text": "Define matrix $\\mathbf { V } ^ { \\prime }$ by: ", + "bbox": [ + 173, + 877, + 313, + 892 + ], + "page_idx": 31 + }, + { + "type": "equation", + "img_path": "images/ff8bcbe3a01291f337438a2b037c0036484ae2cf7969d8640388c124be6054a7.jpg", + "text": "$$\n\\begin{array} { r } { \\mathbf { V } ^ { \\prime } [ k , : ] = \\left\\{ \\begin{array} { l l } { \\mathbf { e } _ { j _ { k } } ^ { T } } & { 0 \\leq k < s } \\\\ { \\mathbf { 0 } } & { \\mathrm { o t h e r w i s e } . } \\end{array} \\right. } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 400, + 891, + 596, + 928 + ], + "page_idx": 31 + }, + { + "type": "text", + "text": "To show that $\\mathbf { S } ^ { \\prime } = \\mathbf { H } \\mathbf { V } ^ { \\prime }$ , we consider an arbitrary entry: ", + "bbox": [ + 173, + 102, + 542, + 118 + ], + "page_idx": 32 + }, + { + "type": "equation", + "img_path": "images/9a0ee6522ceafb8907365528743cc60af391240dfdb2856dd136af9d3fa79b76.jpg", + "text": "$$\n\\begin{array} { r l } { \\mathbf { H } \\mathbf { V } ^ { \\prime } [ i , j ] = \\displaystyle \\sum _ { k = 0 } ^ { n } \\mathbf { H } [ i , k ] \\cdot \\mathbf { V } ^ { \\prime } [ k , j ] \\qquad } & { \\mathrm { ~ b y ~ d e f i n i t i o n ~ o f ~ m a t r i x ~ m u l t i p l i c a t i o n } } \\\\ { = \\displaystyle \\sum _ { k = 0 } ^ { s } \\mathbf { H } [ i , k ] \\cdot \\mathbf { V } ^ { \\prime } [ k , j ] \\qquad } & { \\mathrm { ~ H ~ i s ~ 0 ~ i n ~ a l l ~ b u t ~ f i r s t ~ s ~ c o l u m n s } } \\\\ { = \\displaystyle \\sum _ { k = 0 } ^ { s } c _ { k } \\cdot \\mathbb { I } _ { i = i _ { k } } \\cdot \\mathbb { I } _ { j = j _ { k } } \\qquad } & { \\mathrm { ~ b y ~ d e f i n i t i o n ~ o f ~ } \\mathbf { H } \\mathrm { ~ a n d ~ } \\mathbf { V } ^ { \\prime } } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 230, + 126, + 777, + 253 + ], + "page_idx": 32 + }, + { + "type": "text", + "text": "Here, we note that $( i , j )$ can equal $( i _ { k } , j _ { k } )$ for at most one value of $k$ since the locations in $\\theta$ are unique. Hence, $\\mathbf { H } \\mathbf { V } ^ { \\prime } [ i , j ] = c _ { k }$ only if $( i , j ) = ( i _ { k } , j _ { k } )$ for some $k$ , which is exactly the definition of $\\mathbf { S } ^ { \\prime }$ . Hence, $\\mathbf { S } ^ { \\prime } = \\mathbf { H } \\mathbf { V } ^ { \\prime }$ . ", + "bbox": [ + 174, + 260, + 825, + 303 + ], + "page_idx": 32 + }, + { + "type": "text", + "text": "We argue that $\\mathbf { H }$ is a horizontal step matrix through a series of assertions. First, note that $\\mathbf { H }$ has exactly one non-zero entry in each of its first $s$ columns. Also, note that since $\\theta$ is in row-major order, these non-zero entries are sorted (any column to the right cannot have a non-zero entry in a higher row). Hence, to show that $\\mathbf { H }$ is a horizontal step matrix, it is sufficient to argue that adjacent columns of $\\mathbf { H }$ have non-zero entries at most one row apart. This is equivalent to $\\mathbf { S } ^ { \\prime }$ having no zero rows between two non-zero rows, which is guaranteed by ${ \\bf P } _ { 1 }$ . Hence, $\\mathbf { H }$ is a horizontal step matrix. ", + "bbox": [ + 173, + 309, + 825, + 393 + ], + "page_idx": 32 + }, + { + "type": "text", + "text": "Since $\\mathbf { V } ^ { \\prime }$ has at most one non-zero entry per row, we may permute the rows of $\\mathbf { V } ^ { \\prime }$ to obtain a matrix $\\mathbf { V }$ , where the non-zero entries of $\\mathbf { V }$ are sorted (any lower row below cannot have a non-zero entry in an earlier column). Hence, for some permutation matrix $( \\mathbf { P } _ { 2 } ) ^ { - 1 }$ , $\\mathbf { V } = ( \\mathbf { P } _ { 2 } ) ^ { - 1 } \\mathbf { V } ^ { \\prime }$ , which implies that $\\mathbf { V } ^ { \\prime } = \\mathbf { P } _ { 2 } \\mathbf { V }$ . It has exactly one non-zero entry in each of its first $s$ columns. From the action of $\\mathbf { P } _ { 2 }$ , these non-zero entries are sorted. Therefore, by the same argument as for $\\mathbf { H }$ above, $\\mathbf { V } ^ { T }$ is a horizontal step matrix. Hence, $\\mathbf { V }$ is a vertical step matrix. ", + "bbox": [ + 173, + 400, + 825, + 486 + ], + "page_idx": 32 + }, + { + "type": "text", + "text": "In all, we have found a decomposition $\\mathbf { S } = \\mathbf { P } _ { 1 } \\mathbf { H } \\mathbf { P } _ { 2 } \\mathbf { V } \\mathbf { P } _ { 3 }$ , where each $P _ { \\ell }$ is a permutation matrix $( \\in B B ^ { * }$ by Theorem 2), $\\mathbf { H }$ is a horizontal step matrix ( $\\in B$ by Lemma I.2), and $\\mathbf { V }$ is a vertical step matrix $( \\in B ^ { * }$ by Corollary I.3). Moreover, by Lemma G.6, $\\mathbf { P } _ { 2 } \\in B ^ { \\ast } B$ , so $\\mathbf { H } , \\mathbf { P } _ { 2 } , \\mathbf { V }$ can be combined to obtain $\\mathbf { H } \\mathbf { P } _ { 2 } \\mathbf { \\bar { V } } \\in ( B B ^ { \\bar { * } } ) ^ { 2 }$ . By Lemma H.2, $\\mathbf { S } \\in ( B B ^ { * } ) ^ { 4 }$ . □ ", + "bbox": [ + 173, + 492, + 825, + 549 + ], + "page_idx": 32 + }, + { + "type": "text", + "text": "Corollary I.4. Let R be an $n \\times n$ matrix of rank $r$ . Then $\\mathbf { R } \\in ( B B ^ { * } ) _ { 4 } ^ { 8 r }$ ", + "bbox": [ + 174, + 560, + 648, + 577 + ], + "page_idx": 32 + }, + { + "type": "text", + "text": "Proof. We can decompose $\\mathbf { R }$ as $\\mathbf { R } = \\mathbf { G } \\mathbf { H } ^ { * }$ where $\\mathbf { G } , \\mathbf { H }$ are $n \\times r$ matrices. With appropriate zero-padding, both of these can be made into $n \\times n$ matrices with at most $r n$ NNZ. The proof follows immediately from Theorem 3 and Lemma H.2. □ ", + "bbox": [ + 174, + 595, + 825, + 638 + ], + "page_idx": 32 + }, + { + "type": "text", + "text": "J EXAMPLE OF K-MATRIX REPRESENTATION OF STRUCTURED MATRICES ANDCOMPARISON TO $B \\mathcal { P }$ HIERARCHY", + "text_level": 1, + "bbox": [ + 176, + 660, + 821, + 694 + ], + "page_idx": 32 + }, + { + "type": "text", + "text": "In this appendix, we show explicitly how some common structured matrices (e.g. originating from fast transforms) can be represented as K-matrices. We also draw comparisons between the $B B ^ { * }$ hierarchy and the $B \\mathcal { P }$ hierarchy introduced by Dao et al. (2019). ", + "bbox": [ + 174, + 710, + 825, + 752 + ], + "page_idx": 32 + }, + { + "type": "text", + "text": "Lemma J.1. Let ${ \\bf F } _ { n }$ be the Discrete Fourier Transform of size $n$ . Then $\\mathbf { F } _ { n } \\in ( B B ^ { * } ) ^ { 2 }$ . ", + "bbox": [ + 174, + 757, + 735, + 772 + ], + "page_idx": 32 + }, + { + "type": "text", + "text": "Proof. From Parker (1995), we can express ${ \\bf F } _ { n }$ as ${ \\bf F } _ { n } = { \\bf B } { \\bf P }$ , where $\\mathbf { B } \\in B$ and $\\mathbf { P }$ is a permutation (the bit reversal permutation). From Theorem 2, $\\mathbf { P } \\in B B ^ { * }$ . Hence, by Lemma H.2, $\\mathbf { F } _ { n } ^ { - } \\in ( B B ^ { * } ) ^ { 2 }$ . ", + "bbox": [ + 173, + 791, + 825, + 823 + ], + "page_idx": 32 + }, + { + "type": "text", + "text": "Lemma J.2. Let $\\mathbf { H } _ { n }$ be the Hadamard Transform of size $n$ . Then $\\mathbf { H } _ { n } \\in B B ^ { * }$ . ", + "bbox": [ + 173, + 847, + 684, + 862 + ], + "page_idx": 32 + }, + { + "type": "text", + "text": "Proof. ${ \\mathbf { H } } _ { n } \\in { \\mathfrak { B } }$ , so trivially $\\mathbf { H } _ { n } \\in B B ^ { * }$ ", + "bbox": [ + 174, + 881, + 437, + 897 + ], + "page_idx": 32 + }, + { + "type": "text", + "text": "Lemma J.3. Let $\\mathbf { S } _ { n }$ be the Discrete Sine Transform of size $n$ . Then $\\mathbf { S } _ { n } \\in ( B B ^ { * } ) ^ { 2 }$ . ", + "bbox": [ + 171, + 909, + 712, + 924 + ], + "page_idx": 32 + }, + { + "type": "text", + "text": "Proof. As described in Makhoul (1980), $\\mathbf { S } _ { n }$ can be performed as a scaled permutation (separating the even and odd indices of the input, and reversing and negating the odd indices) composed with $\\mathbb { F } _ { n }$ . Therefore, we may decompose $\\mathbf { S } _ { n }$ as $\\mathbf { S } _ { n } = \\mathbf { B } ~ \\mathbf { P } _ { 2 } ~ \\mathbf { D } ~ \\mathbf { P } _ { 1 } ,$ , where $\\mathbf { P } _ { 1 } , \\mathbf { P } _ { 2 }$ are permutations, $\\mathbf { B } \\in B$ , and $\\mathbf { D }$ is a diagonal matrix. $\\mathbf { P } _ { 2 }$ D $\\mathbf { P } _ { 1 }$ is simply a permutation matrix with scaled entries, which can be equivalently expressed as $\\mathbf { D } ^ { \\prime } \\mathbf { P } ^ { \\prime }$ for some diagonal matrix $\\mathbf { D ^ { \\prime } }$ and permutation $\\mathbf { P ^ { \\prime } }$ . By Lemma H.1, $\\mathbf { B } \\bar { \\mathbf { D ^ { \\prime } } } \\in B B ^ { * }$ . By Theorem 2, $\\mathbf { P } ^ { \\prime } \\in B B ^ { * }$ . Hence, by Lemma H.2, $\\bar { \\mathbf { S } _ { n } } \\in ( B B ^ { * } ) ^ { 2 }$ . □ ", + "bbox": [ + 173, + 102, + 826, + 188 + ], + "page_idx": 33 + }, + { + "type": "text", + "text": "Remark J.4. An analogous argument shows that the Discrete Cosine Transform is also in $( B B ^ { * } ) ^ { 2 }$ ", + "bbox": [ + 173, + 204, + 818, + 220 + ], + "page_idx": 33 + }, + { + "type": "text", + "text": "Lemma J.5. Let $\\mathbf { C } _ { n }$ be an $n \\times n$ circulant (convolution) matrix. Then $\\mathbf { C } _ { n } \\in B B ^ { * }$ . ", + "bbox": [ + 174, + 226, + 717, + 241 + ], + "page_idx": 33 + }, + { + "type": "text", + "text": "Proof. Using Theorem 2.6.4 of Pan (2001), we can express $\\mathbf { C } _ { n }$ as $\\mathbf { C } _ { n } = ( \\mathbf { F } _ { n } ) ^ { - 1 } \\mathbf { D } \\mathbf { F } _ { n }$ where ${ \\bf F } _ { n }$ is the Discrete Fourier Transform and $\\mathbf { D }$ is a diagonal matrix. $\\left( \\mathbf { F } _ { n } \\right) ^ { - 1 } = \\mathbf { B } \\mathbf { \\Lambda } \\mathbf { P }$ (with $\\mathbf { B } \\in B$ , $\\mathbf { P }$ a permutation), which implies that ${ \\bf F } _ { n } = \\left( { \\bf P } \\right) ^ { - 1 } \\left( { \\bf B } \\right) ^ { - 1 }$ . Therefore ", + "bbox": [ + 173, + 266, + 825, + 315 + ], + "page_idx": 33 + }, + { + "type": "equation", + "img_path": "images/2093a282bede9cd9f85441018ef319fec57948e6c681841b22b1dae5d248b8fc.jpg", + "text": "$$\n{ \\bf C } _ { n } = { \\bf B } { \\bf P } { \\bf D } { \\bf \\Lambda } \\left( { \\bf P } \\right) ^ { - 1 } \\left( { \\bf B } \\right) ^ { - 1 } .\n$$", + "text_format": "latex", + "bbox": [ + 398, + 324, + 599, + 344 + ], + "page_idx": 33 + }, + { + "type": "text", + "text": "The middle three factors have the effect of performing a permutation, scaling each element, and undoing the permutation, which is equivalent to simply scaling by some diagonal matrix $\\mathbf { D ^ { \\prime } }$ . Hence, we are left with ", + "bbox": [ + 174, + 353, + 826, + 396 + ], + "page_idx": 33 + }, + { + "type": "equation", + "img_path": "images/bc6f96999e33bad0feb03e49b8f25e8891094d64091f542f506f8ec7dd10f1e3.jpg", + "text": "$$\n\\mathbf { C } _ { n } = \\mathbf { B } \\mathbf { D } ^ { \\prime } \\left( \\mathbf { B } \\right) ^ { - 1 } .\n$$", + "text_format": "latex", + "bbox": [ + 429, + 396, + 568, + 416 + ], + "page_idx": 33 + }, + { + "type": "text", + "text": "By Lemma H.1, B $\\mathbf { D } ^ { \\prime } \\in B$ . By Lemma H.6, $\\left( \\mathbf { B } \\right) ^ { - 1 } \\in B ^ { * }$ . Hence, $\\mathbf { C } _ { n } \\in B B ^ { * }$ ", + "bbox": [ + 173, + 424, + 687, + 443 + ], + "page_idx": 33 + }, + { + "type": "text", + "text": "Remark J.6. We can expand any $n \\times n$ Toeplitz matrix $\\mathbf { T } _ { n }$ into a $2 n \\times 2 n$ circulant matrix (with upper left $n \\times n$ submatrix equal to $\\mathbf { T } _ { n }$ ). Hence, $\\mathbf { T } _ { n } \\in ( B B ^ { * } ) _ { 2 } ^ { 1 }$ by Lemma J.5. ", + "bbox": [ + 174, + 458, + 826, + 488 + ], + "page_idx": 33 + }, + { + "type": "text", + "text": "The Fastfood matrix class (Le et al., 2013) can be tightly captured in the $B B ^ { * }$ hierarchy: ", + "bbox": [ + 173, + 500, + 750, + 515 + ], + "page_idx": 33 + }, + { + "type": "text", + "text": "Lemma J.7. The product SHDPHB where $\\mathbf { S } , \\mathbf { D } , \\mathbf { B }$ are diagonal matrices, $\\mathbf { H }$ is the Hadamard transform, and $\\mathbf { P }$ is a permutation matrix, is in $( B B ^ { * } ) ^ { 3 }$ . ", + "bbox": [ + 173, + 520, + 826, + 550 + ], + "page_idx": 33 + }, + { + "type": "text", + "text": "Proof. We have shown in Lemma J.2 that $\\mathbf { H } \\in B B ^ { * }$ , and in Theorem 2 that $\\mathbf { P } \\in B B ^ { * }$ . Since $B B ^ { * }$ is closed under diagonal multiplication (Lemma H.1), we conclude that $\\mathbf { S H D P H B } \\in ( B B ^ { * } ) ^ { 3 }$ . ", + "bbox": [ + 173, + 574, + 825, + 603 + ], + "page_idx": 33 + }, + { + "type": "text", + "text": "The two classes of matrices introduced in Moczulski et al. (2016), called AFDF and ACDC, are also tightly captured in the $B B ^ { * }$ hierarchy: ", + "bbox": [ + 176, + 627, + 825, + 657 + ], + "page_idx": 33 + }, + { + "type": "text", + "text": "Lemma J.8. Let $\\mathbf { A } \\mathbf { F } ^ { - 1 } \\mathbf { D } \\mathbf { F }$ be a product of a diagonal matrix A, the inverse Fourier transform $\\mathbf { F } ^ { - 1 }$ another diagonal matrix D, and the Fourier transform F. Then $\\mathbf { A } \\mathbf { F } ^ { - 1 } \\mathbf { D } \\mathbf { F } \\in B B ^ { * }$ . ", + "bbox": [ + 174, + 662, + 823, + 691 + ], + "page_idx": 33 + }, + { + "type": "text", + "text": "Let $\\mathbf { A C } ^ { - 1 } \\mathbf { D C }$ be a product of a diagonal matrix A, the inverse cosine transform $\\mathbf { C } ^ { - 1 }$ , another diagonal matrix D, and the cosine transform C. Then $\\mathbf { A } \\mathbf { C } ^ { - 1 } \\mathbf { D } \\mathbf { C } \\in ( B B ^ { * } ) ^ { 4 }$ . ", + "bbox": [ + 173, + 696, + 825, + 727 + ], + "page_idx": 33 + }, + { + "type": "text", + "text": "Proof. We have argued in Lemma J.5 that $\\mathbf { F } ^ { - 1 } \\mathbf { D } \\mathbf { F } \\in B B ^ { * }$ . Since $B B ^ { * }$ is closed under diagonal multiplication (Lemma H.1), we conclude that $\\mathbf { A } \\mathbf { F } ^ { - 1 } \\mathbf { D } \\mathbf { F } \\in B B ^ { * }$ . ", + "bbox": [ + 174, + 751, + 823, + 780 + ], + "page_idx": 33 + }, + { + "type": "text", + "text": "We have shown that $\\mathbf { C } \\in ( B B ^ { * } ) ^ { 2 }$ , so $\\mathbf { C } ^ { - 1 } \\in ( B B ^ { * } ) ^ { 2 }$ as well. Since $B B ^ { * }$ is closed under diagonal multiplication (Lemma H.1), we conclude that $\\mathbf { A C } ^ { - 1 } \\mathbf { D C } \\in ( B B ^ { * } ) ^ { 4 }$ . ", + "bbox": [ + 173, + 785, + 820, + 816 + ], + "page_idx": 33 + }, + { + "type": "text", + "text": "Remark J.9. Within each butterfly factor matrix of the DFT (excluding the bit reversal permutation) and the Hadamard transform, the columns are pairwise orthogonal and have norm 2. Hence, we can√ divide all factors by $\\sqrt { 2 }$ to make orthogonal factor matrices. To counteract this scaling, we can add a diagonal matrix with 2log2(n) ${ \\sqrt { 2 } } ^ { \\log _ { 2 } ( n ) } = { \\sqrt { n } }$ = n in all entries to the factorization. By doing this we can place all of the above transforms in the OBB hierarchy (defined in Appendix $K _ { \\cdot }$ ) with the same width and expansion factor. ", + "bbox": [ + 173, + 833, + 825, + 924 + ], + "page_idx": 33 + }, + { + "type": "text", + "text": "J.1 MULTI-DIMENSIONAL TRANSFORMS ", + "text_level": 1, + "bbox": [ + 176, + 103, + 464, + 117 + ], + "page_idx": 34 + }, + { + "type": "text", + "text": "Here, we show that, using larger matrices, we are able to similarly capture multi-dimensional versions of the above transforms. ", + "bbox": [ + 171, + 128, + 823, + 159 + ], + "page_idx": 34 + }, + { + "type": "text", + "text": "Lemma J.10. Let ${ \\bf F } _ { n } ^ { 2 }$ be the 2-dimensional Discrete Fourier Transform (represented as an $n ^ { 2 } \\times n ^ { 2 }$ matrix). Then $\\mathbf { F } _ { n } ^ { 2 } \\in \\tilde { ( B B ^ { \\ast } ) } ^ { 2 }$ . ", + "bbox": [ + 174, + 161, + 823, + 190 + ], + "page_idx": 34 + }, + { + "type": "text", + "text": "Proof. The separation property of the 2-D DFT allows us to express its action on an $n \\times n$ matrix as the composition of a 1-D DFT on each of its rows and a 1-D DFT on each of its columns. If we view the 2-D DFT as an $n ^ { 2 } \\times n ^ { 2 }$ matrix, its input and outputs will both be column vectors of size $n ^ { 2 }$ . As our convention, we list the entries of the input vector in the row-major order corresponding to the $n \\times n$ input matrix. Then, we consider the 2-D DFT in four steps, where the first two steps perform the 1-D DFT row-wise, and the second two steps perform the 1-D DFT column-wise: ", + "bbox": [ + 174, + 205, + 825, + 290 + ], + "page_idx": 34 + }, + { + "type": "text", + "text": "Step 1: Permute the columns: ", + "bbox": [ + 174, + 296, + 367, + 310 + ], + "page_idx": 34 + }, + { + "type": "text", + "text": "We permute the columns (with a bit reversal permutation), which performs a bit reversal permutation on each row. Viewing the input as a vector, this step corresponds to left multiplication by a permutation matrix $\\mathbf { P } _ { c }$ that permutes the entries of each chunk of size $n$ of the input vector. Step 2: Multiply each row by a butterfly matrix ", + "bbox": [ + 174, + 318, + 825, + 373 + ], + "page_idx": 34 + }, + { + "type": "text", + "text": "Since the entries of the input were listed in row major order, this step is achieved through multiplication by a block diagonal matrix of $n$ butterfly matrices of size $n$ , which can be viewed as a product of butterfly factor matrices $\\mathbf { B } _ { n } ^ { ( n ^ { 2 } ) } \\dots \\mathbf { B } _ { \\frac { n } { 2 } } ^ { ( n ^ { 2 } ) } \\mathbf { \\dot { B } } _ { 2 } ^ { ( n ^ { 2 } ) }$ . ", + "bbox": [ + 174, + 380, + 825, + 430 + ], + "page_idx": 34 + }, + { + "type": "text", + "text": "Step 3: Permute the rows: ", + "bbox": [ + 174, + 436, + 344, + 452 + ], + "page_idx": 34 + }, + { + "type": "text", + "text": "We permute the rows (with a bit reversal permutation), which performs a bit reversal permutation on each column. This corresponds to left multiplication by a permutation matrix $\\mathbf { P } _ { r }$ . Since we are permuting the rows, $\\mathbf { P } _ { r }$ permutes the entries at the granularity of each $n$ -chunk. Since Steps 1 and 2 each performed an identical computation to each $n$ -chunk we can move this row permutation before Step 2, combining $\\mathbf { P } _ { c }$ and $\\mathbf { P } _ { r }$ into a single permutation $\\mathbf { P }$ . ", + "bbox": [ + 174, + 458, + 825, + 529 + ], + "page_idx": 34 + }, + { + "type": "text", + "text": "Step 4: Multiply each column by a butterfly matrix ", + "bbox": [ + 174, + 535, + 509, + 549 + ], + "page_idx": 34 + }, + { + "type": "text", + "text": "Consider multiplication by the first factor matrix. In each row, this matrix is taking linear combinations of adjacent column entries. In our length- $\\cdot n ^ { 2 }$ vector, these entries will be exactly $n$ indices apart. Therefore this multiplication can be handled by a butterfly factor matrix $\\mathbf { B } _ { 2 n } ^ { ( n ^ { 2 } ) }$ . Similarly, we find that this butterfly multiplication can be expressed as multiplication by a product of butterfly factor matrices $\\mathbf { B } _ { n ^ { 2 } } ^ { ( n ^ { 2 } ) } \\cdot \\cdot . . \\mathbf { B } _ { \\frac { n ^ { 2 } } { 2 } } ^ { ( n ^ { 2 } ) } \\mathbf { B } _ { 2 n } ^ { ( n ^ { 2 } ) }$ . Combined with the factor matrices from Step 2, these form a butterfly matrix $\\mathbf { B }$ of size $n ^ { 2 }$ . ", + "bbox": [ + 173, + 556, + 825, + 656 + ], + "page_idx": 34 + }, + { + "type": "text", + "text": "In all, we see that the 2-D DFT may be realized as multiplication by a permutation matrix $\\mathbf { P }$ followed by multiplication by a butterfly matrix B. The same argument as Lemma J.1 shows that $\\mathbf { F } _ { n } ^ { 2 } \\in ( B B ^ { * } ) ^ { 2 }$ . □ ", + "bbox": [ + 174, + 662, + 825, + 707 + ], + "page_idx": 34 + }, + { + "type": "text", + "text": "Remark J.11. An analogous argument (using the separation property of the respective transforms) can be used to argue that 2-D Discrete Sine and Discrete Cosine transforms are in $( B B ^ { * } ) ^ { 2 }$ , and that 2-D Hadamard Transforms are in $B B ^ { * }$ . ", + "bbox": [ + 176, + 713, + 825, + 756 + ], + "page_idx": 34 + }, + { + "type": "text", + "text": "Lemma J.12. Let ${ \\bf C } _ { n } ^ { 2 }$ be a 2-dimensional convolution matrix. Then $\\mathbf { C } _ { n } ^ { 2 } \\in B B ^ { * }$ ", + "bbox": [ + 174, + 758, + 697, + 775 + ], + "page_idx": 34 + }, + { + "type": "text", + "text": "Proof. We can express a 2-D convis the 2-D Fourier transform and matrix as is the in ${ \\bf C } _ { n } ^ { 2 } = ( { \\bf F } _ { n } ^ { 2 } ) ^ { - 1 } { \\bf D } { \\bf F } _ { n } ^ { 2 }$ , where ansform $\\mathbf { D }$ is diagonal, From the pr ${ \\bf F } _ { n } ^ { 2 }$ $( \\mathbf { F } _ { n } ^ { 2 } ) ^ { - 1 }$ of Lemma J.10, we see that that we can express ${ \\bf F } _ { n } ^ { 2 }$ (and similarly $( \\mathbf { F } _ { n } ^ { 2 } ) ^ { - 1 } )$ as the product of a butterfly matrix and a permutation matrix. The rest of the argument is analogous to the proof of Lemma J.5. □ ", + "bbox": [ + 173, + 789, + 825, + 859 + ], + "page_idx": 34 + }, + { + "type": "text", + "text": "Remark J.13. Using an inductive argument, we can show that all $k$ -dimensional $( k \\in \\mathbb { Z } )$ variants of the above transforms, expressed as $n ^ { k } \\times n ^ { k }$ matrices are contained in $B B ^ { * }$ or $( B B ^ { * } ) ^ { 2 }$ . To do this, we use the separation property of the transforms to break them into a $k - 1$ -dimensional transform (the inductive hypothesis) followed by a $^ { l }$ -dimensional transform. ", + "bbox": [ + 174, + 867, + 825, + 924 + ], + "page_idx": 34 + }, + { + "type": "text", + "text": "K THE ORTHOGONAL KALEIDOSCOPE HIERARCHY ", + "text_level": 1, + "bbox": [ + 174, + 102, + 614, + 118 + ], + "page_idx": 35 + }, + { + "type": "text", + "text": "Through practical application of the butterfly matrices, it has been found useful to constrain them in orthogonality. In Section K.1 we will modify the existing kaleidoscope hierarchy to create the orthogonal kaleidoscope hierarchy OBB. Then, in Section K.2, we will argue that all orthogonal matrices, and as a result all matrices, can also be expressed in this hierarchy in $O ( n )$ width. Lastly, in Section K.3, we will argue that permutation matrices and sparse matrices also exist in this hierarchy in $O ( 1 )$ width, which in turn implies a corresponding result for matrices with low-depth arithmetic circuits. ", + "bbox": [ + 174, + 132, + 825, + 231 + ], + "page_idx": 35 + }, + { + "type": "text", + "text": "K.1 DEFINITION ", + "text_level": 1, + "bbox": [ + 174, + 247, + 302, + 262 + ], + "page_idx": 35 + }, + { + "type": "text", + "text": "The definition of the orthogonal butterfly is identical to the original butterfly, with the constraint that all butterfly factors are orthogonal. We specify this definition below: ", + "bbox": [ + 174, + 273, + 823, + 303 + ], + "page_idx": 35 + }, + { + "type": "text", + "text": "Definition K.1 (Analog of Definition 2.1). An orthogonal butterfly factor of size $k \\geq 2$ (denoted as $\\widetilde { \\mathbf B } _ { k }$ ) is a butterfly factor that is also orthogonal. ", + "bbox": [ + 176, + 306, + 820, + 337 + ], + "page_idx": 35 + }, + { + "type": "text", + "text": "Definition K.2 (Analog of Definition 2.3). An orthogonal butterfly matrix of size $n$ (denoted as $\\widetilde { \\mathbf { B } } ^ { ( n ) }$ ) is a butterfly matrix with all butterfly factor matrices being orthogonal. ", + "bbox": [ + 176, + 340, + 823, + 372 + ], + "page_idx": 35 + }, + { + "type": "text", + "text": "Note that the above definition implies that an orthogonal butterfly matrix, as well as its conjugate transpose, is orthogonal. ", + "bbox": [ + 176, + 382, + 823, + 411 + ], + "page_idx": 35 + }, + { + "type": "text", + "text": "The orthogonal hierarchy definition nearly mimics the original hierarchy Definition 2.4, as follows: ", + "bbox": [ + 176, + 417, + 820, + 434 + ], + "page_idx": 35 + }, + { + "type": "text", + "text": "Definition K.3. ", + "text_level": 1, + "bbox": [ + 174, + 436, + 277, + 452 + ], + "page_idx": 35 + }, + { + "type": "text", + "text": "• We say that an $n \\times n$ matrix $\\mathbf { M } \\in \\widetilde { B }$ if we can express $\\mathbf { M } = \\widetilde { \\mathbf { B } } ^ { ( n ) }$ . • We say that an $n \\times n$ matrix $\\mathbf { M } \\in \\widetilde { B } ^ { \\ast }$ if we can express $\\mathbf { M } = \\left[ \\widetilde { \\mathbf { B } } ^ { ( n ) } \\right] ^ { * }$ . • We say that an $n \\times n$ matrix $\\mathbf { M } \\in \\mathcal { O B B }$ if we can express $\\mathbf { M } = \\mathbf { M } _ { 1 } \\mathbf { D } \\mathbf { M } _ { 2 }$ for some $\\mathbf { M } _ { 1 } \\in \\widetilde { B } , \\mathbf { M } _ { 2 } \\in \\widetilde { B } ^ { * }$ , and diagonal matrix D. Note that $\\mathbf { D }$ need not be full rank. • Width w and expansion e in $( O B B ) _ { e } ^ { w }$ mimic the same definition as in the original hierarchy, using OBB instead of $B B ^ { * }$ , such that $\\mathbf { E } \\in ( \\mathcal { O } B B ) ^ { w }$ . ", + "bbox": [ + 212, + 460, + 826, + 590 + ], + "page_idx": 35 + }, + { + "type": "text", + "text": "By padding if necessary, we will assume that $n$ is a power of 2. ", + "bbox": [ + 173, + 601, + 588, + 614 + ], + "page_idx": 35 + }, + { + "type": "text", + "text": "K.2 EXPRESSIVITY ", + "text_level": 1, + "bbox": [ + 174, + 631, + 321, + 646 + ], + "page_idx": 35 + }, + { + "type": "text", + "text": "In this subsection we prove that all orthogonal (resp. unitary) matrices are contained in $O B B ^ { n }$ . To do this, we consider the class of Householder reflections, given by $\\mathbf { I } - 2 \\mathbf { u } \\mathbf { u } ^ { * }$ for any unit vector u (Householder, 1958): ", + "bbox": [ + 174, + 657, + 825, + 699 + ], + "page_idx": 35 + }, + { + "type": "text", + "text": "Lemma K.1. All Householder reflections are in OBB with inner diagonal matrix I. ", + "bbox": [ + 173, + 703, + 725, + 719 + ], + "page_idx": 35 + }, + { + "type": "text", + "text": "We will prove this lemma shortly. First, we use this lemma to present a decomposition for all orthogonal (resp. unitary) matrices. ", + "bbox": [ + 173, + 729, + 821, + 758 + ], + "page_idx": 35 + }, + { + "type": "text", + "text": "Lemma K.2. Let M be an $n \\times n$ orthogonal/unitary matrix. Then $\\mathbf { M } \\in ( O B B ) ^ { n - 1 }$ . ", + "bbox": [ + 178, + 762, + 735, + 777 + ], + "page_idx": 35 + }, + { + "type": "text", + "text": "Proof. We consider the QR decomposition of M. It is known that we can compose $\\mathbf { M }$ into a product of $n - 1$ Householder reflections and an orthogonal/unitary diagonal matrix (Householder, 1958).10 From Lemma K.1, each Householder reflection is in $o s B$ . ", + "bbox": [ + 174, + 791, + 825, + 835 + ], + "page_idx": 35 + }, + { + "type": "text", + "text": "To complete the proof, we argue that $\\mathbf { R }$ can be folded into the rightmost butterfly matrix. Let $\\mathbf { Q } _ { 1 }$ be the rightmost butterfly factor matrix in $\\mathbf { Q } ~ ( \\in \\widetilde { \\mathbf { B } } _ { n } ^ { ( n ) } )$ . Right multiplication of $\\mathbf { Q } _ { 1 }$ by $\\mathbf { R }$ scales each columns of $\\mathbf { Q } _ { 1 }$ by some $c \\in \\mathbb { C }$ with $| | c | | = 1$ ( $\\mathbf { R }$ is unitary diagonal). This preserves both the sparsity ", + "bbox": [ + 174, + 842, + 825, + 888 + ], + "page_idx": 35 + }, + { + "type": "text", + "text": "pattern of $\\mathbf { Q } _ { 1 }$ and the orthogonality of its columns. Moreover, the norm of each column of $\\mathbf { Q } _ { 1 } \\mathbf { R }$ is 1. \nTherefore, $\\mathbf { Q } _ { 1 } \\mathbf { R }$ is an orthogonal butterfly factor matrix, so $\\mathbf { M } = \\mathbf { Q } \\mathbf { R } \\in ( { \\mathcal { O } } B B ) ^ { n - 1 }$ , as desired. ", + "bbox": [ + 171, + 103, + 825, + 133 + ], + "page_idx": 36 + }, + { + "type": "text", + "text": "We now return to the proof of Lemma K.1 ", + "bbox": [ + 176, + 147, + 450, + 161 + ], + "page_idx": 36 + }, + { + "type": "text", + "text": "Proof of Lemma K.1. Given $\\mathbf { u } \\in \\mathbb { C } ^ { n }$ $\\scriptstyle { \\dot { n } }$ a power of 2), let $\\mathbf { u } _ { 0 } = \\mathbf { u } [ : n / 2 ] \\in \\mathbb { C } ^ { n / 2 } , \\mathbf { u } _ { 1 } = \\mathbf { u } [ n / 2 : ] \\in$ $\\mathbb { C } ^ { n / 2 }$ denote the first and second halves of $\\mathbf { u }$ . ", + "bbox": [ + 173, + 176, + 823, + 208 + ], + "page_idx": 36 + }, + { + "type": "text", + "text": "To show that $\\mathbf { H } \\in \\mathcal { O } B B$ with inner diagonal matrix $\\mathbf { I }$ , we proceed by induction. The base case for $n = 2$ is trivial. It suffices to show that there exist unitary butterfly factors $\\mathbf { L } , \\mathbf { R }$ such that LHR has the form $\\begin{array}{c} \\begin{array} { r l } { \\left[ { \\displaystyle { \\bf { I } } _ { n / 2 } - 2 { \\bf { v } } _ { 0 } { \\bf { v } } _ { 0 } ^ { * } } & { { \\bf { 0 } } } \\\\ { { \\bf { 0 } } } & { { \\bf { I } } _ { n / 2 } - 2 { \\bf { v } } _ { 1 } { \\bf { v } } _ { 1 } ^ { * } } \\end{array} \\right] } & { { } } \\end{array}$ for some unit vectors $\\mathbf { v } _ { 0 } , \\mathbf { v } _ { 1 } \\in \\mathbb { C } ^ { n / 2 }$ . ", + "bbox": [ + 173, + 214, + 825, + 276 + ], + "page_idx": 36 + }, + { + "type": "text", + "text": "Define ", + "bbox": [ + 173, + 280, + 220, + 295 + ], + "page_idx": 36 + }, + { + "type": "equation", + "img_path": "images/dab1a7d7eeb5ddd6790a51a75c9d0c076905ff45f6bf6e7282ee24cacc1386ba.jpg", + "text": "$$\n\\left( \\mathbf { v } _ { 0 } [ i ] , \\mathbf { v } _ { 1 } [ i ] \\right) = \\left\\{ \\begin{array} { l l } { \\left( \\frac { \\mathbf { u } _ { 0 } [ i ] } { \\sqrt { | \\mathbf { u } _ { 0 } [ i ] | ^ { 2 } + | \\mathbf { u } _ { 1 } [ i ] | ^ { 2 } } } , \\frac { \\mathbf { u } _ { 1 } [ i ] } { \\sqrt { | \\mathbf { u } _ { 0 } [ i ] | ^ { 2 } + | \\mathbf { u } _ { 1 } [ i ] | ^ { 2 } } } \\right) } & { \\mathrm { ~ i f ~ } | \\mathbf { u } _ { 0 } [ i ] | ^ { 2 } + | \\mathbf { u } _ { 1 } [ i ] | ^ { 2 } \\neq 0 } \\\\ { ( 1 , 0 ) } & { \\mathrm { ~ o t h e r w i s e } } \\end{array} \\right. .\n$$", + "text_format": "latex", + "bbox": [ + 215, + 292, + 782, + 344 + ], + "page_idx": 36 + }, + { + "type": "text", + "text": "It is easily checked that ", + "bbox": [ + 173, + 344, + 330, + 359 + ], + "page_idx": 36 + }, + { + "type": "equation", + "img_path": "images/17934c44a6bc21e7d3e38c4314de2ffa56b998c532c893fbd39c394a4069234c.jpg", + "text": "$$\n\\begin{array} { r l } & { \\mathbf { v } _ { 0 } [ i ] ^ { * } \\mathbf { v } _ { 0 } [ i ] + \\mathbf { v } _ { 1 } [ i ] ^ { * } \\mathbf { v } _ { 1 } [ i ] = 1 } \\\\ & { \\mathbf { v } _ { 0 } [ i ] ^ { * } \\mathbf { u } _ { 0 } [ i ] + \\mathbf { v } _ { 1 } [ i ] ^ { * } \\mathbf { u } _ { 1 } [ i ] = \\sqrt { | \\mathbf { u } _ { 0 } [ i ] | ^ { 2 } + | \\mathbf { u } _ { 1 } [ i ] | ^ { 2 } } . } \\\\ & { \\mathbf { v } _ { 1 } [ i ] \\mathbf { u } _ { 0 } [ i ] - \\mathbf { v } _ { 0 } [ i ] ^ { * } \\mathbf { u } _ { 1 } [ i ] = 0 } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 334, + 359, + 663, + 420 + ], + "page_idx": 36 + }, + { + "type": "text", + "text": "We choose ", + "bbox": [ + 173, + 426, + 246, + 440 + ], + "page_idx": 36 + }, + { + "type": "equation", + "img_path": "images/cfd68d858526984b2fd0965d062dde71143e8ada7dee9824ea11b7517bd2259b.jpg", + "text": "$$\n\\mathbf { L } = \\left[ \\begin{array} { c c } { \\mathrm { D i a g } ( \\mathbf { v } _ { 0 } ^ { * } ) } & { \\mathrm { D i a g } ( \\mathbf { v } _ { 1 } ^ { * } ) } \\\\ { \\mathrm { D i a g } ( \\mathbf { v } _ { 1 } ) } & { \\mathrm { D i a g } ( - \\mathbf { v } _ { 0 } ) } \\end{array} \\right]\n$$", + "text_format": "latex", + "bbox": [ + 393, + 435, + 602, + 470 + ], + "page_idx": 36 + }, + { + "type": "text", + "text": "and ${ \\bf R } = { \\bf L } ^ { * } . { \\bf L } , { \\bf R }$ are (permuted) direct sums of blocks of the form $\\begin{array} { r l } { \\left[ \\mathbf { v } _ { 0 } [ i ] ^ { * } } & { { } \\mathbf { v } _ { 1 } [ i ] ^ { * } \\right] } \\\\ { \\mathbf { v } _ { 1 } [ i ] } & { { } - \\mathbf { v } _ { 0 } [ i ] } \\end{array}$ , which are orthogonal by construction (via (5)). Hence, $\\mathbf { L } \\in \\widetilde { \\mathbf { B } } _ { n } ^ { ( n ) }$ and $\\mathbf { R } \\in ( \\widetilde { \\mathbf { B } } ^ { * } ) _ { n } ^ { ( n ) }$ . Further, ", + "bbox": [ + 174, + 478, + 825, + 529 + ], + "page_idx": 36 + }, + { + "type": "equation", + "img_path": "images/9f96645ba18326c50e66e5bbaa4589dafa71d2fcc6c14d05c9b802b7d4b4108c.jpg", + "text": "$$\n\\begin{array} { r l } & { \\mathbf { L H R } = \\left[ \\mathrm { D i a g } ( \\mathbf { v } _ { 0 } ^ { * } ) \\quad \\mathrm { D i a g } ( \\mathbf { v } _ { 1 } ^ { * } ) \\right] \\left( \\mathbf { I } - 2 \\left[ \\mathbf { u } _ { 0 } \\right] \\left[ \\mathbf { u } _ { 0 } \\right] ^ { * } \\right) \\left[ \\mathrm { D i a g } ( \\mathbf { v } _ { 0 } ^ { * } ) \\quad \\mathrm { D i a g } ( \\mathbf { v } _ { 1 } ^ { * } ) \\right] ^ { * } } \\\\ & { \\qquad = \\mathbf { I } - 2 \\left[ \\mathrm { D i a g } ( \\mathbf { v } _ { 0 } ^ { * } ) \\quad \\mathrm { D i a g } ( - \\mathbf { v } _ { 0 } ) \\right] \\left[ \\mathbf { u } _ { 0 } \\right] \\left[ \\mathbf { u } _ { 0 } \\right] ^ { * } \\left[ \\mathrm { D i a g } ( \\mathbf { v } _ { 0 } ^ { * } ) \\quad \\mathrm { D i a g } ( - \\mathbf { v } _ { 0 } ) \\right] ^ { * } } \\\\ & { \\qquad = \\mathbf { I } - 2 \\left[ \\mathrm { D i a g } ( \\mathbf { v } _ { 1 } ) \\quad \\mathrm { D i a g } ( - \\mathbf { v } _ { 0 } ) \\right] \\left[ \\mathbf { u } _ { 1 } \\right] \\left[ \\mathbf { u } _ { 1 } \\right] ^ { * } \\left[ \\mathrm { D i a g } ( \\mathbf { v } _ { 0 } ^ { * } ) \\quad \\mathrm { D i a g } ( \\mathbf { v } _ { 1 } ^ { * } ) \\right] ^ { * } } \\\\ & { \\qquad = \\mathbf { I } - 2 \\underbrace { \\left[ \\mathbf { v } _ { 0 } ^ { * } \\circ \\mathbf { u } _ { 0 } + \\mathbf { v } _ { 1 } ^ { * } \\circ \\mathbf { u } _ { 1 } \\right] } _ { \\mathbf { W } } \\left[ \\underbrace { \\mathbf { v } _ { 0 } ^ { * } \\circ \\mathbf { u } _ { 0 } + \\mathbf { v } _ { 1 } ^ { * } \\circ \\mathbf { u } _ { 1 } } _ { \\mathbf { W } } \\right] ^ { * } , } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 215, + 529, + 779, + 655 + ], + "page_idx": 36 + }, + { + "type": "text", + "text": "where $\\circ$ denotes the Hadamard product. From (6) ", + "bbox": [ + 174, + 656, + 498, + 670 + ], + "page_idx": 36 + }, + { + "type": "equation", + "img_path": "images/a668234d32d753cdfbc43faa1543c8a3a8c6b8cdc7114e28aeb66753ac325c57.jpg", + "text": "$$\n\\mathbf { w } [ i ] = \\left\\{ \\begin{array} { l l } { \\sqrt { | \\mathbf { u } _ { 0 } [ i ] | ^ { 2 } + | \\mathbf { u } _ { 1 } [ i ] | ^ { 2 } } } & { i \\in [ n / 2 ] } \\\\ { 0 } & { i \\in [ n / 2 : n ] } \\end{array} \\right.\n$$", + "text_format": "latex", + "bbox": [ + 346, + 671, + 650, + 708 + ], + "page_idx": 36 + }, + { + "type": "text", + "text": "Denoting the first half of this vector by $\\mathbf { w } _ { 0 } \\in \\mathbb { C } ^ { n / 2 }$ , we have ", + "bbox": [ + 173, + 712, + 573, + 727 + ], + "page_idx": 36 + }, + { + "type": "equation", + "img_path": "images/f8972fb6a76a451dd7b4c661a4db147ea4f4157077150d7b044b6e93bb74647e.jpg", + "text": "$$\n\\mathbf { L H R } = \\left[ \\begin{array} { c c } { \\mathbf { I } - 2 \\mathbf { w } _ { 0 } \\mathbf { w } _ { 0 } ^ { * } } & { \\mathbf { 0 } } \\\\ { \\mathbf { 0 } } & { \\mathbf { I } } \\end{array} \\right] ,\n$$", + "text_format": "latex", + "bbox": [ + 400, + 728, + 594, + 763 + ], + "page_idx": 36 + }, + { + "type": "text", + "text": "where $\\| \\mathbf { w } _ { 0 } \\| _ { 2 } = \\| \\mathbf { u } \\| _ { 2 } = 1$ . The result follows inductively. ", + "bbox": [ + 173, + 763, + 557, + 780 + ], + "page_idx": 36 + }, + { + "type": "text", + "text": "As an immediate corollary, we can use Singular Value Decomposition to obtain a factorization for an arbitrary $n \\times n$ matrix. ", + "bbox": [ + 174, + 792, + 823, + 821 + ], + "page_idx": 36 + }, + { + "type": "text", + "text": "Corollary K.3. Let M be an arbitrary $n \\times n$ matrix. Then, $\\mathbf { M } \\in ( { \\mathcal { O } } B B ) ^ { 2 n - 1 }$ , where all but one matrix in the decomposition is orthogonal (unitary). ", + "bbox": [ + 174, + 824, + 825, + 853 + ], + "page_idx": 36 + }, + { + "type": "text", + "text": "Proof. By employing Singular Value Decomposition, we can decompose $\\mathbf { M }$ as $\\mathbf { M } = \\mathbf { U } \\pmb { \\Sigma } \\mathbf { V } ^ { * }$ , where $\\mathbf { U } , \\mathbf { V } ^ { * }$ are orthogonal and $\\pmb { \\Sigma }$ is diagonal. By Lemma $\\mathbf { K } . 2 , \\mathbf { U } , \\bar { \\mathbf { V } } ^ { * } \\in ( \\mathcal { O B } \\mathcal { B } ) ^ { n - 1 }$ , and trivially $\\pmb { \\Sigma } \\in \\mathcal { O } B B$ . Hence, $\\mathbf { M } \\in ( \\mathcal { O } B B ) ^ { 2 n - 1 }$ . Note that $\\pmb { \\Sigma }$ is the only matrix in the decomposition that is not orthogonal (unitary). □ ", + "bbox": [ + 174, + 867, + 825, + 924 + ], + "page_idx": 36 + }, + { + "type": "text", + "text": "K.3 CONSTRUCTIONS ", + "text_level": 1, + "bbox": [ + 176, + 103, + 339, + 118 + ], + "page_idx": 37 + }, + { + "type": "text", + "text": "We show that we can construct $s$ -sparse matrices in the OBB hierarchy with the same width as the $B B ^ { * }$ hierarchy. The proof follows a structure to that of Theorem 3. We begin by arguing about permutation and step matrices, then using the same factorization to argue that matrices with at most $n$ NNZ are contained in $( B B ^ { * } ) ^ { 4 }$ . Then, we will appeal to a modified sum closure lemma to extend the argument to matrices of general $s$ NNZ. Similar to Appendix F, we can use these results to place all matrices with low-depth circuits for matrix vector multiplication in the $\\ O B B$ hierarchy. ", + "bbox": [ + 173, + 128, + 825, + 213 + ], + "page_idx": 37 + }, + { + "type": "text", + "text": "K.3.1 PERMUTATIONS ", + "text_level": 1, + "bbox": [ + 176, + 228, + 343, + 243 + ], + "page_idx": 37 + }, + { + "type": "text", + "text": "We begin by presenting the argument that permutations are included in OBB as a corollary to Theorem 2. ", + "bbox": [ + 178, + 252, + 823, + 281 + ], + "page_idx": 37 + }, + { + "type": "text", + "text": "Corollary K.4. Let P be a permutation matrix. Then $\\mathbf { P } \\in \\widetilde { B } \\widetilde { B } ^ { * }$ . ", + "bbox": [ + 173, + 284, + 598, + 301 + ], + "page_idx": 37 + }, + { + "type": "text", + "text": "Proof. We appeal to the decomposition from Theorem 2, noting that all butterfly factor matrices constructed in the proofs of Lemmas G.3 and G.1 are permutation matrices, and thus are orthogonal. Hence, $\\mathbf { P } \\in { \\mathcal { O } } B B$ where the inner diagonal matrix is I. □ ", + "bbox": [ + 174, + 315, + 826, + 358 + ], + "page_idx": 37 + }, + { + "type": "text", + "text": "Similarly, the construction of Lemma G.6 also show that permutations are included in $\\widetilde { B } ^ { * } \\widetilde { B }$ . ", + "bbox": [ + 171, + 375, + 772, + 391 + ], + "page_idx": 37 + }, + { + "type": "text", + "text": "Corollary K.5. Let $\\mathbf { P }$ be a permutation matrix. Then $\\mathbf { P } \\in \\widetilde { B } ^ { * } \\widetilde { B }$ . ", + "bbox": [ + 174, + 396, + 598, + 412 + ], + "page_idx": 37 + }, + { + "type": "text", + "text": "To prove the containment of sparse matrices within the $o s B$ hierarchy, we make use of the following lemma. ", + "bbox": [ + 174, + 424, + 825, + 452 + ], + "page_idx": 37 + }, + { + "type": "text", + "text": "Lemma K.6. Let $\\mathbf { P }$ be a permutation matrix and $\\mathbf { D }$ a diagonal matrix. Then there exist diagonal matrices $\\mathbf { D ^ { \\prime } }$ and $\\mathbf { D } ^ { \\prime \\prime }$ such that: ", + "bbox": [ + 171, + 457, + 823, + 484 + ], + "page_idx": 37 + }, + { + "type": "equation", + "img_path": "images/fe8c4adc1543d5353b474daacb64f8be5e56c9e4bae6f964697a824a5c0bcdee.jpg", + "text": "$$\n\\mathbf { P D } = \\mathbf { D } ^ { \\prime } \\mathbf { P } \\qquad \\mathbf { D P } = \\mathbf { P D } ^ { \\prime \\prime } .\n$$", + "text_format": "latex", + "bbox": [ + 377, + 491, + 619, + 508 + ], + "page_idx": 37 + }, + { + "type": "text", + "text": "Proof. Let $\\sigma$ be the permutation such that $\\mathbf { P } [ i , j ] = \\delta _ { i , \\sigma ( j ) }$ ", + "bbox": [ + 173, + 523, + 563, + 540 + ], + "page_idx": 37 + }, + { + "type": "text", + "text": "Define $\\mathbf { D ^ { \\prime } }$ such that $\\mathbf { D } ^ { \\prime } [ \\sigma ( j ) , \\sigma ( j ) ] = \\mathbf { D } [ j , j ]$ . Then, if $i = \\sigma ( j )$ : ", + "bbox": [ + 173, + 545, + 598, + 563 + ], + "page_idx": 37 + }, + { + "type": "equation", + "img_path": "images/90be0a73847e56d57f8128affbb600568993047e9c5010abca684d5e3e906638.jpg", + "text": "$$\n\\begin{array} { r } { ( \\mathbf { P D } ) [ i , j ] = \\mathbf { P } [ i , j ] \\mathbf { D } [ j , j ] = \\mathbf { D } ^ { \\prime } [ \\sigma ( j ) , \\sigma ( j ) ] \\mathbf { P } [ \\sigma ( j ) , j ] = ( \\mathbf { D } ^ { \\prime } \\mathbf { P } ) [ \\sigma ( j ) , j ] = ( \\mathbf { D } ^ { \\prime } \\mathbf { P } ) [ i , j ] . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 204, + 568, + 789, + 587 + ], + "page_idx": 37 + }, + { + "type": "text", + "text": "Otherwise, if $i \\neq \\sigma ( j )$ , then $( { \\bf P D } ) [ i , j ] = 0 = ( { \\bf D ^ { \\prime } P } ) [ i , j ]$ . Hence, $\\mathbf { P D } = \\mathbf { D } ^ { \\prime } \\mathbf { P }$ . ", + "bbox": [ + 174, + 590, + 704, + 608 + ], + "page_idx": 37 + }, + { + "type": "text", + "text": "Define $\\mathbf { D } ^ { \\prime \\prime }$ such that ${ \\bf D } ^ { \\prime \\prime } [ j , j ] = { \\bf D } [ \\sigma ( j ) , \\sigma ( j ) ]$ . An analogous argument to above shows that $\\mathbf { D P } = \\mathbf { P D ^ { \\prime \\prime } }$ . ", + "bbox": [ + 173, + 613, + 825, + 642 + ], + "page_idx": 37 + }, + { + "type": "text", + "text": "K.3.2 STEP MATRICES ", + "text_level": 1, + "bbox": [ + 174, + 657, + 346, + 672 + ], + "page_idx": 37 + }, + { + "type": "text", + "text": "In the $B B ^ { * }$ hierarchy (Lemma I.2), we were able to show that horizontal step matrices are butterfly matrices. Here, we present a similar result for the $\\ O B B$ hierarchy. ", + "bbox": [ + 173, + 681, + 823, + 710 + ], + "page_idx": 37 + }, + { + "type": "text", + "text": "Lemma K.7. Let $\\mathbf { H }$ be an $n \\times n$ horizontal step matrix. Then we can decompose ${ \\bf H } = { \\bf D } { \\bf O }$ , where $\\mathbf { D }$ is a diagonal matrix and $\\mathbf { O } \\in \\widetilde { B }$ . ", + "bbox": [ + 173, + 714, + 823, + 744 + ], + "page_idx": 37 + }, + { + "type": "text", + "text": "Proof. Throughout the proof, we make reference to the original horizontal step matrix construction given in Lemma I.2 and its proof. ", + "bbox": [ + 174, + 761, + 823, + 790 + ], + "page_idx": 37 + }, + { + "type": "text", + "text": "To begin, we show that an arbitrary $2 ^ { k } \\times 2 ^ { k }$ butterfly factor $\\mathbf { H } _ { 2 ^ { k } }$ in the decomposition of $\\mathbf { H }$ can be expressed as the product of a diagonal matrix and an orthogonal butterfly factor. Since a butterfly factor is direct sum of $2 \\times 2$ matrices, there is a permutation matrix $\\mathbf { P } _ { 2 ^ { k } }$ such that conjugation of $\\mathbf { H } _ { 2 ^ { k } }$ by $\\mathbf { P } _ { 2 ^ { k } }$ gives a block diagonal matrix $\\mathbf { H } _ { 2 ^ { k } } ^ { \\prime }$ of ${ \\frac { n } { 2 } } \\ 2 \\times 2$ matrices, i.e. ", + "bbox": [ + 174, + 795, + 825, + 853 + ], + "page_idx": 37 + }, + { + "type": "equation", + "img_path": "images/a4cb9b1d7f35443fb39d0293b250e9717c463e164f18db57e633ea12a96fa94e.jpg", + "text": "$$\n\\mathbf { P } _ { 2 ^ { k } } \\mathbf { H } _ { 2 ^ { k } } \\mathbf { P } _ { 2 ^ { k } } ^ { * } = \\mathbf { H } _ { 2 ^ { k } } ^ { \\prime } .\n$$", + "text_format": "latex", + "bbox": [ + 428, + 861, + 570, + 878 + ], + "page_idx": 37 + }, + { + "type": "text", + "text": "(See Figure 10 for an illustration.) Specifically, $\\mathbf { P } _ { 2 ^ { k } }$ is the permutation where: ", + "bbox": [ + 171, + 883, + 684, + 900 + ], + "page_idx": 37 + }, + { + "type": "equation", + "img_path": "images/2f7c163aae1a9c1f7639cc3e31b77728c067fe128716b6f52400cb4c059c2759.jpg", + "text": "$$\n{ \\bf P } _ { s } [ 2 i , : ] = { \\bf e } _ { i } ^ { T } { \\bf \\mathrm { ~ ~ { ~ \\cal ~ ~ { ~ P ~ } ~ } ~ } } _ { s } [ 2 i + 1 , : ] = { \\bf e } _ { i + \\frac { n } { 2 } } ^ { T } .\n$$", + "text_format": "latex", + "bbox": [ + 344, + 906, + 651, + 928 + ], + "page_idx": 37 + }, + { + "type": "image", + "img_path": "images/a4607d3febb414e4ea1a9f3500d9dc6ca268a2509f0b7cd81b00a8b90f87d6e6.jpg", + "image_caption": [ + "Figure 10: Block diagonalization of ${ \\bf { H } } _ { 8 }$ " + ], + "image_footnote": [], + "bbox": [ + 176, + 111, + 892, + 287 + ], + "page_idx": 38 + }, + { + "type": "text", + "text": "We argue that each of these $2 \\times 2$ blocks can be decomposed into a diagonal matrix times an orthogonal matrix. Note that the butterfly factor matrices constructed in the proof of Lemma I.2 each have at most one non-zero entry per column. Hence, there are 4 cases to consider. Note that matrices with at most one non-zero entry are exhausted by Cases 1 and 2. ", + "bbox": [ + 173, + 348, + 826, + 405 + ], + "page_idx": 38 + }, + { + "type": "equation", + "img_path": "images/09b0b65bd89e123e7438424fe136eb0b9b11d952dedecd55f05ad3381597b1c7.jpg", + "text": "$$\n\\mathfrak { e } \\ 1 : \\left[ 0 \\atop 0 \\right] = \\underbrace { \\left[ 0 \\begin{array} { l l } { 0 } & { 0 } \\\\ { 0 } & { b } \\end{array} \\right] } _ { \\textbf { D } } \\underbrace { \\left[ 0 \\begin{array} { l l } { 1 } & { 0 } \\\\ { 0 } & { 1 } \\end{array} \\right] } _ { \\textbf { O } }\n$$", + "text_format": "latex", + "bbox": [ + 202, + 409, + 415, + 467 + ], + "page_idx": 38 + }, + { + "type": "text", + "text": "Case 2: $\\left[ \\begin{array} { l l } { 0 } & { a } \\\\ { b } & { 0 } \\end{array} \\right] = \\underbrace { \\left[ \\begin{array} { l l } { a } & { 0 } \\\\ { 0 } & { b } \\end{array} \\right] } _ { \\textbf { D } } \\underbrace { \\left[ \\begin{array} { l l } { 0 } & { 1 } \\\\ { 1 } & { 0 } \\end{array} \\right] } _ { \\textbf { O } }$ ", + "bbox": [ + 173, + 469, + 413, + 527 + ], + "page_idx": 38 + }, + { + "type": "image", + "img_path": "images/7f28d28d7daab240206962aaddca12847be0539a1b0d4e28308bd7d0b076eccb.jpg", + "image_caption": [], + "image_footnote": [], + "bbox": [ + 173, + 598, + 648, + 661 + ], + "page_idx": 38 + }, + { + "type": "text", + "text": "In the last two cases, $\\mathbf { O }$ is a $2 \\times 2$ rotation matrix, which is commonly known to be orthogonal. Assume that we perform the above decomposition on all of the blocks of $\\mathbf { H } _ { 2 ^ { k } } ^ { \\prime }$ in parallel, therefore expressing $\\mathbf { H } _ { 2 ^ { k } } ^ { \\prime } \\doteq \\mathbf { D } ^ { \\prime } \\mathbf { O } ^ { \\prime }$ . We now have ", + "bbox": [ + 174, + 665, + 826, + 708 + ], + "page_idx": 38 + }, + { + "type": "equation", + "img_path": "images/89a88ce963a6e85b3c2811f0c74bd30d0508fbae2288d51f86f5118bce0d4745.jpg", + "text": "$$\n{ \\bf H } _ { 2 ^ { k } } = { \\bf P } _ { 2 ^ { k } } ^ { * } { \\bf D } ^ { \\prime } { \\bf O } ^ { \\prime } { \\bf P } _ { 2 ^ { k } } .\n$$", + "text_format": "latex", + "bbox": [ + 423, + 712, + 573, + 729 + ], + "page_idx": 38 + }, + { + "type": "text", + "text": "By Lemma K.6, we can rewrite this as ", + "bbox": [ + 174, + 732, + 428, + 747 + ], + "page_idx": 38 + }, + { + "type": "equation", + "img_path": "images/cf9a8c7c3b1979c8c7b0b2ac95af55918af29275013ff34084fa276a7eea87fc.jpg", + "text": "$$\n{ \\bf H } _ { 2 ^ { k } } = { \\bf D } ^ { \\prime \\prime } { \\bf P } _ { 2 ^ { k } } ^ { * } { \\bf O } ^ { \\prime } { \\bf P } _ { 2 ^ { k } } .\n$$", + "text_format": "latex", + "bbox": [ + 421, + 748, + 575, + 767 + ], + "page_idx": 38 + }, + { + "type": "text", + "text": "Note that $\\mathbf { P } _ { 2 ^ { k } } ^ { * } \\mathbf { O } ^ { \\prime } \\mathbf { P } _ { 2 ^ { k } }$ is the product of three orthogonal matrices, and thus orthogonal. Additionally, the construction of $\\mathbf { P } _ { 2 ^ { k } }$ ensures that $\\mathbf { P } _ { 2 ^ { k } } ^ { * } \\mathbf { O } ^ { \\prime } \\mathbf { P } _ { 2 ^ { k } }$ is butterfly factor.11 Hence, $\\mathbf { H } _ { 2 ^ { k } }$ can be expressed as the product of a diagonal matrix and an orthogonal butterfly factor, as desired. ", + "bbox": [ + 173, + 768, + 826, + 814 + ], + "page_idx": 38 + }, + { + "type": "text", + "text": "Now, we show that this decomposition of butterfly factors implies Lemma K.7. By performing this decomposition in parallel on each butterfly factor, we conclude that any butterfly factor matrix $\\mathbf { \\bar { H } } _ { 2 ^ { k } } ^ { ( n ) }$ of $\\mathbf { H }$ can be decomposed as ) = D2k O(n)2k . 12 ", + "bbox": [ + 173, + 820, + 825, + 873 + ], + "page_idx": 38 + }, + { + "type": "text", + "text": "We complete the argument by induction on $n$ . The base case $n = 2$ holds by the observation about butterfly factor matrices above. Assume that any horizontal step matrix of size ${ \\frac { n } { 2 } } \\times { \\frac { n } { 2 } }$ can be expressed as a diagonal matrix times an orthogonal butterfly matrix. Now, consider the $n \\times n$ horizontal step matrix $\\mathbf { H }$ . From Lemma I.2, $\\mathbf { H }$ can be expressed as ", + "bbox": [ + 173, + 103, + 825, + 160 + ], + "page_idx": 39 + }, + { + "type": "equation", + "img_path": "images/f416386e5bb8c0718c80ca863428e5a987c30b37d2bf57e1c04266c4a695868d.jpg", + "text": "$$\n\\mathbf { H } = \\mathbf { B } _ { n } ^ { ( n ) } \\left[ \\begin{array} { c c } { \\mathbf { H } _ { 1 } } & { 0 } \\\\ { 0 } & { \\mathbf { H } _ { 2 } } \\end{array} \\right] ,\n$$", + "text_format": "latex", + "bbox": [ + 418, + 166, + 578, + 202 + ], + "page_idx": 39 + }, + { + "type": "text", + "text": "where $\\mathbf { H } _ { 1 } , \\mathbf { H } _ { 2 }$ are ${ \\frac { n } { 2 } } \\times { \\frac { n } { 2 } }$ horizontal step matrices. By our inductive hypothesis, ", + "bbox": [ + 171, + 208, + 699, + 224 + ], + "page_idx": 39 + }, + { + "type": "equation", + "img_path": "images/136728a2ff3ccef05087ea5f6c333207cca10b3291a4750d2d1d162a6f2c4119.jpg", + "text": "$$\n\\mathbf { H } = \\mathbf { B } _ { n } ^ { ( n ) } \\mathbf { D } _ { 1 } \\left[ \\begin{array} { c c } { \\mathbf { O } _ { 1 } } & { 0 } \\\\ { 0 } & { \\mathbf { O } _ { 2 } } \\end{array} \\right] ,\n$$", + "text_format": "latex", + "bbox": [ + 408, + 231, + 588, + 267 + ], + "page_idx": 39 + }, + { + "type": "text", + "text": "where $\\mathbf { D } _ { 1 }$ is diagonal and $\\mathbf { O } _ { 1 } , \\mathbf { O } _ { 2 }$ are ${ \\frac { n } { 2 } } \\times { \\frac { n } { 2 } }$ matrices in $\\widetilde { B }$ . However, ${ \\bf B } _ { n } ^ { ( n ) } { \\bf D } _ { 1 }$ is a butterfly factor, and therefore can be expressed as $\\mathbf { D } _ { n } \\mathbf { O } _ { n } ^ { ( n ) }$ . Therefore, ", + "bbox": [ + 173, + 276, + 826, + 311 + ], + "page_idx": 39 + }, + { + "type": "equation", + "img_path": "images/df417c49b7d774cb2d559d0f61e552401521f85e7e356657ff6094a2d734c2ed.jpg", + "text": "$$\n{ \\bf H } = { \\bf D } _ { n } { \\bf O } _ { n } ^ { ( n ) } \\left[ \\begin{array} { c c } { { \\bf O } _ { 1 } } & { 0 } \\\\ { 0 } & { { \\bf O } _ { 2 } } \\end{array} \\right] = { \\bf D } _ { n } { \\bf O } ,\n$$", + "text_format": "latex", + "bbox": [ + 379, + 318, + 617, + 353 + ], + "page_idx": 39 + }, + { + "type": "text", + "text": "with $\\mathbf { O } \\in \\widetilde { B }$ , as desired. ", + "bbox": [ + 173, + 362, + 330, + 378 + ], + "page_idx": 39 + }, + { + "type": "text", + "text": "Just as with the $B B ^ { * }$ hierarchy, the decomposition of vertical step matrices falls out as an immediate corollary to the horizontal step matrix proof. ", + "bbox": [ + 174, + 396, + 823, + 425 + ], + "page_idx": 39 + }, + { + "type": "text", + "text": "Corollary K.8. Let $\\mathbf { V }$ be a vertical step matrix. Then we can decompose $\\mathbf { V } = \\mathbf { O } ^ { * } \\mathbf { D }$ , where $\\mathbf { D }$ is $a$ diagonal matrix and $\\mathbf { O } ^ { * } \\in \\widetilde { B } ^ { * }$ . ", + "bbox": [ + 173, + 429, + 825, + 460 + ], + "page_idx": 39 + }, + { + "type": "text", + "text": "K.3.3 SPARSE MATRICES ", + "text_level": 1, + "bbox": [ + 174, + 476, + 364, + 491 + ], + "page_idx": 39 + }, + { + "type": "text", + "text": "Now that we have argued about the decomposition of permutation and step matrices in the OBB hierarchy, we can leverage the construction from Lemma I.1 to argue about matrices with at most $n$ NNZ. ", + "bbox": [ + 173, + 501, + 826, + 542 + ], + "page_idx": 39 + }, + { + "type": "text", + "text": "Corollary K.9. Let S be an $n \\times n$ matrix with at most n NNZ. Then, $\\mathbf { S } \\in ( \\mathcal { O } B B ) ^ { 4 }$ . ", + "bbox": [ + 173, + 546, + 720, + 563 + ], + "page_idx": 39 + }, + { + "type": "text", + "text": "Proof. We use the construction from Lemma I.1, along with Lemma K.7 and Corollary K.8, to express S as: ", + "bbox": [ + 171, + 579, + 823, + 609 + ], + "page_idx": 39 + }, + { + "type": "equation", + "img_path": "images/f4ddc3bdf875d54dacd9d6aaa9113a9957824b1947c2e065861b9fefe4c697cf.jpg", + "text": "$$\n\\mathbf { S } = \\underbrace { \\mathbf { O } _ { 1 } \\mathbf { O } _ { 1 } ^ { \\prime } } _ { \\mathbf { P } _ { 1 } } \\underbrace { \\mathbf { D } _ { 2 } \\mathbf { O } _ { 2 } } _ { \\mathbf { H } } \\underbrace { \\mathbf { O } _ { 3 } \\mathbf { O } _ { 3 } ^ { \\prime } } _ { \\mathbf { P } _ { 2 } } \\underbrace { \\mathbf { O } _ { 4 } ^ { \\prime } \\mathbf { D } _ { 4 } } _ { \\mathbf { V } } \\underbrace { \\mathbf { O } _ { 5 } \\mathbf { O } _ { 5 } ^ { \\prime } } _ { \\mathbf { P } _ { 3 } } ,\n$$", + "text_format": "latex", + "bbox": [ + 364, + 616, + 632, + 657 + ], + "page_idx": 39 + }, + { + "type": "text", + "text": "with each $\\mathbf { O } _ { i } \\in \\widetilde { B }$ , each $\\mathbf { O } _ { j } ^ { \\prime } \\in \\widetilde { B } ^ { * }$ , and each $\\mathbf { D } _ { k }$ diagonal. Since $\\mathbf { P } _ { 2 }$ is a permutation, by Corollary K.5, we can write it as $\\tilde { \\bf { O } } _ { 3 } ^ { ' } \\tilde { \\bf { O } } _ { 3 }$ for some $\\tilde { \\mathbf { O } _ { 3 } } ^ { \\prime } \\in \\widetilde { B } ^ { * }$ and $\\tilde { \\mathbf { O } } _ { 3 } \\in \\tilde { B }$ . Moreover, noting that $\\mathbf { O } _ { 1 } ^ { \\prime }$ and $\\mathbf { O } _ { 5 }$ are permutations, we make use of Lemma K.6 to re-express $\\mathbf { S }$ as: ", + "bbox": [ + 173, + 665, + 826, + 715 + ], + "page_idx": 39 + }, + { + "type": "equation", + "img_path": "images/798b4deb3104d6592027cdd284c5e253530e2eea6db0a3a2033a500f4a83478f.jpg", + "text": "$$\n\\mathbf { S } = \\underbrace { \\mathbf { O } _ { 1 } \\mathbf { D } _ { 2 } ^ { \\prime } \\mathbf { O } _ { 1 } ^ { \\prime } } _ { \\mathbf { M } _ { 1 } } \\underbrace { \\mathbf { O } _ { 2 } \\tilde { \\mathbf { O } _ { 3 } } } _ { \\mathbf { M } _ { 2 } } \\underbrace { \\mathbf { \\tilde { O } _ { 3 } O _ { 4 } ^ { \\prime } } } _ { \\mathbf { M } _ { 3 } } \\underbrace { \\mathbf { O } _ { 5 } \\mathbf { D } _ { 4 } ^ { \\prime } \\mathbf { O } _ { 5 } ^ { \\prime } } _ { \\mathbf { M } _ { 4 } } .\n$$", + "text_format": "latex", + "bbox": [ + 362, + 722, + 633, + 767 + ], + "page_idx": 39 + }, + { + "type": "text", + "text": "Note that each $\\mathbf { M } _ { \\ell } \\in \\mathcal { O B B }$ . Hence, $\\mathbf { S } \\in ( \\mathcal { O } B B ) ^ { 4 }$ , as desired. ", + "bbox": [ + 173, + 775, + 576, + 791 + ], + "page_idx": 39 + }, + { + "type": "text", + "text": "Just as in Appendix I, we would like to extend this orthogonal-based construction to capture matrices of general sparsity. To accomplish this, we introduce an addition closure lemma analogous to Lemma K.10 for the $\\ O B B$ hierarchy. ", + "bbox": [ + 174, + 808, + 826, + 852 + ], + "page_idx": 39 + }, + { + "type": "text", + "text": "Lemma K.10. Let $\\mathbf { A } _ { 1 } , \\ldots , \\mathbf { A } _ { m }$ be $k \\times k$ matrices in $( O B B ) _ { e } ^ { w }$ then $\\begin{array} { r } { \\sum _ { i = 1 } ^ { m } \\mathbf { A } _ { i } \\in ( \\mathcal { O } B \\mathcal { B } ) _ { 4 e } ^ { m w } . } \\end{array}$ ", + "bbox": [ + 174, + 854, + 781, + 873 + ], + "page_idx": 39 + }, + { + "type": "text", + "text": "With Lemma K.10, we arrive at the following Corollary on general orthogonal sparsity. ", + "bbox": [ + 173, + 882, + 741, + 898 + ], + "page_idx": 39 + }, + { + "type": "text", + "text": "Corollary K.11. Let S be an $n \\times n$ matrix with s NNZ. Then, $\\mathbf { S } \\in \\left( \\mathcal { O } B B \\right) _ { 4 } ^ { 4 \\left\\lceil \\frac { s } { n } \\right\\rceil }$ . ", + "bbox": [ + 173, + 902, + 702, + 925 + ], + "page_idx": 39 + }, + { + "type": "text", + "text": "Proof. Just as in the proof of Theorem 3, we accomplish this using a sum of $\\textstyle { \\left\\lceil { \\frac { s } { n } } \\right\\rceil }$ matrices of at most $n$ NNZ. For handling the sum of matrices, we need to appeal to Lemma K.10. □ ", + "bbox": [ + 171, + 102, + 825, + 133 + ], + "page_idx": 40 + }, + { + "type": "text", + "text": "To conclude the argument, we give the proof of Lemma K.10. ", + "bbox": [ + 173, + 146, + 578, + 162 + ], + "page_idx": 40 + }, + { + "type": "text", + "text": "Proof of Lemma $K . I O$ . For each $1 \\leq i \\leq m$ , let $\\mathbf { E } _ { \\mathbf { A } _ { i } } \\in \\mathbb { F } ^ { e k \\times e k }$ be defined such that $\\mathbf { A } _ { i } = \\mathbf { S } \\mathbf { E } _ { \\mathbf { A } _ { i } } \\mathbf { S } ^ { * }$ (with $\\mathbf { S }$ as in Definition 2.4). Note that $\\mathbf { E } _ { \\mathbf { A } _ { i } } \\in ( \\mathcal { O } B B ) ^ { w }$ . Consider matrices of the form: ", + "bbox": [ + 173, + 175, + 823, + 205 + ], + "page_idx": 40 + }, + { + "type": "equation", + "img_path": "images/a6f5893d143e676eed143ff48dd675d388e90e6be20c7e28536a229b5218b751.jpg", + "text": "$$\n\\begin{array} { r l } & { \\left[ \\mathbf { I } _ { e k } \\quad 0 \\quad 0 \\quad 0 \\quad \\mathbf { E } _ { \\mathbf { A } _ { i } } \\right] } \\\\ & { \\left[ \\begin{array} { l l l l } { \\mathbf { 0 } } & { \\mathbf { I } _ { e k } \\quad 0 \\quad } & { \\mathbf { 0 } \\quad } & { 0 } \\\\ { 0 \\quad \\mathbf { I } _ { e k } \\quad 0 \\quad } & { 0 \\quad } & { \\mathbf { 0 } } \\\\ { \\mathbf { I } _ { e k } \\quad 0 \\quad } & { 0 \\quad } & { \\mathbf { 0 } } \\end{array} \\right] } \\\\ & { \\underbrace { \\left[ \\begin{array} { l l l l } { \\mathbf { I } _ { e k } \\quad 0 \\quad } & { 0 \\quad } & { \\mathbf { 0 } } \\\\ { 0 \\quad } & { \\mathbf { I } _ { e k } \\quad 0 \\quad } & { 0 } \\end{array} \\right] } _ { \\mathbf { M } _ { i } \\in \\mathbb { F } ^ { 4 e k \\times 4 e k } } = \\sqrt { 2 } \\underbrace { \\left[ \\frac { 1 } { \\sqrt { 2 } } \\mathbf { I } _ { 2 e k } \\quad \\frac { 1 } { \\sqrt { 2 } } \\mathbf { I } _ { 2 e k } \\right] } _ { \\mathbf { O } \\in \\widetilde { \\mathbf { B } } _ { 4 e k } ^ { ( 4 e k ) } } \\underbrace { \\left[ \\begin{array} { l l l l l } { \\mathbf { I } _ { e k } \\quad 0 \\quad } & { 0 \\quad } & { 0 } & { 0 } \\\\ { 0 \\quad } & { \\mathbf { I } _ { e k } \\quad } & { 0 \\quad } & { 0 } \\\\ { 0 \\quad } & { 0 \\quad } & { \\mathbf { E } _ { \\mathbf { A } _ { i } } \\quad 0 } \\\\ { 0 \\quad } & { 0 \\quad } & { 0 } & { 0 } \\end{array} \\right] } _ { \\mathbf { K } } \\underbrace { \\left[ \\begin{array} { l l l l } { \\mathbf { I } _ { e k } \\quad 0 \\quad } & { 0 \\quad } & { 0 } & { 0 } \\\\ { 0 \\quad } & { \\mathbf { I } _ { e k } \\quad } & { 0 \\quad } & { 0 } \\\\ { 0 \\quad } & { 0 \\quad } & { \\mathbf { I } _ { e k } \\quad } \\\\ { 0 \\quad } & { 0 \\quad } & { \\mathbf { I } _ { e k } \\quad 0 } \\end{array} \\right] } _ { \\mathbf { P } \\in \\widetilde { \\mathbf { B } } _ { 2 e k } ^ { ( 4 e k ) } } } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 181, + 207, + 831, + 295 + ], + "page_idx": 40 + }, + { + "type": "text", + "text": "Note that $\\mathbf { K }$ , a block diagonal matrix composed of matrices in $( \\mathcal { O } B B ) ^ { w }$ , is itself in $( \\mathcal { O } B B ) ^ { w }$ since ", + "bbox": [ + 174, + 304, + 818, + 320 + ], + "page_idx": 40 + }, + { + "type": "equation", + "img_path": "images/3fefd4ce7a53f74ea8450efb101b6a13e084c41eddec66dfffd188082168cb4c.jpg", + "text": "$$\n\\mathbf { K } = \\prod _ { j = 1 } ^ { w } \\underbrace { \\left[ \\begin{array} { c c c c } { \\mathbf { I } _ { e k } } & { 0 } & { 0 } & { 0 } \\\\ { 0 } & { \\mathbf { I } _ { e k } } & { 0 } & { 0 } \\\\ { 0 } & { 0 } & { \\mathbf { O } _ { j } } & { 0 } \\\\ { 0 } & { 0 } & { 0 } & { \\mathbf { I } _ { e k } } \\end{array} \\right] } _ { \\mathbf { L } _ { j } \\in \\widetilde { B } } \\underbrace { \\left[ \\begin{array} { c c c c } { \\mathbf { I } _ { e k } } & { 0 } & { 0 } & { 0 } \\\\ { 0 } & { \\mathbf { I } _ { e k } } & { 0 } & { 0 } \\\\ { 0 } & { 0 } & { \\mathbf { D } _ { j } } & { 0 } \\\\ { 0 } & { 0 } & { 0 } & { 0 } \\end{array} \\right] } _ { \\mathrm { D i a g o n a l } } \\underbrace { \\left[ \\begin{array} { c c c c } { \\mathbf { I } _ { e k } } & { 0 } & { 0 } & { 0 } \\\\ { 0 } & { \\mathbf { I } _ { e k } } & { 0 } & { 0 } \\\\ { 0 } & { 0 } & { \\mathbf { O } _ { j } ^ { \\prime } } & { 0 } \\\\ { 0 } & { 0 } & { 0 } & { \\mathbf { I } _ { e k } } \\end{array} \\right] } _ { \\mathbf { R } _ { j } \\in \\widetilde { B } ^ { * } } ,\n$$", + "text_format": "latex", + "bbox": [ + 232, + 321, + 764, + 410 + ], + "page_idx": 40 + }, + { + "type": "text", + "text": "where each $\\mathbf { O } _ { j }$ is a $e k \\times e k$ matrix in $\\widetilde { B }$ , and each ${ \\bf O } _ { j } ^ { \\prime }$ is a $e k \\times e k$ matrix in ${ \\widetilde { B } } ^ { * }$ . $\\mathbf { L } _ { w }$ (the leftmost factor) is a block diagonal matrix composed of $4 ~ e k \\times e k$ matrices in $\\widetilde { B }$ . Therefore, we can fold $\\mathbf { O }$ into this factor (since a butterfly factor in Be (4ek)4ek was not yet used in $\\mathbf { L } _ { w }$ ) to conclude that $\\mathbf { O L } _ { w } \\in \\widetilde { B }$ . Similarly, since no btterfly factor from Be (4ek)2ek has been used in ${ \\bf R } _ { 1 }$ , we may fold $\\mathbf { P }$ into ${ \\bf R } _ { 1 }$ to conclude that $\\mathbf { R } _ { 1 } \\mathbf { P } \\in \\widetilde { B } ^ { * }$ . Finally, we address the scalar multiple of $\\sqrt { 2 }$ by multiplying all entries of any diagonal matrix in the decomposition of $\\mathbf { K }$ by $\\sqrt { 2 }$ . Hence, we may conclude that $\\mathbf { M } _ { i } \\in ( \\mathcal { O } B B ) ^ { w }$ . ", + "bbox": [ + 173, + 412, + 828, + 517 + ], + "page_idx": 40 + }, + { + "type": "text", + "text": "Through repeated application $\\textbar { m }$ times) of the identity ", + "bbox": [ + 176, + 522, + 532, + 537 + ], + "page_idx": 40 + }, + { + "type": "equation", + "img_path": "images/d404d30cb377d762fed7671a3ee62dbe74810b1d76ae2370e32f9d86d3c535b5.jpg", + "text": "$$\n\\left[ \\begin{array} { c c c c } { \\mathbf { I } } & { \\mathbf { A } _ { 1 } } & { 0 } & { \\mathbf { B } _ { 1 } } \\\\ { 0 } & { \\mathbf { I } } & { 0 } & { 0 } \\\\ { \\mathbf { I } } & { \\mathbf { A } _ { 2 } } & { 0 } & { \\mathbf { B } _ { 2 } } \\\\ { 0 } & { \\mathbf { I } } & { 0 } & { 0 } \\end{array} \\right] \\left[ \\begin{array} { c c c c } { \\mathbf { I } } & { 0 } & { 0 } & { \\mathbf { C } _ { 1 } } \\\\ { 0 } & { \\mathbf { I } } & { 0 } & { 0 } \\\\ { \\mathbf { I } } & { 0 } & { 0 } & { \\mathbf { C } _ { 2 } } \\\\ { 0 } & { \\mathbf { I } } & { 0 } & { 0 } \\end{array} \\right] = \\left[ \\begin{array} { c c c c } { \\mathbf { I } } & { \\mathbf { A } _ { 1 } + \\mathbf { B } _ { 1 } } & { 0 } & { \\mathbf { C } _ { 1 } } \\\\ { 0 } & { \\mathbf { I } } & { 0 } & { 0 } \\\\ { \\mathbf { I } } & { \\mathbf { A } _ { 2 } + \\mathbf { B } _ { 2 } } & { 0 } & { \\mathbf { C } _ { 1 } } \\\\ { 0 } & { \\mathbf { I } } & { 0 } & { 0 } \\end{array} \\right] ,\n$$", + "text_format": "latex", + "bbox": [ + 274, + 537, + 722, + 598 + ], + "page_idx": 40 + }, + { + "type": "text", + "text": "we see that ", + "bbox": [ + 173, + 599, + 250, + 613 + ], + "page_idx": 40 + }, + { + "type": "equation", + "img_path": "images/63c34d019267ce0bfd8b56d0651ac58fef7dc2acd6d7ab9e4facfed8095ae461.jpg", + "text": "$$\n\\prod _ { i = 1 } ^ { m } \\mathbf { M } _ { i } = \\left[ \\begin{array} { c c c c } { \\mathbf { I } _ { e k } } & { \\sum _ { i = 2 } ^ { m } \\mathbf { E } _ { \\mathbf { A } _ { i } } } & { 0 } & { \\mathbf { E } _ { \\mathbf { A } _ { 1 } } } \\\\ { 0 } & { \\mathbf { I } _ { e k } } & { 0 } & { 0 } \\\\ { \\mathbf { I } _ { e k } } & { \\mathbf { - E } _ { \\mathbf { A } _ { m } } + \\sum _ { i = 2 } ^ { m - 1 } \\mathbf { E } _ { \\mathbf { A } _ { i } } } & { 0 } & { \\mathbf { E } _ { \\mathbf { A } _ { 1 } } } \\\\ { 0 } & { \\mathbf { I } _ { e k } } & { 0 } & { 0 } \\end{array} \\right] .\n$$", + "text_format": "latex", + "bbox": [ + 321, + 607, + 673, + 669 + ], + "page_idx": 40 + }, + { + "type": "text", + "text": "Therefore, $\\mathbf { M } \\in ( \\mathcal { O } B B ) ^ { m w }$ . Next, we note that ", + "bbox": [ + 173, + 700, + 485, + 717 + ], + "page_idx": 40 + }, + { + "type": "equation", + "img_path": "images/3b4bb35e6e3f120d2c581d877677469bda57fe8faada5eab6f72b4b12f945384.jpg", + "text": "$$\n\\sum _ { i = 1 } ^ { m } \\mathbf { A } _ { i } = \\mathbf { S } \\mathbf { M } \\underbrace { \\left[ \\begin{array} { c c c c } { 0 } & { \\mathbf { I } _ { e k } } & { 0 } & { \\mathbf { I } _ { e k } } \\\\ { \\mathbf { I } _ { e k } } & { 0 } & { \\mathbf { I } _ { e k } } & { 0 } \\\\ { 0 } & { \\mathbf { I } _ { e k } } & { 0 } & { - \\mathbf { I } _ { e k } } \\\\ { \\mathbf { I } _ { e k } } & { 0 } & { - \\mathbf { I } _ { e k } } & { 0 } \\end{array} \\right] } _ { \\mathbf { Q } } \\mathbf { S } ^ { T } .\n$$", + "text_format": "latex", + "bbox": [ + 348, + 718, + 648, + 803 + ], + "page_idx": 40 + }, + { + "type": "text", + "text": "We would like to show that we can fold $\\mathbf { Q }$ into the rightmost $\\ O B B$ factor of M. The rightmost matrix in the decomposition of $\\mathbf { M }$ is $\\mathbf { P }$ . Note that ", + "bbox": [ + 173, + 809, + 826, + 838 + ], + "page_idx": 40 + }, + { + "type": "equation", + "img_path": "images/8a5b6e6f1eed59213df86fb1481a447a0ed2ffb571f1e2dab42f3699c8fa5273.jpg", + "text": "$$\n\\mathbf { P Q } = \\left[ \\begin{array} { c c c c } { 0 } & { \\mathbf { I } _ { e k } } & { 0 } & { \\mathbf { I } _ { e k } } \\\\ { \\mathbf { I } _ { e k } } & { 0 } & { \\mathbf { I } _ { e k } } & { 0 } \\\\ { \\mathbf { I } _ { e k } } & { \\mathbf { I } _ { e k } } & { \\mathbf { I } _ { e k } } & { 0 } \\\\ { 0 } & { \\mathbf { I } _ { e k } } & { 0 } & { \\mathbf { - I } _ { e k } } \\end{array} \\right] = \\sqrt { 2 } \\underbrace { \\left[ \\begin{array} { c c c c } { 0 } & { \\mathbf { I } _ { e k } } & { 0 } & { 0 } \\\\ { \\mathbf { I } _ { e k } } & { 0 } & { 0 } & { 0 } \\\\ { 0 } & { 0 } & { \\mathbf { I } _ { e k } } & { 0 } \\\\ { 0 } & { 0 } & { 0 } & { \\mathbf { I } _ { e k } } \\end{array} \\right] } _ { \\widetilde { \\mathbf { B } } _ { 2 e k } ^ { ( 4 e k ) } } \\underbrace { \\left[ \\frac { 1 } { \\sqrt { 2 } } \\mathbf { I } _ { 2 e k } \\quad \\frac { 1 } { \\sqrt { 2 } } \\mathbf { I } _ { 2 e k } \\right] } _ { \\widetilde { \\mathbf { B } } _ { 4 e k } ^ { ( 4 e k ) } } .\n$$", + "text_format": "latex", + "bbox": [ + 212, + 838, + 784, + 929 + ], + "page_idx": 40 + }, + { + "type": "text", + "text": "Just as earlier, the factor of $\\sqrt { 2 }$ can be multiplied through any diagonal matrix. Also, these two orthogonal butterfly factor matrices can be folded into the the rightmost $\\mathbf { R }$ matrix (the decomposition of $\\mathbf { K }$ above does not use these two, rightmost butterfly factors). Hence, $\\begin{array} { r } { \\sum _ { i = 1 } ^ { m } \\mathbf { A } _ { i } \\in ( \\mathcal { O B } \\dot { B } ) _ { 4 e } ^ { m w } } \\end{array}$ )mw4e , as desired. □ ", + "bbox": [ + 174, + 102, + 825, + 160 + ], + "page_idx": 41 + }, + { + "type": "text", + "text": "K.3.4 ARITHMETIC CIRCUITS ", + "text_level": 1, + "bbox": [ + 176, + 178, + 397, + 193 + ], + "page_idx": 41 + }, + { + "type": "text", + "text": "Just as in Theorem 1, we can use the sparsity result in Lemma K.10 to place matrices with low-depth (linear) arithmetic circuits for matrix vector multiplication in the $\\ O B B$ hierarchy. ", + "bbox": [ + 173, + 203, + 823, + 231 + ], + "page_idx": 41 + }, + { + "type": "text", + "text": "Corollary K.12. Let M be an $n \\times n$ matrix such that matrix-vector multiplication of M times an \narbitrary vector v can be represented as a be a linear arithmetic circuit $C$ comprised of s gates \n(including inputs) and having depth $d .$ . Then, $\\mathbf { M } \\in ( \\mathcal { O } B B ) _ { O ( \\frac { s } { n } ) } ^ { O ( d ) }$ . ", + "bbox": [ + 174, + 236, + 825, + 285 + ], + "page_idx": 41 + }, + { + "type": "text", + "text": "Proof. We use the construction given in the proof of Theorem 1. Corollaries K.9 and K.4 allow us to recover the same width and expansion factor with the $o s B$ hierarchy. □ ", + "bbox": [ + 171, + 303, + 823, + 332 + ], + "page_idx": 41 + }, + { + "type": "text", + "text": "L RELU NETWORK WITH STRUCTURED WEIGHT MATRICES", + "text_level": 1, + "bbox": [ + 173, + 353, + 678, + 369 + ], + "page_idx": 41 + }, + { + "type": "text", + "text": "We show that for any neural network with ReLU nonlinearities and whose weight matrices have arithmetic circuits with few gates, its linear network counterpart (obtained by removing all the ReLU’s) also has an arithmetic circuit with not too many more gates. This implies that in trying to find the smallest arithmetic circuit augmented with ReLU gates to represent a ReLU network, one might as well try to find the smallest arithmetic circuits that represent the matrix-vector multiplication of each weight matrix. ", + "bbox": [ + 174, + 385, + 825, + 468 + ], + "page_idx": 41 + }, + { + "type": "text", + "text": "Proposition 2. Consider a neural network architecture consisting of $L$ layers with weight matrices $\\mathbf { W } _ { 1 } , \\dots , \\mathbf { W } _ { L } \\in \\mathbb { F } ^ { n \\times n }$ and ReLU nonlinearity in between. ", + "bbox": [ + 174, + 473, + 823, + 502 + ], + "page_idx": 41 + }, + { + "type": "text", + "text": "Suppose that matrix-vector multiplication of $\\mathbf { W } _ { i }$ times an arbitrary vector v can be represented as a linear arithmetic circuit with $s _ { i }$ gates (including inputs). Then there exists an arithmetic circuit augmented with ReLU gates with PLi=1 si + Ln total gates that computes the output ReLU $( \\mathbf { W } _ { L } ( \\dots \\mathrm { R e L U } ( \\mathbf { W } _ { 1 } \\mathbf { v } ) ) )$ ) of the network for an arbitrary input vector v. ", + "bbox": [ + 174, + 508, + 825, + 568 + ], + "page_idx": 41 + }, + { + "type": "text", + "text": "Conversely, if there is an arithmetic circuit augmented with ReLU gates with s total gates that computes all the activations of the network ReL $\\operatorname { \\mathrm { . U } } ( \\mathbf { W } _ { 1 } \\mathbf { v } ) , \\ldots , \\operatorname { R e L U } ( \\mathbf { W } _ { L } \\ldots \\operatorname { \\mathrm { . R e L U } } ( \\mathbf { W } _ { 1 } \\mathbf { v } ) )$ for an arbitrary input v, then there exists an arithmetic circuit augmented with ReLU gates with $2 s + 2 L n$ total gates that computes the activations of the network without ReLU $\\mathbf { W } _ { 1 } \\mathbf { v } , \\dots , \\mathbf { W } _ { L } \\dots \\mathbf { W } _ { 1 } \\mathbf { v }$ . ", + "bbox": [ + 173, + 574, + 825, + 631 + ], + "page_idx": 41 + }, + { + "type": "text", + "text": "Proof of Proposition 2. To compute the output of the network Re $\\operatorname { \\mathrm { . U } } ( \\mathbf { W } _ { L } ( \\dots \\operatorname { R e L U } ( \\mathbf { W } _ { 1 } \\mathbf { v } ) ) )$ ), we first compute the matrix-vector product $\\mathbf { W } _ { 1 } \\mathbf { v }$ with an arithmetic circuit of $s _ { 1 }$ gates by assumption, and use $n$ other ReLU gates to compute the pointwise ReLU. Then we repeat the process for layer $2 , 3 , \\ldots , L$ , using the arithmetic circuits of $\\mathbf { W } _ { 1 } , \\dots , \\mathbf { W } _ { L }$ and $L n$ additional gates for ReLU. In total we obtain an arithmetic circuit augmented with ReLU gates with $\\textstyle \\sum _ { i = 1 } ^ { L } s _ { i } + L n$ total gates. ", + "bbox": [ + 173, + 648, + 825, + 722 + ], + "page_idx": 41 + }, + { + "type": "text", + "text": "Conversely, to build an arithmetic circuit augmented with ReLU gates to compute $\\mathbf { W } _ { 1 } \\mathbf { v } , \\dots , \\mathbf { W } _ { L } \\dots \\mathbf { W } _ { 1 } \\mathbf { v }$ , we pass $\\mathbf { v }$ and then $- \\mathbf { v }$ through the circuit that computes $\\mathrm { R e L U } ( \\mathbf { W } _ { 1 } \\mathbf { x } )$ for an arbitrary $\\mathbf { x }$ to get ReLU $\\left( \\mathbf { W } _ { 1 } \\mathbf { v } \\right)$ and ReLU $\\left( - \\mathbf { W } _ { 1 } \\mathbf { v } \\right)$ . Noting that $x = \\mathrm { R e L U } ( x ) - \\mathrm { R e L U } ( - x )$ , we can use $n$ additional gates to compute $\\mathbf { W } _ { 1 } \\mathbf { v }$ from ReLU $\\left( \\mathbf { W } _ { 1 } \\mathbf { v } \\right)$ and $\\operatorname { R e L U } ( - \\mathbf { W } _ { 1 } \\mathbf { v } )$ . ", + "bbox": [ + 174, + 728, + 825, + 785 + ], + "page_idx": 41 + }, + { + "type": "text", + "text": "Repeat the process for layer $2 , 3 , \\ldots , L$ (for example, pass $\\mathbf { W } _ { 1 } \\mathbf { v }$ and $- \\mathbf { W } _ { 1 } \\mathbf { v }$ to the circuit that computes $\\mathbf { W } _ { \\mathrm { 2 } } \\mathbf { x }$ for an arbitrary $\\mathbf { x }$ on layer 2). Overall we need to double the circuits that computes all the activations of the network $\\operatorname { R e L U } ( \\mathbf { \\dot { W } } _ { 1 } \\mathbf { v } ) , \\dots , \\operatorname { R e L U } ( \\mathbf { W } _ { L } \\dots \\operatorname { R e L U } ( \\mathbf { W } _ { 1 } \\mathbf { v } ) )$ , requiring $2 s$ gates. We also need $n$ additional gates per layer to compute the negation of the input to that layer (e.g. computing $- \\mathbf { v }$ from v), and $n$ additional gates per layer to subtract the output of the ReLU circuit (e.g. computing $\\mathbf { W } _ { 1 } \\mathbf { v }$ from $\\mathrm { R e L U } ( \\mathbf { W } _ { 1 } \\mathbf { v } )$ and $\\operatorname { R e L U } ( - \\mathbf { W } _ { 1 } \\mathbf { v } )$ .) Therefore we can construct an arithmetic circuit augmented with ReLU gates with $2 s + 2 L$ total gates that computes the activations of the network without ReLU $\\mathbf { W } _ { 1 } \\mathbf { v } , \\dots , \\mathbf { W } _ { L } \\dots \\mathbf { W } _ { 1 } \\mathbf { v }$ . ", + "bbox": [ + 173, + 790, + 825, + 904 + ], + "page_idx": 41 + }, + { + "type": "text", + "text": "We now prove an asymptotic bound on the VC dimension of a ReLU network whose weight matrices are kaleidoscope matrices with bounded width and expansion. ", + "bbox": [ + 171, + 103, + 823, + 132 + ], + "page_idx": 42 + }, + { + "type": "text", + "text": "Proposition 3. Let $\\mathcal { F }$ be the class of ReLU neural networks consisting of $L$ layers, where each layer is a $K$ -matrix with width and expansion bounded by some constant $C$ . Suppose that the network has $W$ total parameters. Let sign $\\mathcal { F }$ denote the corresponding classification functions: $\\{ x \\mapsto \\operatorname { s i g n } f ( x ) : f \\in { \\mathcal { F } } \\}$ . Then this class has VC dimension: ", + "bbox": [ + 173, + 133, + 825, + 190 + ], + "page_idx": 42 + }, + { + "type": "equation", + "img_path": "images/b07fcddd39796f289207532c245eb6a0f0438676278f9347655be1f0eef21dc2.jpg", + "text": "$$\n\\operatorname { V C d i m } ( \\operatorname { s i g n } { \\mathcal { F } } ) = O ( L W \\log W ) .\n$$", + "text_format": "latex", + "bbox": [ + 380, + 193, + 616, + 209 + ], + "page_idx": 42 + }, + { + "type": "text", + "text": "We leverage the result from Thomas et al. (2018) for the case where the entries of the weight matrices interact multiplicatively, but with polynomially bounded degrees. This proof is similar to the VC bound for ReLU networks whose weight matrices are butterfly matrices (Dao et al., 2019). ", + "bbox": [ + 174, + 217, + 825, + 260 + ], + "page_idx": 42 + }, + { + "type": "text", + "text": "Proof. To use Theorem 3 of Thomas et al. (2018), we simply need to check that the entries of the linear layer, as polynomials of the parameters, has degree at most $c _ { 1 } m _ { l } ^ { c _ { 2 } }$ for some universal constant $c _ { 1 } , c _ { 2 } > 0$ , where $m _ { l }$ is the size of output of the $l$ -th layer. If the network weight matrices are K-matrices with bounded width and expansion, each weight matrix is a product of at most $c _ { 3 } \\log m _ { l }$ sparse factors, for some universal constant $c _ { 3 } > 0$ . This means that the degree is polynomially bounded, which satisfies the condition of the theorem. Therefore the VC dimension is bounded to be almost linear in the number of parameters: ", + "bbox": [ + 173, + 273, + 825, + 372 + ], + "page_idx": 42 + }, + { + "type": "equation", + "img_path": "images/f76055338f318f1bde28f1efe8b39133be9d88513589e943c274bfb13b326720.jpg", + "text": "$$\n\\operatorname { V C d i m } ( \\operatorname { s i g n } { \\mathcal { F } } ) = O ( L W \\log W ) .\n$$", + "text_format": "latex", + "bbox": [ + 380, + 375, + 617, + 391 + ], + "page_idx": 42 + }, + { + "type": "text", + "text": "M ARITHMETIC CIRCUIT PRIMER ", + "text_level": 1, + "bbox": [ + 176, + 426, + 470, + 443 + ], + "page_idx": 42 + }, + { + "type": "text", + "text": "We give a quick overview of arithmetic circuits. This is a model of computation that has been studied for numerous computational problems (and is the basic model for algebraic complexity theory). For our purposes, we will exclusively focus on arithmetic circuits for the matrix-vector multiplication problem. For a more detailed exposition, the reader is referred to the standard book on this topic (Bürgisser et al., 2013). ", + "bbox": [ + 173, + 457, + 825, + 527 + ], + "page_idx": 42 + }, + { + "type": "text", + "text": "Definition M.1 (Arithmetic Circuits). An arithmetic circuit that computes $\\mathbf { y } = \\mathbf { A x }$ (for $\\mathbf { A } \\in \\mathbb { F } ^ { m \\times n } ,$ ) has $n$ input gates (corresponding to $\\mathbf { x } [ 0 ] , \\ldots , \\mathbf { x } [ n - 1 ] )$ and $m$ output gates (corresponding to $\\mathbf { y } [ 0 ] , \\ldots , \\mathbf { y } [ m - 1 ] )$ . All the internal gates correspond to addition, subtraction, multiplication and division13 over the underlying field F. The circuit is also allowed to use constants from F for ‘free.’ The definition of the internal gates can depend on A (as well as x of course). In other words, one can ‘bake’ the knowledge about A into the circuit. ", + "bbox": [ + 173, + 529, + 825, + 613 + ], + "page_idx": 42 + }, + { + "type": "text", + "text": "The size s of a circuit is n plus the number of addition, multiplication, subtraction and division gates used in the circuit. The depth d of a circuit is the minimum number of layers such that all gates in a given layer take as its input gates from previous layers.14 ", + "bbox": [ + 174, + 619, + 825, + 662 + ], + "page_idx": 42 + }, + { + "type": "text", + "text": "One drawback of arithmetic circuits (especially for infinite fields e.g. $\\mathbb { F } = \\mathbb { R }$ , which is our preferred choice in this work) is that they assume operations over $\\mathbb { F }$ can be performed exactly. In particular, it ignores precision issues involved with real arithmetic. Nonetheless, this model turns out to be a very useful model in reasoning about the complexity of doing matrix-vector multiplication for any family of matrices. ", + "bbox": [ + 174, + 671, + 825, + 741 + ], + "page_idx": 42 + }, + { + "type": "text", + "text": "Perhaps the strongest argument in support of arithmetic circuits is that a large (if not an overwhelming) majority of matrix-vector multiplication algorithm also imply an arithmetic circuit of size comparable to the runtime of the algorithm (and the depth of the circuit roughly correponds to the time taken to for each compute it by a parallel algorithm). For example consider the obvious algorithm to compute $i \\in [ m ]$ , compute $\\mathbf { y } [ i ]$ as the sum $\\begin{array} { r } { \\bar { \\sum } _ { i = 0 } ^ { n - 1 } \\mathbf { A } [ i , j ] \\mathbf { x } [ j ] ) } \\end{array}$ . It is easy to see that this algorithm $\\mathbf { A x }$ (i.e. implies an arithmetic circuit of size $O ( n m )$ and depth $O ( \\log n )$ .15 ", + "bbox": [ + 173, + 748, + 826, + 837 + ], + "page_idx": 42 + }, + { + "type": "text", + "text": "One thing to note about the arithmetic circuit above is that all the multiplications involve at least one input that is a constant from $\\mathbb { F }$ (recall that we can assume that the entries of A are constants that can be used to build the circuit). This leads to the following important sub-class of arithmetic circuits: ", + "bbox": [ + 174, + 103, + 823, + 146 + ], + "page_idx": 43 + }, + { + "type": "text", + "text": "Definition M.2 (Linear Arithmetic Circuits). An arithmetic circuit is called a linear arithmetic circuit if it only uses addition, subtraction and multiplication. Further, every multiplcation has a fixed constant from $\\mathbb { F }$ as at least one of its two inputs. In other words, all gates in the circuit are linear functions of their inputs (i.e. of the form $a x +$ by for fixed constants $a , b \\in \\mathbb { F }$ ). ", + "bbox": [ + 173, + 164, + 825, + 220 + ], + "page_idx": 43 + }, + { + "type": "text", + "text": "Intuitively for the matrix-vector multiplication, it makes sense to consider linear arithmetic circuits since the final function we want to compute $\\mathbf { A x }$ is indeed a linear function of its inputs. For inifinite fields (e.g. $\\mathbb { F } = \\mathbb { R }$ or $\\mathbb { F } = \\mathbb { C }$ ), it turns out that this is essentially without loss of generality: ", + "bbox": [ + 174, + 247, + 825, + 289 + ], + "page_idx": 43 + }, + { + "type": "text", + "text": "Theorem 4 ((Bürgisser et al., 2013)). Let F be an infinite field. Any (general) arithmetic circuit to compute Ax over $\\mathbb { F }$ of size s and depth d can be converted into a linear arithmetic circuit of size $O ( s )$ and depth $O ( d )$ . ", + "bbox": [ + 174, + 306, + 825, + 349 + ], + "page_idx": 43 + }, + { + "type": "text", + "text": "The above result implies that for asymptotic considerations, linear arithmetic circuits for matrix-vector multiplication are equivalent to general arithmetic circuits.16 ", + "bbox": [ + 174, + 376, + 823, + 404 + ], + "page_idx": 43 + }, + { + "type": "text", + "text": "One important property of linear arithmetic circuits of depth $d$ , which we will use in our arguments, is that such a circuit can be equivalently represented as product of $d$ sparse matrices (see the proof of Theorem 1 for the precise derivation17). ", + "bbox": [ + 174, + 410, + 825, + 452 + ], + "page_idx": 43 + }, + { + "type": "text", + "text": "As mentioned earlier, a vast majority of efficient matrix vector multiplication algorithms are equivalent to small (both in size and depth) linear arithmetic circuit. For example the FFT can be thought of as an efficient arithmetic circuit to compute the Discrete Fourier Transform (indeed when one converts the linear arithmetic circuit for FFT into a matrix decomposition,18 then each matrix in the decomposition is a butterfly factor, with each block matrix in each factor being the same). For an illustration of this consider the DFT with $n = 4$ as illustrated in Figure 11. ", + "bbox": [ + 174, + 459, + 825, + 544 + ], + "page_idx": 43 + }, + { + "type": "image", + "img_path": "images/a3cbb522b99a7e6b2265f68e5eb9f6e0cce9cdc88c2032108159d5ac677d7182.jpg", + "image_caption": [ + "Figure 11: DFT of order 4. " + ], + "image_footnote": [], + "bbox": [ + 434, + 590, + 562, + 689 + ], + "page_idx": 43 + }, + { + "type": "text", + "text": "Figure 12 represent the arithmetic circuit corresponding to FFT with $n = 4$ ", + "bbox": [ + 173, + 775, + 668, + 790 + ], + "page_idx": 43 + }, + { + "type": "image", + "img_path": "images/ed59117b4e076a126cd83314d57abe6b306ececc3fd8efefc8ff242d70ff5fe8.jpg", + "image_caption": [ + "Figure 12: Arithmetic circuit for 4-DFT from Figure 11. " + ], + "image_footnote": [], + "bbox": [ + 173, + 101, + 823, + 313 + ], + "page_idx": 44 + }, + { + "type": "text", + "text": "Finally, Figure 13 is representation of the arithmetic circuit of Figure 12 as a product of a butterfly matrix and (the bit-reversal) permutation. We note that our generic arithmetic circuit to decomposition into $B B ^ { * }$ is not as tight as in Figure 13. ", + "bbox": [ + 174, + 367, + 825, + 410 + ], + "page_idx": 44 + }, + { + "type": "image", + "img_path": "images/1cb545a085d2ee937c721349e630bd97a789faba060e50b28a39b329f3ede1fd.jpg", + "image_caption": [ + "Figure 13: Decomposition of DFT of Figure 11 via the arithmetic circuit of Figure 12. " + ], + "image_footnote": [], + "bbox": [ + 303, + 428, + 692, + 568 + ], + "page_idx": 44 + }, + { + "type": "text", + "text": "One reason for the vast majority of existing efficient matrix vector algorithms leading to (linear) arithmetic circuits is that they generally are divide and conquer algorithms that use polynomial operations such as polynomial multiplication or evaluation (both of which themselves are divide and conquer algorithms that use FFT as a blackbox) or polynomial addition. Each of these pieces are well known to have small (depth and size) linear arithmetic circuits (since FFT has these properties). Finally, the divide and conquer structure of the algorithms leads to the circuit being of low depth. See the book of Pan (Pan, 2001) for a more elaborate description of this connection. ", + "bbox": [ + 173, + 627, + 825, + 724 + ], + "page_idx": 44 + }, + { + "type": "text", + "text": "In fact, the recent work of De Sa et al. (De Sa et al., 2018) makes this fact explicit and presents the most general known structure on matrices that imply near-linear size linear arithmetic circuits for the corresponding matrix vector multiplication. Their work combines two separate classes of structures matrices– orthogonal polynomial transforms (Driscoll et al., 1997; Szegö, 1967) as well as matrices with low displacement rank (Kailath et al., 1979; Olshevsky & Shokrollahi, 2000)– and presents a linear class of linear arithmetic circuits to solve their matrix vector multiplication problem. We note that structured matrices with low displacement rank have been used to replace fully connected layers in some neural network architectures (Sainath et al., 2013; Thomas et al., 2018). ", + "bbox": [ + 174, + 731, + 825, + 843 + ], + "page_idx": 44 + } +] \ No newline at end of file diff --git a/parse/train/BkgrBgSYDS/BkgrBgSYDS_middle.json b/parse/train/BkgrBgSYDS/BkgrBgSYDS_middle.json new file mode 100644 index 0000000000000000000000000000000000000000..c3e821b20dd1b9a9cbf0bf407eacca298c8e9a7e --- /dev/null +++ b/parse/train/BkgrBgSYDS/BkgrBgSYDS_middle.json @@ -0,0 +1,157052 @@ +{ + "pdf_info": [ + { + "preproc_blocks": [ + { + "type": "title", + "bbox": [ + 108, + 78, + 507, + 116 + ], + "lines": [ + { + "bbox": [ + 106, + 78, + 508, + 98 + ], + "spans": [ + { + "bbox": [ + 106, + 78, + 508, + 98 + ], + "score": 1.0, + "content": "KALEIDOSCOPE: AN EFFICIENT, LEARNABLE REPRE-", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 99, + 478, + 118 + ], + "spans": [ + { + "bbox": [ + 105, + 99, + 478, + 118 + ], + "score": 1.0, + "content": "SENTATION FOR ALL STRUCTURED LINEAR MAPS", + "type": "text" + } + ], + "index": 1 + } + ], + "index": 0.5 + }, 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Their effi-", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 564, + 505, + 580 + ], + "spans": [ + { + "bbox": [ + 105, + 564, + 505, + 580 + ], + "score": 1.0, + "content": "ciency in speed (fast algorithms) and space (few parameters) can reduce computation and memory", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 106, + 577, + 506, + 589 + ], + "spans": [ + { + "bbox": [ + 106, + 577, + 506, + 589 + ], + "score": 1.0, + "content": "usage. The class of structured linear maps includes fixed specialized transforms such as the discrete", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 587, + 506, + 600 + ], + "spans": [ + { + "bbox": [ + 105, + 587, + 506, + 600 + ], + "score": 1.0, + "content": "Fourier transform (DFT) and Hadamard transform used in signal processing (Cooley et al., 1969),", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 106, + 599, + 505, + 611 + ], + "spans": [ + { + "bbox": [ + 106, + 599, + 505, + 611 + ], + "score": 1.0, + "content": "convolutions for image, language, and speech modeling (Gu et al., 2018), and low-rank and sparse", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 106, + 609, + 505, + 622 + ], + "spans": [ + { + "bbox": [ + 106, + 609, + 505, + 622 + ], + "score": 1.0, + "content": "matrices for efficient storage and inference on edge devices (Yu et al., 2017). Forms of structure", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 621, + 505, + 633 + ], + "spans": [ + { + "bbox": [ + 105, + 621, + 505, + 633 + ], + "score": 1.0, + "content": "such as sparsity have been at the forefront of recent advances in ML (Frankle & Carbin, 2019), and", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 106, + 631, + 506, + 644 + ], + "spans": [ + { + "bbox": [ + 106, + 631, + 506, + 644 + ], + "score": 1.0, + "content": "are critical for on-device and energy-efficient models, two application areas of tremendous recent", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 106, + 642, + 299, + 654 + ], + "spans": [ + { + "bbox": [ + 106, + 642, + 299, + 654 + ], + "score": 1.0, + "content": "interest (Tsidulko, 2019; Schwartz et al., 2019).", + "type": "text" + } + ], + "index": 41 + } + ], + "index": 37, + "bbox_fs": [ + 105, + 555, + 506, + 654 + ] + }, + { + "type": "text", + "bbox": [ + 108, + 659, + 505, + 714 + ], + "lines": [ + { + "bbox": [ + 106, + 658, + 506, + 672 + ], + "spans": [ + { + "bbox": [ + 106, + 658, + 506, + 672 + ], + "score": 1.0, + "content": "There are a plethora of classes of structured linear maps, each with a significantly different repre-", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 106, + 670, + 505, + 682 + ], + "spans": [ + { + "bbox": [ + 106, + 670, + 505, + 682 + ], + "score": 1.0, + "content": "sentation, algorithm, and implementation. They have different tradeoffs in terms of inference speed,", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 681, + 505, + 693 + ], + "spans": [ + { + "bbox": [ + 105, + 681, + 505, + 693 + ], + "score": 1.0, + "content": "training speed, and accuracy, and the conventional wisdom is that no one class works uniformly well", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 692, + 505, + 704 + ], + "spans": [ + { + "bbox": [ + 105, + 692, + 505, + 704 + ], + "score": 1.0, + "content": "across all applications. As a result, ML practitioners currently hand-pick specific classes of structured", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 106, + 703, + 444, + 715 + ], + "spans": [ + { + "bbox": [ + 106, + 703, + 444, + 715 + ], + "score": 1.0, + "content": "linear maps for each of their applications. This is a difficult and labor-intensive task.", + "type": "text" + } + ], + "index": 46 + } + ], + "index": 44, + "bbox_fs": [ + 105, + 658, + 506, + 715 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 106, + 82, + 505, + 171 + ], + "lines": [ + { + "bbox": [ + 105, + 82, + 506, + 95 + ], + "spans": [ + { + "bbox": [ + 105, + 82, + 506, + 95 + ], + "score": 1.0, + "content": "Ideally, these problems should be addressed with a universal representation for structured linear maps:", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 93, + 506, + 107 + ], + "spans": [ + { + "bbox": [ + 105, + 93, + 506, + 107 + ], + "score": 1.0, + "content": "(i) Such a parameterization should be expressive enough to capture important classes of structure,", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 104, + 506, + 118 + ], + "spans": [ + { + "bbox": [ + 105, + 104, + 506, + 118 + ], + "score": 1.0, + "content": "with a nearly tight parameter count and runtime: the space required to represent the linear map should", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 106, + 115, + 505, + 128 + ], + "spans": [ + { + "bbox": [ + 106, + 115, + 505, + 128 + ], + "score": 1.0, + "content": "be close to optimal, and the resulting algorithm for matrix vector multiplication should be close to the", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 126, + 506, + 140 + ], + "spans": [ + { + "bbox": [ + 105, + 126, + 506, + 140 + ], + "score": 1.0, + "content": "fastest possible algorithm. (ii) The parameterization should be differentiable in order to be learned", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 137, + 506, + 150 + ], + "spans": [ + { + "bbox": [ + 105, + 137, + 506, + 150 + ], + "score": 1.0, + "content": "as a component of end-to-end ML pipelines, enabling it to easily be used as a drop-in replacement", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 148, + 505, + 162 + ], + "spans": [ + { + "bbox": [ + 105, + 148, + 505, + 162 + ], + "score": 1.0, + "content": "for manually engineered structured components. (iii) The parameterization should admit practically", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 159, + 438, + 173 + ], + "spans": [ + { + "bbox": [ + 105, + 159, + 438, + 173 + ], + "score": 1.0, + "content": "efficient algorithms for training and inference, in terms of both speed and memory.", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 3.5 + }, + { + "type": "text", + "bbox": [ + 106, + 176, + 505, + 362 + ], + "lines": [ + { + "bbox": [ + 106, + 177, + 505, + 188 + ], + "spans": [ + { + "bbox": [ + 106, + 177, + 505, + 188 + ], + "score": 1.0, + "content": "Currently, no class of structured linear maps satisfies all of these criteria. Most existing classes of", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 187, + 506, + 200 + ], + "spans": [ + { + "bbox": [ + 105, + 187, + 506, + 200 + ], + "score": 1.0, + "content": "structured matrices—such as the class of low-rank matrices—fail to tightly capture other important", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 198, + 506, + 212 + ], + "spans": [ + { + "bbox": [ + 105, + 198, + 457, + 212 + ], + "score": 1.0, + "content": "types of structure. For example, the DFT has an efficient structured representation of size", + "type": "text" + }, + { + "bbox": [ + 458, + 198, + 502, + 210 + ], + "score": 0.93, + "content": "O ( n \\log n )", + "type": "inline_equation" + }, + { + "bbox": [ + 502, + 198, + 506, + 212 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 104, + 208, + 506, + 223 + ], + "spans": [ + { + "bbox": [ + 104, + 208, + 375, + 223 + ], + "score": 1.0, + "content": "yet cannot be well-approximated by a low-rank transform of size", + "type": "text" + }, + { + "bbox": [ + 376, + 208, + 401, + 220 + ], + "score": 0.88, + "content": "\\dot { \\ll } n ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 401, + 208, + 506, + 223 + ], + "score": 1.0, + "content": ". Another important type", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 221, + 505, + 232 + ], + "spans": [ + { + "bbox": [ + 106, + 221, + 505, + 232 + ], + "score": 1.0, + "content": "of structure is sparsity; lots of exciting recent work has focused on the design of sparse neural", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 231, + 505, + 244 + ], + "spans": [ + { + "bbox": [ + 105, + 231, + 505, + 244 + ], + "score": 1.0, + "content": "networks. For instance, sparse networks of comparable quality to their dense counterparts—yet an", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 104, + 240, + 506, + 256 + ], + "spans": [ + { + "bbox": [ + 104, + 240, + 506, + 256 + ], + "score": 1.0, + "content": "order of magnitude fewer parameters—may be created via pruning (Han et al., 2016) or by identifying", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 104, + 253, + 506, + 266 + ], + "spans": [ + { + "bbox": [ + 104, + 253, + 506, + 266 + ], + "score": 1.0, + "content": "“winning lottery tickets” (Frankle & Carbin, 2019). In parallel, recent theoretical results by De Sa", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 264, + 506, + 277 + ], + "spans": [ + { + "bbox": [ + 105, + 264, + 506, + 277 + ], + "score": 1.0, + "content": "et al. (2018) show that sparsity and the notion of structure in linear maps are fundamentally linked:", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 275, + 506, + 288 + ], + "spans": [ + { + "bbox": [ + 105, + 275, + 506, + 288 + ], + "score": 1.0, + "content": "any given matrix can be factored into a product of sparse matrices with total parameter count equal", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 286, + 506, + 299 + ], + "spans": [ + { + "bbox": [ + 105, + 286, + 506, + 299 + ], + "score": 1.0, + "content": "to the efficiency (i.e. minimum arithmetic circuit complexity) of the matrix. In other words, the", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 297, + 506, + 309 + ], + "spans": [ + { + "bbox": [ + 105, + 297, + 506, + 309 + ], + "score": 1.0, + "content": "representation of linear maps as products of sparse matrices tightly captures all forms of structure.", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 307, + 505, + 321 + ], + "spans": [ + { + "bbox": [ + 105, + 307, + 505, + 321 + ], + "score": 1.0, + "content": "Unfortunately, it is difficult to actually learn these sparse factorizations, because it requires finding", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 319, + 506, + 331 + ], + "spans": [ + { + "bbox": [ + 105, + 319, + 506, + 331 + ], + "score": 1.0, + "content": "the sparsity patterns of the factors—a discrete, nondifferentiable search problem. Thus, current", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 329, + 505, + 342 + ], + "spans": [ + { + "bbox": [ + 106, + 329, + 505, + 342 + ], + "score": 1.0, + "content": "methods for training sparse neural networks are either expensive (Frankle & Carbin, 2019) or rely", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 340, + 506, + 353 + ], + "spans": [ + { + "bbox": [ + 106, + 340, + 506, + 353 + ], + "score": 1.0, + "content": "on highly hand-tuned heuristics for evolving the sparsity patterns throughout training (Dettmers &", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 352, + 188, + 364 + ], + "spans": [ + { + "bbox": [ + 106, + 352, + 188, + 364 + ], + "score": 1.0, + "content": "Zettlemoyer, 2019).", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 16 + }, + { + "type": "text", + "bbox": [ + 107, + 368, + 505, + 522 + ], + "lines": [ + { + "bbox": [ + 106, + 368, + 505, + 381 + ], + "spans": [ + { + "bbox": [ + 106, + 368, + 505, + 381 + ], + "score": 1.0, + "content": "By contrast, we propose a representation of linear maps as products of sparse matrices with specific", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 380, + 505, + 392 + ], + "spans": [ + { + "bbox": [ + 105, + 380, + 505, + 392 + ], + "score": 1.0, + "content": "predefined sparsity patterns (Section 2), and show that it does satisfy our desiderata: it retains the", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 390, + 506, + 403 + ], + "spans": [ + { + "bbox": [ + 105, + 390, + 506, + 403 + ], + "score": 1.0, + "content": "expressiveness of unstructured sparsity, while being differentiably learnable and efficient like other", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 104, + 400, + 506, + 415 + ], + "spans": [ + { + "bbox": [ + 104, + 400, + 506, + 415 + ], + "score": 1.0, + "content": "structured representations. Concretely, our representation is based on products of a particular building", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 411, + 505, + 425 + ], + "spans": [ + { + "bbox": [ + 105, + 411, + 505, + 425 + ], + "score": 1.0, + "content": "block known as a butterfly matrix (Parker, 1995; Dao et al., 2019); we term such products kaleidoscope", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 424, + 505, + 435 + ], + "spans": [ + { + "bbox": [ + 106, + 424, + 505, + 435 + ], + "score": 1.0, + "content": "matrices (K-matrices for short).1 (i) Our main theoretical contribution (Section 2.3) concerns the", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 434, + 506, + 447 + ], + "spans": [ + { + "bbox": [ + 105, + 434, + 506, + 447 + ], + "score": 1.0, + "content": "expressiveness of this representation: we show that any structured linear map (i.e. one that can be", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 444, + 506, + 458 + ], + "spans": [ + { + "bbox": [ + 105, + 444, + 163, + 458 + ], + "score": 1.0, + "content": "applied using", + "type": "text" + }, + { + "bbox": [ + 164, + 444, + 196, + 456 + ], + "score": 0.91, + "content": "s \\ll n ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 196, + 444, + 506, + 458 + ], + "score": 1.0, + "content": "arithmetic operations) can be represented as a K-matrix, with a nearly tight", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 457, + 506, + 468 + ], + "spans": [ + { + "bbox": [ + 106, + 457, + 393, + 468 + ], + "score": 1.0, + "content": "number of parameters and algorithmic complexity (both on the order of", + "type": "text" + }, + { + "bbox": [ + 393, + 458, + 399, + 466 + ], + "score": 0.47, + "content": "s", + "type": "inline_equation" + }, + { + "bbox": [ + 400, + 457, + 506, + 468 + ], + "score": 1.0, + "content": "up to logarithmic factors).", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 467, + 506, + 480 + ], + "spans": [ + { + "bbox": [ + 106, + 467, + 506, + 480 + ], + "score": 1.0, + "content": "(ii) The kaleidoscope representation is fully differentiable; thus, all the parameters of a K-matrix", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 477, + 506, + 492 + ], + "spans": [ + { + "bbox": [ + 105, + 477, + 506, + 492 + ], + "score": 1.0, + "content": "can be learned using standard optimization algorithms such as SGD. (iii) Because of their simple,", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 489, + 505, + 502 + ], + "spans": [ + { + "bbox": [ + 105, + 489, + 505, + 502 + ], + "score": 1.0, + "content": "regular structure, K-matrices are practical and easy to use. We provide memory- and runtime-efficient", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 499, + 506, + 513 + ], + "spans": [ + { + "bbox": [ + 105, + 499, + 506, + 513 + ], + "score": 1.0, + "content": "implementations of K-matrix multiplication on CPU and GPU for training and inference, with a", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 106, + 511, + 210, + 523 + ], + "spans": [ + { + "bbox": [ + 106, + 511, + 210, + 523 + ], + "score": 1.0, + "content": "simple PyTorch interface.", + "type": "text" + } + ], + "index": 38 + } + ], + "index": 31.5 + }, + { + "type": "text", + "bbox": [ + 107, + 528, + 505, + 659 + ], + "lines": [ + { + "bbox": [ + 106, + 528, + 506, + 540 + ], + "spans": [ + { + "bbox": [ + 106, + 528, + 506, + 540 + ], + "score": 1.0, + "content": "We empirically validate that, due to their expressiveness, learnability, and efficiency, we can use", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 106, + 540, + 506, + 551 + ], + "spans": [ + { + "bbox": [ + 106, + 540, + 506, + 551 + ], + "score": 1.0, + "content": "K-matrices as a drop-in replacement for linear components in deep learning models. In Section 3.1,", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 549, + 506, + 563 + ], + "spans": [ + { + "bbox": [ + 105, + 549, + 506, + 563 + ], + "score": 1.0, + "content": "we use K-matrices to replace hand-crafted structure in two different settings. We simplify the six steps", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 560, + 505, + 573 + ], + "spans": [ + { + "bbox": [ + 105, + 560, + 505, + 573 + ], + "score": 1.0, + "content": "of filter bank computation in speech preprocessing into a single learnable K-matrix step, with only an", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 106, + 571, + 506, + 585 + ], + "spans": [ + { + "bbox": [ + 106, + 572, + 129, + 582 + ], + "score": 0.87, + "content": "0 . 4 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 129, + 571, + 506, + 585 + ], + "score": 1.0, + "content": "accuracy drop on the TIMIT speech recognition task. We use K-matrices to replace channel", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 582, + 506, + 595 + ], + "spans": [ + { + "bbox": [ + 105, + 582, + 411, + 595 + ], + "score": 1.0, + "content": "shuffles in ShuffleNet, improving ImageNet classification accuracy by up to", + "type": "text" + }, + { + "bbox": [ + 411, + 583, + 426, + 593 + ], + "score": 0.86, + "content": "5 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 426, + 582, + 506, + 595 + ], + "score": 1.0, + "content": ". In Section 3.2, we", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 594, + 505, + 606 + ], + "spans": [ + { + "bbox": [ + 105, + 594, + 505, + 606 + ], + "score": 1.0, + "content": "show that K-matrices can successfully recover latent structure; a K-matrix is used to learn latent", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 604, + 506, + 618 + ], + "spans": [ + { + "bbox": [ + 105, + 604, + 506, + 618 + ], + "score": 1.0, + "content": "permutations in a permuted image dataset (Permuted CIFAR), resulting in 9 points higher accuracy", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 615, + 505, + 628 + ], + "spans": [ + { + "bbox": [ + 105, + 615, + 505, + 628 + ], + "score": 1.0, + "content": "in a downstream CNN model. 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(iii) The parameterization should admit practically", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 159, + 438, + 173 + ], + "spans": [ + { + "bbox": [ + 105, + 159, + 438, + 173 + ], + "score": 1.0, + "content": "efficient algorithms for training and inference, in terms of both speed and memory.", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 3.5, + "bbox_fs": [ + 105, + 82, + 506, + 173 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 176, + 505, + 362 + ], + "lines": [ + { + "bbox": [ + 106, + 177, + 505, + 188 + ], + "spans": [ + { + "bbox": [ + 106, + 177, + 505, + 188 + ], + "score": 1.0, + "content": "Currently, no class of structured linear maps satisfies all of these criteria. Most existing classes of", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 187, + 506, + 200 + ], + "spans": [ + { + "bbox": [ + 105, + 187, + 506, + 200 + ], + "score": 1.0, + "content": "structured matrices—such as the class of low-rank matrices—fail to tightly capture other important", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 198, + 506, + 212 + ], + "spans": [ + { + "bbox": [ + 105, + 198, + 457, + 212 + ], + "score": 1.0, + "content": "types of structure. For example, the DFT has an efficient structured representation of size", + "type": "text" + }, + { + "bbox": [ + 458, + 198, + 502, + 210 + ], + "score": 0.93, + "content": "O ( n \\log n )", + "type": "inline_equation" + }, + { + "bbox": [ + 502, + 198, + 506, + 212 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 104, + 208, + 506, + 223 + ], + "spans": [ + { + "bbox": [ + 104, + 208, + 375, + 223 + ], + "score": 1.0, + "content": "yet cannot be well-approximated by a low-rank transform of size", + "type": "text" + }, + { + "bbox": [ + 376, + 208, + 401, + 220 + ], + "score": 0.88, + "content": "\\dot { \\ll } n ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 401, + 208, + 506, + 223 + ], + "score": 1.0, + "content": ". Another important type", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 221, + 505, + 232 + ], + "spans": [ + { + "bbox": [ + 106, + 221, + 505, + 232 + ], + "score": 1.0, + "content": "of structure is sparsity; lots of exciting recent work has focused on the design of sparse neural", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 231, + 505, + 244 + ], + "spans": [ + { + "bbox": [ + 105, + 231, + 505, + 244 + ], + "score": 1.0, + "content": "networks. For instance, sparse networks of comparable quality to their dense counterparts—yet an", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 104, + 240, + 506, + 256 + ], + "spans": [ + { + "bbox": [ + 104, + 240, + 506, + 256 + ], + "score": 1.0, + "content": "order of magnitude fewer parameters—may be created via pruning (Han et al., 2016) or by identifying", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 104, + 253, + 506, + 266 + ], + "spans": [ + { + "bbox": [ + 104, + 253, + 506, + 266 + ], + "score": 1.0, + "content": "“winning lottery tickets” (Frankle & Carbin, 2019). In parallel, recent theoretical results by De Sa", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 264, + 506, + 277 + ], + "spans": [ + { + "bbox": [ + 105, + 264, + 506, + 277 + ], + "score": 1.0, + "content": "et al. (2018) show that sparsity and the notion of structure in linear maps are fundamentally linked:", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 275, + 506, + 288 + ], + "spans": [ + { + "bbox": [ + 105, + 275, + 506, + 288 + ], + "score": 1.0, + "content": "any given matrix can be factored into a product of sparse matrices with total parameter count equal", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 286, + 506, + 299 + ], + "spans": [ + { + "bbox": [ + 105, + 286, + 506, + 299 + ], + "score": 1.0, + "content": "to the efficiency (i.e. minimum arithmetic circuit complexity) of the matrix. In other words, the", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 297, + 506, + 309 + ], + "spans": [ + { + "bbox": [ + 105, + 297, + 506, + 309 + ], + "score": 1.0, + "content": "representation of linear maps as products of sparse matrices tightly captures all forms of structure.", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 307, + 505, + 321 + ], + "spans": [ + { + "bbox": [ + 105, + 307, + 505, + 321 + ], + "score": 1.0, + "content": "Unfortunately, it is difficult to actually learn these sparse factorizations, because it requires finding", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 319, + 506, + 331 + ], + "spans": [ + { + "bbox": [ + 105, + 319, + 506, + 331 + ], + "score": 1.0, + "content": "the sparsity patterns of the factors—a discrete, nondifferentiable search problem. Thus, current", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 329, + 505, + 342 + ], + "spans": [ + { + "bbox": [ + 106, + 329, + 505, + 342 + ], + "score": 1.0, + "content": "methods for training sparse neural networks are either expensive (Frankle & Carbin, 2019) or rely", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 340, + 506, + 353 + ], + "spans": [ + { + "bbox": [ + 106, + 340, + 506, + 353 + ], + "score": 1.0, + "content": "on highly hand-tuned heuristics for evolving the sparsity patterns throughout training (Dettmers &", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 352, + 188, + 364 + ], + "spans": [ + { + "bbox": [ + 106, + 352, + 188, + 364 + ], + "score": 1.0, + "content": "Zettlemoyer, 2019).", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 16, + "bbox_fs": [ + 104, + 177, + 506, + 364 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 368, + 505, + 522 + ], + "lines": [ + { + "bbox": [ + 106, + 368, + 505, + 381 + ], + "spans": [ + { + "bbox": [ + 106, + 368, + 505, + 381 + ], + "score": 1.0, + "content": "By contrast, we propose a representation of linear maps as products of sparse matrices with specific", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 380, + 505, + 392 + ], + "spans": [ + { + "bbox": [ + 105, + 380, + 505, + 392 + ], + "score": 1.0, + "content": "predefined sparsity patterns (Section 2), and show that it does satisfy our desiderata: it retains the", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 390, + 506, + 403 + ], + "spans": [ + { + "bbox": [ + 105, + 390, + 506, + 403 + ], + "score": 1.0, + "content": "expressiveness of unstructured sparsity, while being differentiably learnable and efficient like other", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 104, + 400, + 506, + 415 + ], + "spans": [ + { + "bbox": [ + 104, + 400, + 506, + 415 + ], + "score": 1.0, + "content": "structured representations. Concretely, our representation is based on products of a particular building", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 411, + 505, + 425 + ], + "spans": [ + { + "bbox": [ + 105, + 411, + 505, + 425 + ], + "score": 1.0, + "content": "block known as a butterfly matrix (Parker, 1995; Dao et al., 2019); we term such products kaleidoscope", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 424, + 505, + 435 + ], + "spans": [ + { + "bbox": [ + 106, + 424, + 505, + 435 + ], + "score": 1.0, + "content": "matrices (K-matrices for short).1 (i) Our main theoretical contribution (Section 2.3) concerns the", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 434, + 506, + 447 + ], + "spans": [ + { + "bbox": [ + 105, + 434, + 506, + 447 + ], + "score": 1.0, + "content": "expressiveness of this representation: we show that any structured linear map (i.e. one that can be", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 444, + 506, + 458 + ], + "spans": [ + { + "bbox": [ + 105, + 444, + 163, + 458 + ], + "score": 1.0, + "content": "applied using", + "type": "text" + }, + { + "bbox": [ + 164, + 444, + 196, + 456 + ], + "score": 0.91, + "content": "s \\ll n ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 196, + 444, + 506, + 458 + ], + "score": 1.0, + "content": "arithmetic operations) can be represented as a K-matrix, with a nearly tight", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 457, + 506, + 468 + ], + "spans": [ + { + "bbox": [ + 106, + 457, + 393, + 468 + ], + "score": 1.0, + "content": "number of parameters and algorithmic complexity (both on the order of", + "type": "text" + }, + { + "bbox": [ + 393, + 458, + 399, + 466 + ], + "score": 0.47, + "content": "s", + "type": "inline_equation" + }, + { + "bbox": [ + 400, + 457, + 506, + 468 + ], + "score": 1.0, + "content": "up to logarithmic factors).", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 467, + 506, + 480 + ], + "spans": [ + { + "bbox": [ + 106, + 467, + 506, + 480 + ], + "score": 1.0, + "content": "(ii) The kaleidoscope representation is fully differentiable; thus, all the parameters of a K-matrix", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 477, + 506, + 492 + ], + "spans": [ + { + "bbox": [ + 105, + 477, + 506, + 492 + ], + "score": 1.0, + "content": "can be learned using standard optimization algorithms such as SGD. (iii) Because of their simple,", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 489, + 505, + 502 + ], + "spans": [ + { + "bbox": [ + 105, + 489, + 505, + 502 + ], + "score": 1.0, + "content": "regular structure, K-matrices are practical and easy to use. We provide memory- and runtime-efficient", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 499, + 506, + 513 + ], + "spans": [ + { + "bbox": [ + 105, + 499, + 506, + 513 + ], + "score": 1.0, + "content": "implementations of K-matrix multiplication on CPU and GPU for training and inference, with a", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 106, + 511, + 210, + 523 + ], + "spans": [ + { + "bbox": [ + 106, + 511, + 210, + 523 + ], + "score": 1.0, + "content": "simple PyTorch interface.", + "type": "text" + } + ], + "index": 38 + } + ], + "index": 31.5, + "bbox_fs": [ + 104, + 368, + 506, + 523 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 528, + 505, + 659 + ], + "lines": [ + { + "bbox": [ + 106, + 528, + 506, + 540 + ], + "spans": [ + { + "bbox": [ + 106, + 528, + 506, + 540 + ], + "score": 1.0, + "content": "We empirically validate that, due to their expressiveness, learnability, and efficiency, we can use", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 106, + 540, + 506, + 551 + ], + "spans": [ + { + "bbox": [ + 106, + 540, + 506, + 551 + ], + "score": 1.0, + "content": "K-matrices as a drop-in replacement for linear components in deep learning models. 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In Section 3.2, we", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 594, + 505, + 606 + ], + "spans": [ + { + "bbox": [ + 105, + 594, + 505, + 606 + ], + "score": 1.0, + "content": "show that K-matrices can successfully recover latent structure; a K-matrix is used to learn latent", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 604, + 506, + 618 + ], + "spans": [ + { + "bbox": [ + 105, + 604, + 506, + 618 + ], + "score": 1.0, + "content": "permutations in a permuted image dataset (Permuted CIFAR), resulting in 9 points higher accuracy", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 615, + 505, + 628 + ], + "spans": [ + { + "bbox": [ + 105, + 615, + 505, + 628 + ], + "score": 1.0, + "content": "in a downstream CNN model. 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We then propose a differentiable family of kaleidoscope matrices, composed of", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 138, + 506, + 152 + ], + "spans": [ + { + "bbox": [ + 105, + 138, + 506, + 152 + ], + "score": 1.0, + "content": "products of butterfly matrices, and prove their expressivity: all structured matrices can be represented", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 106, + 150, + 359, + 163 + ], + "spans": [ + { + "bbox": [ + 106, + 150, + 359, + 163 + ], + "score": 1.0, + "content": "in this form, with almost optimal parameter count and runtime.", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 2 + }, + { + "type": "title", + "bbox": [ + 108, + 175, + 403, + 186 + ], + "lines": [ + { + "bbox": [ + 105, + 174, + 405, + 187 + ], + "spans": [ + { + "bbox": [ + 105, + 174, + 405, + 187 + ], + "score": 1.0, + "content": "2.1 BACKGROUND: SPARSE FACTORIZATION, BUTTERFLY MATRICES", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 5 + }, + { + "type": "text", + "bbox": [ + 107, + 194, + 505, + 306 + ], + "lines": [ + { + "bbox": [ + 105, + 194, + 506, + 208 + ], + "spans": [ + { + "bbox": [ + 105, + 194, + 506, + 208 + ], + "score": 1.0, + "content": "Sparse factorization One method of constructing matrices with theoretically fast matrix-vector", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 206, + 505, + 219 + ], + "spans": [ + { + "bbox": [ + 106, + 206, + 505, + 219 + ], + "score": 1.0, + "content": "multiplication algorithms is as a product of sparse matrices, so that multiplication by an arbitrary", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 218, + 506, + 229 + ], + "spans": [ + { + "bbox": [ + 106, + 218, + 506, + 229 + ], + "score": 1.0, + "content": "vector has cost proportional to the total number of nonzeros (NNZ) of the matrices in the product.", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 228, + 506, + 241 + ], + "spans": [ + { + "bbox": [ + 106, + 228, + 506, + 241 + ], + "score": 1.0, + "content": "Surprisingly, the converse is also true. De Sa et al. (2018) introduce the concept of sparse product", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 106, + 239, + 505, + 251 + ], + "spans": [ + { + "bbox": [ + 106, + 239, + 505, + 251 + ], + "score": 1.0, + "content": "width (SPW), which roughly corresponds to the total NNZ in a factorization of a matrix, and", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 249, + 506, + 263 + ], + "spans": [ + { + "bbox": [ + 105, + 249, + 506, + 263 + ], + "score": 1.0, + "content": "show that it is an asymptotically optimal descriptor of the algorithmic complexity of matrix-vector", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 261, + 506, + 273 + ], + "spans": [ + { + "bbox": [ + 106, + 261, + 506, + 273 + ], + "score": 1.0, + "content": "multiplication (Bürgisser et al., 2013). We use a similar argument in the proof of our main theorem", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 272, + 504, + 284 + ], + "spans": [ + { + "bbox": [ + 106, + 272, + 504, + 284 + ], + "score": 1.0, + "content": "(Section 2.3). However, attempting to learn such a factorization of a given matrix is difficult, as the", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 283, + 506, + 295 + ], + "spans": [ + { + "bbox": [ + 105, + 283, + 506, + 295 + ], + "score": 1.0, + "content": "sparsity constraint is not continuous. Moreover, because of the possibly irregular sparsity patterns, it", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 294, + 493, + 306 + ], + "spans": [ + { + "bbox": [ + 105, + 294, + 493, + 306 + ], + "score": 1.0, + "content": "is difficult to realize the theoretical speedups in practice (Gray et al., 2017; Gahvari et al., 2007).", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 10.5 + }, + { + "type": "text", + "bbox": [ + 106, + 317, + 505, + 372 + ], + "lines": [ + { + "bbox": [ + 106, + 317, + 505, + 329 + ], + "spans": [ + { + "bbox": [ + 106, + 317, + 505, + 329 + ], + "score": 1.0, + "content": "Butterfly matrices Butterfly matrices, encoding the recursive divide-and-conquer structure of the", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 327, + 506, + 341 + ], + "spans": [ + { + "bbox": [ + 105, + 327, + 506, + 341 + ], + "score": 1.0, + "content": "fast Fourier transform (FFT) algorithm, have long been used in numerical linear algebra (Parker,", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 338, + 506, + 352 + ], + "spans": [ + { + "bbox": [ + 105, + 338, + 506, + 352 + ], + "score": 1.0, + "content": "1995; Li et al., 2015) and machine learning (Mathieu & LeCun, 2014; Jing et al., 2017; Munkhoeva", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 348, + 505, + 363 + ], + "spans": [ + { + "bbox": [ + 105, + 348, + 505, + 363 + ], + "score": 1.0, + "content": "et al., 2018; Dao et al., 2019; Choromanski et al., 2019). Here we define butterfly matrices, which we", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 360, + 374, + 374 + ], + "spans": [ + { + "bbox": [ + 106, + 360, + 374, + 374 + ], + "score": 1.0, + "content": "use as a building block for our hierarchy of kaleidoscope matrices.", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 18 + }, + { + "type": "text", + "bbox": [ + 106, + 374, + 505, + 412 + ], + "lines": [ + { + "bbox": [ + 105, + 373, + 505, + 388 + ], + "spans": [ + { + "bbox": [ + 105, + 373, + 277, + 388 + ], + "score": 1.0, + "content": "Definition 2.1. A butterfly factor of size", + "type": "text" + }, + { + "bbox": [ + 277, + 375, + 306, + 387 + ], + "score": 0.89, + "content": "k \\geq 2", + "type": "inline_equation" + }, + { + "bbox": [ + 306, + 373, + 359, + 388 + ], + "score": 1.0, + "content": "(denoted as", + "type": "text" + }, + { + "bbox": [ + 359, + 375, + 374, + 387 + ], + "score": 0.85, + "content": "\\mathbf { B } _ { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 374, + 373, + 477, + 388 + ], + "score": 1.0, + "content": ") is a matrix of the form", + "type": "text" + }, + { + "bbox": [ + 477, + 375, + 505, + 387 + ], + "score": 0.86, + "content": "\\mathbf { B } _ { k } \\ =", + "type": "inline_equation" + } + ], + "index": 21 + }, + { + "bbox": [ + 108, + 385, + 464, + 413 + ], + "spans": [ + { + "bbox": [ + 108, + 386, + 154, + 412 + ], + "score": 0.9, + "content": "\\left[ \\begin{array} { c c } { \\mathbf { D } _ { 1 } } & { \\mathbf { D } _ { 2 } } \\\\ { \\mathbf { D } _ { 3 } } & { \\mathbf { D } _ { 4 } } \\end{array} \\right]", + "type": "inline_equation" + }, + { + "bbox": [ + 155, + 385, + 203, + 413 + ], + "score": 1.0, + "content": "where each", + "type": "text" + }, + { + "bbox": [ + 204, + 392, + 217, + 404 + ], + "score": 0.83, + "content": "\\mathbf { D } _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 217, + 385, + 235, + 413 + ], + "score": 1.0, + "content": "is a", + "type": "text" + }, + { + "bbox": [ + 235, + 391, + 262, + 406 + ], + "score": 0.87, + "content": "{ \\begin{array} { l } { { \\frac { k } { 2 } } \\times { \\frac { k } { 2 } } } \\end{array} }", + "type": "inline_equation" + }, + { + "bbox": [ + 262, + 385, + 378, + 413 + ], + "score": 1.0, + "content": "diagonal matrix. We restrict", + "type": "text" + }, + { + "bbox": [ + 378, + 393, + 385, + 403 + ], + "score": 0.45, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 385, + 385, + 464, + 413 + ], + "score": 1.0, + "content": "to be a power of 2.", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 21.5 + }, + { + "type": "text", + "bbox": [ + 107, + 415, + 505, + 440 + ], + "lines": [ + { + "bbox": [ + 104, + 412, + 507, + 431 + ], + "spans": [ + { + "bbox": [ + 104, + 412, + 377, + 431 + ], + "score": 1.0, + "content": "Definition 2.2. A butterfly factor matrix of size n with block size", + "type": "text" + }, + { + "bbox": [ + 377, + 417, + 384, + 427 + ], + "score": 0.79, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 385, + 412, + 436, + 431 + ], + "score": 1.0, + "content": "(denoted as", + "type": "text" + }, + { + "bbox": [ + 437, + 414, + 457, + 430 + ], + "score": 0.9, + "content": "\\mathbf { B } _ { k } ^ { \\left( n \\right) }", + "type": "inline_equation" + }, + { + "bbox": [ + 458, + 412, + 507, + 431 + ], + "score": 1.0, + "content": ") is a block", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 427, + 375, + 442 + ], + "spans": [ + { + "bbox": [ + 105, + 427, + 183, + 442 + ], + "score": 1.0, + "content": "diagonal matrix of", + "type": "text" + }, + { + "bbox": [ + 184, + 429, + 191, + 441 + ], + "score": 0.83, + "content": "\\frac { n } { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 192, + 427, + 363, + 442 + ], + "score": 1.0, + "content": "(possibly different) butterfly factors of size", + "type": "text" + }, + { + "bbox": [ + 364, + 429, + 370, + 438 + ], + "score": 0.8, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 371, + 427, + 375, + 442 + ], + "score": 1.0, + "content": ":", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 23.5 + }, + { + "type": "interline_equation", + "bbox": [ + 220, + 445, + 390, + 466 + ], + "lines": [ + { + "bbox": [ + 220, + 445, + 390, + 466 + ], + "spans": [ + { + "bbox": [ + 220, + 445, + 390, + 466 + ], + "score": 0.93, + "content": "\\mathbf B _ { k } ^ { ( n ) } = \\operatorname { d i a g } \\left( \\left[ \\mathbf B _ { k } \\right] _ { 1 } , \\left[ \\mathbf B _ { k } \\right] _ { 2 } , \\ldots , \\left[ \\mathbf B _ { k } \\right] _ { \\frac n k } \\right)", + "type": "interline_equation", + "image_path": "7e206429ba9471ddd994e66e0577cf3ee0242b5ee73bf922a44600792ee16eb1.jpg" + } + ] + } + ], + "index": 25, + "virtual_lines": [ + { + "bbox": [ + 220, + 445, + 390, + 466 + ], + "spans": [], + "index": 25 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 470, + 506, + 510 + ], + "lines": [ + { + "bbox": [ + 102, + 468, + 506, + 505 + ], + "spans": [ + { + "bbox": [ + 102, + 480, + 334, + 502 + ], + "score": 1.0, + "content": "product of butterfly factor matrices: B(n) = B(n)n B(n)n2 .", + "type": "text" + }, + { + "bbox": [ + 104, + 468, + 273, + 485 + ], + "score": 1.0, + "content": "Definition 2.3. A butterfly matrix of size", + "type": "text" + }, + { + "bbox": [ + 273, + 473, + 280, + 481 + ], + "score": 0.62, + "content": "n", + "type": "inline_equation" + }, + { + "bbox": [ + 281, + 468, + 330, + 485 + ], + "score": 1.0, + "content": "(denoted as", + "type": "text" + }, + { + "bbox": [ + 330, + 470, + 350, + 482 + ], + "score": 0.85, + "content": "\\mathbf { B } ^ { ( n ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 351, + 468, + 506, + 485 + ], + "score": 1.0, + "content": ") is a matrix that can be expressed as a", + "type": "text" + }, + { + "bbox": [ + 364, + 476, + 483, + 505 + ], + "score": 1.0, + "content": ". Equivalently, we may define", + "type": "text" + }, + { + "bbox": [ + 484, + 484, + 504, + 495 + ], + "score": 0.88, + "content": "\\mathbf { B } ^ { ( n ) }", + "type": "inline_equation" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 497, + 377, + 511 + ], + "spans": [ + { + "bbox": [ + 106, + 497, + 377, + 511 + ], + "score": 1.0, + "content": "recursively as a matrix that can be expressed in the following form:", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 26.5 + }, + { + "type": "interline_equation", + "bbox": [ + 235, + 513, + 376, + 541 + ], + "lines": [ + { + "bbox": [ + 235, + 513, + 376, + 541 + ], + "spans": [ + { + "bbox": [ + 235, + 513, + 376, + 541 + ], + "score": 0.94, + "content": "\\mathbf { B } ^ { ( n ) } = \\mathbf { B } _ { n } ^ { ( n ) } \\left[ { \\big [ } { \\mathbf { B } ^ { ( \\frac { n } { 2 } ) } } { \\big ] } _ { 1 } \\right. \\qquad 0 \\qquad { \\big ] }", + "type": "interline_equation", + "image_path": "cbc52fb80e487a19f7e27a87dd8dcdac4c78d5c32d90781bbe0b3c788ff03326.jpg" + } + ] + } + ], + "index": 28.5, + "virtual_lines": [ + { + "bbox": [ + 235, + 513, + 376, + 527.0 + ], + "spans": [], + "index": 28 + }, + { + "bbox": [ + 235, + 527.0, + 376, + 541.0 + ], + "spans": [], + "index": 29 + } + ] + }, + { + "type": "text", + "bbox": [ + 105, + 544, + 307, + 559 + ], + "lines": [ + { + "bbox": [ + 104, + 543, + 308, + 561 + ], + "spans": [ + { + "bbox": [ + 104, + 543, + 149, + 561 + ], + "score": 1.0, + "content": "(Note that", + "type": "text" + }, + { + "bbox": [ + 149, + 545, + 181, + 558 + ], + "score": 0.92, + "content": "\\big [ \\mathbf { B } ^ { ( \\frac { n } { 2 } ) } \\big ] _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 181, + 543, + 200, + 561 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 201, + 545, + 233, + 558 + ], + "score": 0.94, + "content": "\\big [ \\mathbf { B } ^ { ( \\frac { n } { 2 } ) } \\big ] _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 233, + 543, + 308, + 561 + ], + "score": 1.0, + "content": "may be different.)", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 30 + }, + { + "type": "title", + "bbox": [ + 108, + 570, + 272, + 582 + ], + "lines": [ + { + "bbox": [ + 106, + 570, + 273, + 583 + ], + "spans": [ + { + "bbox": [ + 106, + 570, + 273, + 583 + ], + "score": 1.0, + "content": "2.2 THE KALEIDOSCOPE HIERARCHY", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 31 + }, + { + "type": "text", + "bbox": [ + 107, + 591, + 505, + 636 + ], + "lines": [ + { + "bbox": [ + 106, + 592, + 504, + 604 + ], + "spans": [ + { + "bbox": [ + 106, + 592, + 439, + 604 + ], + "score": 1.0, + "content": "Using the building block of butterfly matrices, we formally define the kaleidoscope", + "type": "text" + }, + { + "bbox": [ + 439, + 592, + 464, + 603 + ], + "score": 0.86, + "content": "( B B ^ { * } )", + "type": "inline_equation" + }, + { + "bbox": [ + 464, + 592, + 504, + 604 + ], + "score": 1.0, + "content": "hierarchy", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 603, + 505, + 615 + ], + "spans": [ + { + "bbox": [ + 105, + 603, + 505, + 615 + ], + "score": 1.0, + "content": "and prove its expressiveness. This class of matrices serves as a fully differentiable alternative to", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 614, + 504, + 626 + ], + "spans": [ + { + "bbox": [ + 105, + 614, + 504, + 626 + ], + "score": 1.0, + "content": "products of sparse matrices (Section 2.1), with similar expressivity. In Appendix J, we show where", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 624, + 409, + 637 + ], + "spans": [ + { + "bbox": [ + 105, + 624, + 409, + 637 + ], + "score": 1.0, + "content": "various common structured matrix classes are located within this hierarchy.", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 33.5 + }, + { + "type": "text", + "bbox": [ + 107, + 641, + 505, + 686 + ], + "lines": [ + { + "bbox": [ + 105, + 640, + 505, + 654 + ], + "spans": [ + { + "bbox": [ + 105, + 640, + 505, + 654 + ], + "score": 1.0, + "content": "The building block for this hierarchy is the product of a butterfly matrix and the (conjugate) transpose", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 652, + 506, + 664 + ], + "spans": [ + { + "bbox": [ + 105, + 652, + 506, + 664 + ], + "score": 1.0, + "content": "of another butterfly matrix (which is simply a product of butterfly factors taken in the opposite order).", + "type": "text" + } 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+ "content": "2 A NEARLY-TIGHT PARAMETERIZATION OF ALL STRUCTURED MATRICES", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 105, + 505, + 161 + ], + "lines": [ + { + "bbox": [ + 106, + 106, + 505, + 118 + ], + "spans": [ + { + "bbox": [ + 106, + 106, + 505, + 118 + ], + "score": 1.0, + "content": "We first present some background on the characterization of all structured matrices (i.e. those with", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 117, + 506, + 129 + ], + "spans": [ + { + "bbox": [ + 105, + 117, + 506, + 129 + ], + "score": 1.0, + "content": "subquadratic multiplication algorithms) as products of sparse factors, along with the definition of", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 128, + 506, + 141 + ], + "spans": [ + { + "bbox": [ + 105, + 128, + 506, + 141 + ], + "score": 1.0, + "content": "butterfly matrices. We then propose a differentiable family of kaleidoscope matrices, composed of", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 138, + 506, + 152 + ], + "spans": [ + { + "bbox": [ + 105, + 138, + 506, + 152 + ], + "score": 1.0, + "content": "products of butterfly matrices, and prove their expressivity: all structured matrices can be represented", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 106, + 150, + 359, + 163 + ], + "spans": [ + { + "bbox": [ + 106, + 150, + 359, + 163 + ], + "score": 1.0, + "content": "in this form, with almost optimal parameter count and runtime.", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 2, + "bbox_fs": [ + 105, + 106, + 506, + 163 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 175, + 403, + 186 + ], + "lines": [ + { + "bbox": [ + 105, + 174, + 405, + 187 + ], + "spans": [ + { + "bbox": [ + 105, + 174, + 405, + 187 + ], + "score": 1.0, + "content": "2.1 BACKGROUND: SPARSE FACTORIZATION, BUTTERFLY MATRICES", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 5 + }, + { + "type": "text", + "bbox": [ + 107, + 194, + 505, + 306 + ], + "lines": [ + { + "bbox": [ + 105, + 194, + 506, + 208 + ], + "spans": [ + { + "bbox": [ + 105, + 194, + 506, + 208 + ], + "score": 1.0, + "content": "Sparse factorization One method of constructing matrices with theoretically fast matrix-vector", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 206, + 505, + 219 + ], + "spans": [ + { + "bbox": [ + 106, + 206, + 505, + 219 + ], + "score": 1.0, + "content": "multiplication algorithms is as a product of sparse matrices, so that multiplication by an arbitrary", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 218, + 506, + 229 + ], + "spans": [ + { + "bbox": [ + 106, + 218, + 506, + 229 + ], + "score": 1.0, + "content": "vector has cost proportional to the total number of nonzeros (NNZ) of the matrices in the product.", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 228, + 506, + 241 + ], + "spans": [ + { + "bbox": [ + 106, + 228, + 506, + 241 + ], + "score": 1.0, + "content": "Surprisingly, the converse is also true. De Sa et al. (2018) introduce the concept of sparse product", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 106, + 239, + 505, + 251 + ], + "spans": [ + { + "bbox": [ + 106, + 239, + 505, + 251 + ], + "score": 1.0, + "content": "width (SPW), which roughly corresponds to the total NNZ in a factorization of a matrix, and", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 249, + 506, + 263 + ], + "spans": [ + { + "bbox": [ + 105, + 249, + 506, + 263 + ], + "score": 1.0, + "content": "show that it is an asymptotically optimal descriptor of the algorithmic complexity of matrix-vector", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 261, + 506, + 273 + ], + "spans": [ + { + "bbox": [ + 106, + 261, + 506, + 273 + ], + "score": 1.0, + "content": "multiplication (Bürgisser et al., 2013). We use a similar argument in the proof of our main theorem", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 272, + 504, + 284 + ], + "spans": [ + { + "bbox": [ + 106, + 272, + 504, + 284 + ], + "score": 1.0, + "content": "(Section 2.3). However, attempting to learn such a factorization of a given matrix is difficult, as the", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 283, + 506, + 295 + ], + "spans": [ + { + "bbox": [ + 105, + 283, + 506, + 295 + ], + "score": 1.0, + "content": "sparsity constraint is not continuous. Moreover, because of the possibly irregular sparsity patterns, it", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 294, + 493, + 306 + ], + "spans": [ + { + "bbox": [ + 105, + 294, + 493, + 306 + ], + "score": 1.0, + "content": "is difficult to realize the theoretical speedups in practice (Gray et al., 2017; Gahvari et al., 2007).", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 10.5, + "bbox_fs": [ + 105, + 194, + 506, + 306 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 317, + 505, + 372 + ], + "lines": [ + { + "bbox": [ + 106, + 317, + 505, + 329 + ], + "spans": [ + { + "bbox": [ + 106, + 317, + 505, + 329 + ], + "score": 1.0, + "content": "Butterfly matrices Butterfly matrices, encoding the recursive divide-and-conquer structure of the", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 327, + 506, + 341 + ], + "spans": [ + { + "bbox": [ + 105, + 327, + 506, + 341 + ], + "score": 1.0, + "content": "fast Fourier transform (FFT) algorithm, have long been used in numerical linear algebra (Parker,", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 338, + 506, + 352 + ], + "spans": [ + { + "bbox": [ + 105, + 338, + 506, + 352 + ], + "score": 1.0, + "content": "1995; Li et al., 2015) and machine learning (Mathieu & LeCun, 2014; Jing et al., 2017; Munkhoeva", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 348, + 505, + 363 + ], + "spans": [ + { + "bbox": [ + 105, + 348, + 505, + 363 + ], + "score": 1.0, + "content": "et al., 2018; Dao et al., 2019; Choromanski et al., 2019). Here we define butterfly matrices, which we", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 360, + 374, + 374 + ], + "spans": [ + { + "bbox": [ + 106, + 360, + 374, + 374 + ], + "score": 1.0, + "content": "use as a building block for our hierarchy of kaleidoscope matrices.", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 18, + "bbox_fs": [ + 105, + 317, + 506, + 374 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 374, + 505, + 412 + ], + "lines": [ + { + "bbox": [ + 105, + 373, + 505, + 388 + ], + "spans": [ + { + "bbox": [ + 105, + 373, + 277, + 388 + ], + "score": 1.0, + "content": "Definition 2.1. A butterfly factor of size", + "type": "text" + }, + { + "bbox": [ + 277, + 375, + 306, + 387 + ], + "score": 0.89, + "content": "k \\geq 2", + "type": "inline_equation" + }, + { + "bbox": [ + 306, + 373, + 359, + 388 + ], + "score": 1.0, + "content": "(denoted as", + "type": "text" + }, + { + "bbox": [ + 359, + 375, + 374, + 387 + ], + "score": 0.85, + "content": "\\mathbf { B } _ { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 374, + 373, + 477, + 388 + ], + "score": 1.0, + "content": ") is a matrix of the form", + "type": "text" + }, + { + "bbox": [ + 477, + 375, + 505, + 387 + ], + "score": 0.86, + "content": "\\mathbf { B } _ { k } \\ =", + "type": "inline_equation" + } + ], + "index": 21 + }, + { + "bbox": [ + 108, + 385, + 464, + 413 + ], + "spans": [ + { + "bbox": [ + 108, + 386, + 154, + 412 + ], + "score": 0.9, + "content": "\\left[ \\begin{array} { c c } { \\mathbf { D } _ { 1 } } & { \\mathbf { D } _ { 2 } } \\\\ { \\mathbf { D } _ { 3 } } & { \\mathbf { D } _ { 4 } } \\end{array} \\right]", + "type": "inline_equation" + }, + { + "bbox": [ + 155, + 385, + 203, + 413 + ], + "score": 1.0, + "content": "where each", + "type": "text" + }, + { + "bbox": [ + 204, + 392, + 217, + 404 + ], + "score": 0.83, + "content": "\\mathbf { D } _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 217, + 385, + 235, + 413 + ], + "score": 1.0, + "content": "is a", + "type": "text" + }, + { + "bbox": [ + 235, + 391, + 262, + 406 + ], + "score": 0.87, + "content": "{ \\begin{array} { l } { { \\frac { k } { 2 } } \\times { \\frac { k } { 2 } } } \\end{array} }", + "type": "inline_equation" + }, + { + "bbox": [ + 262, + 385, + 378, + 413 + ], + "score": 1.0, + "content": "diagonal matrix. We restrict", + "type": "text" + }, + { + "bbox": [ + 378, + 393, + 385, + 403 + ], + "score": 0.45, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 385, + 385, + 464, + 413 + ], + "score": 1.0, + "content": "to be a power of 2.", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 21.5, + "bbox_fs": [ + 105, + 373, + 505, + 413 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 415, + 505, + 440 + ], + "lines": [ + { + "bbox": [ + 104, + 412, + 507, + 431 + ], + "spans": [ + { + "bbox": [ + 104, + 412, + 377, + 431 + ], + "score": 1.0, + "content": "Definition 2.2. A butterfly factor matrix of size n with block size", + "type": "text" + }, + { + "bbox": [ + 377, + 417, + 384, + 427 + ], + "score": 0.79, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 385, + 412, + 436, + 431 + ], + "score": 1.0, + "content": "(denoted as", + "type": "text" + }, + { + "bbox": [ + 437, + 414, + 457, + 430 + ], + "score": 0.9, + "content": "\\mathbf { B } _ { k } ^ { \\left( n \\right) }", + "type": "inline_equation" + }, + { + "bbox": [ + 458, + 412, + 507, + 431 + ], + "score": 1.0, + "content": ") is a block", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 427, + 375, + 442 + ], + "spans": [ + { + "bbox": [ + 105, + 427, + 183, + 442 + ], + "score": 1.0, + "content": "diagonal matrix of", + "type": "text" + }, + { + "bbox": [ + 184, + 429, + 191, + 441 + ], + "score": 0.83, + "content": "\\frac { n } { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 192, + 427, + 363, + 442 + ], + "score": 1.0, + "content": "(possibly different) butterfly factors of size", + "type": "text" + }, + { + "bbox": [ + 364, + 429, + 370, + 438 + ], + "score": 0.8, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 371, + 427, + 375, + 442 + ], + "score": 1.0, + "content": ":", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 23.5, + "bbox_fs": [ + 104, + 412, + 507, + 442 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 220, + 445, + 390, + 466 + ], + "lines": [ + { + "bbox": [ + 220, + 445, + 390, + 466 + ], + "spans": [ + { + "bbox": [ + 220, + 445, + 390, + 466 + ], + "score": 0.93, + "content": "\\mathbf B _ { k } ^ { ( n ) } = \\operatorname { d i a g } \\left( \\left[ \\mathbf B _ { k } \\right] _ { 1 } , \\left[ \\mathbf B _ { k } \\right] _ { 2 } , \\ldots , \\left[ \\mathbf B _ { k } \\right] _ { \\frac n k } \\right)", + "type": "interline_equation", + "image_path": "7e206429ba9471ddd994e66e0577cf3ee0242b5ee73bf922a44600792ee16eb1.jpg" + } + ] + } + ], + "index": 25, + "virtual_lines": [ + { + "bbox": [ + 220, + 445, + 390, + 466 + ], + "spans": [], + "index": 25 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 470, + 506, + 510 + ], + "lines": [ + { + "bbox": [ + 102, + 468, + 506, + 505 + ], + "spans": [ + { + "bbox": [ + 102, + 480, + 334, + 502 + ], + "score": 1.0, + "content": "product of butterfly factor matrices: B(n) = B(n)n B(n)n2 .", + "type": "text" + }, + { + "bbox": [ + 104, + 468, + 273, + 485 + ], + "score": 1.0, + "content": "Definition 2.3. A butterfly matrix of size", + "type": "text" + }, + { + "bbox": [ + 273, + 473, + 280, + 481 + ], + "score": 0.62, + "content": "n", + "type": "inline_equation" + }, + { + "bbox": [ + 281, + 468, + 330, + 485 + ], + "score": 1.0, + "content": "(denoted as", + "type": "text" + }, + { + "bbox": [ + 330, + 470, + 350, + 482 + ], + "score": 0.85, + "content": "\\mathbf { B } ^ { ( n ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 351, + 468, + 506, + 485 + ], + "score": 1.0, + "content": ") is a matrix that can be expressed as a", + "type": "text" + }, + { + "bbox": [ + 364, + 476, + 483, + 505 + ], + "score": 1.0, + "content": ". Equivalently, we may define", + "type": "text" + }, + { + "bbox": [ + 484, + 484, + 504, + 495 + ], + "score": 0.88, + "content": "\\mathbf { B } ^ { ( n ) }", + "type": "inline_equation" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 497, + 377, + 511 + ], + "spans": [ + { + "bbox": [ + 106, + 497, + 377, + 511 + ], + "score": 1.0, + "content": "recursively as a matrix that can be expressed in the following form:", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 26.5, + "bbox_fs": [ + 102, + 468, + 506, + 511 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 235, + 513, + 376, + 541 + ], + "lines": [ + { + "bbox": [ + 235, + 513, + 376, + 541 + ], + "spans": [ + { + "bbox": [ + 235, + 513, + 376, + 541 + ], + "score": 0.94, + "content": "\\mathbf { B } ^ { ( n ) } = \\mathbf { B } _ { n } ^ { ( n ) } \\left[ { \\big [ } { \\mathbf { B } ^ { ( \\frac { n } { 2 } ) } } { \\big ] } _ { 1 } \\right. \\qquad 0 \\qquad { \\big ] }", + "type": "interline_equation", + "image_path": "cbc52fb80e487a19f7e27a87dd8dcdac4c78d5c32d90781bbe0b3c788ff03326.jpg" + } + ] + } + ], + "index": 28.5, + "virtual_lines": [ + { + "bbox": [ + 235, + 513, + 376, + 527.0 + ], + "spans": [], + "index": 28 + }, + { + "bbox": [ + 235, + 527.0, + 376, + 541.0 + ], + "spans": [], + "index": 29 + } + ] + }, + { + "type": "text", + "bbox": [ + 105, + 544, + 307, + 559 + ], + "lines": [ + { + "bbox": [ + 104, + 543, + 308, + 561 + ], + "spans": [ + { + "bbox": [ + 104, + 543, + 149, + 561 + ], + "score": 1.0, + "content": "(Note that", + "type": "text" + }, + { + "bbox": [ + 149, + 545, + 181, + 558 + ], + "score": 0.92, + "content": "\\big [ \\mathbf { B } ^ { ( \\frac { n } { 2 } ) } \\big ] _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 181, + 543, + 200, + 561 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 201, + 545, + 233, + 558 + ], + "score": 0.94, + "content": "\\big [ \\mathbf { B } ^ { ( \\frac { n } { 2 } ) } \\big ] _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 233, + 543, + 308, + 561 + ], + "score": 1.0, + "content": "may be different.)", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 30, + "bbox_fs": [ + 104, + 543, + 308, + 561 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 570, + 272, + 582 + ], + "lines": [ + { + "bbox": [ + 106, + 570, + 273, + 583 + ], + "spans": [ + { + "bbox": [ + 106, + 570, + 273, + 583 + ], + "score": 1.0, + "content": "2.2 THE KALEIDOSCOPE HIERARCHY", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 31 + }, + { + "type": "text", + "bbox": [ + 107, + 591, + 505, + 636 + ], + "lines": [ + { + "bbox": [ + 106, + 592, + 504, + 604 + ], + "spans": [ + { + "bbox": [ + 106, + 592, + 439, + 604 + ], + "score": 1.0, + "content": "Using the building block of butterfly matrices, we formally define the kaleidoscope", + "type": "text" + }, + { + "bbox": [ + 439, + 592, + 464, + 603 + ], + "score": 0.86, + "content": "( B B ^ { * } )", + "type": "inline_equation" + }, + { + "bbox": [ + 464, + 592, + 504, + 604 + ], + "score": 1.0, + "content": "hierarchy", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 603, + 505, + 615 + ], + "spans": [ + { + "bbox": [ + 105, + 603, + 505, + 615 + ], + "score": 1.0, + "content": "and prove its expressiveness. This class of matrices serves as a fully differentiable alternative to", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 614, + 504, + 626 + ], + "spans": [ + { + "bbox": [ + 105, + 614, + 504, + 626 + ], + "score": 1.0, + "content": "products of sparse matrices (Section 2.1), with similar expressivity. 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This", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 307, + 490, + 320 + ], + "spans": [ + { + "bbox": [ + 105, + 307, + 208, + 320 + ], + "score": 1.0, + "content": "hierarchy generalizes the", + "type": "text" + }, + { + "bbox": [ + 209, + 308, + 224, + 317 + ], + "score": 0.81, + "content": "B \\mathcal { P }", + "type": "inline_equation" + }, + { + "bbox": [ + 225, + 307, + 490, + 320 + ], + "score": 1.0, + "content": "hierarchy proposed by Dao et al. (2019), as shown in Appendix J.", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 12.5 + }, + { + "type": "text", + "bbox": [ + 106, + 330, + 505, + 407 + ], + "lines": [ + { + "bbox": [ + 105, + 329, + 506, + 344 + ], + "spans": [ + { + "bbox": [ + 105, + 329, + 302, + 344 + ], + "score": 1.0, + "content": "Efficiency in space and speed Each matrix in", + "type": "text" + }, + { + "bbox": [ + 302, + 330, + 335, + 342 + ], + "score": 0.92, + "content": "( B B ^ { * } ) _ { e } ^ { w }", + "type": "inline_equation" + }, + { + "bbox": [ + 335, + 329, + 397, + 344 + ], + "score": 1.0, + "content": "is a product of", + "type": "text" + }, + { + "bbox": [ + 397, + 331, + 410, + 340 + ], + "score": 0.83, + "content": "2 w", + "type": "inline_equation" + }, + { + "bbox": [ + 411, + 329, + 506, + 344 + ], + "score": 1.0, + "content": "total butterfly matrices", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 340, + 505, + 353 + ], + "spans": [ + { + "bbox": [ + 105, + 340, + 419, + 353 + ], + "score": 1.0, + "content": "and transposes of butterfly matrices, each of which is in turn a product of", + "type": "text" + }, + { + "bbox": [ + 419, + 341, + 451, + 353 + ], + "score": 0.9, + "content": "\\log ( n e )", + "type": "inline_equation" + }, + { + "bbox": [ + 451, + 340, + 505, + 353 + ], + "score": 1.0, + "content": "factors with", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 351, + 506, + 365 + ], + "spans": [ + { + "bbox": [ + 106, + 352, + 123, + 363 + ], + "score": 0.61, + "content": "2 n e", + "type": "inline_equation" + }, + { + "bbox": [ + 124, + 351, + 325, + 365 + ], + "score": 1.0, + "content": "nonzeros (NNZ) each. Therefore, each matrix in", + "type": "text" + }, + { + "bbox": [ + 325, + 352, + 357, + 364 + ], + "score": 0.91, + "content": "( B B ^ { * } ) _ { e } ^ { w }", + "type": "inline_equation" + }, + { + "bbox": [ + 358, + 351, + 375, + 365 + ], + "score": 1.0, + "content": "has", + "type": "text" + }, + { + "bbox": [ + 375, + 352, + 432, + 364 + ], + "score": 0.5, + "content": "4 w n e \\log ( n e )", + "type": "inline_equation" + }, + { + "bbox": [ + 432, + 351, + 506, + 365 + ], + "score": 1.0, + "content": "parameters and a", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 363, + 505, + 376 + ], + "spans": [ + { + "bbox": [ + 106, + 363, + 317, + 376 + ], + "score": 1.0, + "content": "matrix-vector multiplication algorithm of complexity", + "type": "text" + }, + { + "bbox": [ + 317, + 364, + 379, + 375 + ], + "score": 0.89, + "content": "O ( w n e \\log n e )", + "type": "inline_equation" + }, + { + "bbox": [ + 379, + 363, + 505, + 376 + ], + "score": 1.0, + "content": "(by multiplying the vector with", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 374, + 505, + 387 + ], + "spans": [ + { + "bbox": [ + 106, + 374, + 505, + 387 + ], + "score": 1.0, + "content": "each sparse factor sequentially). We prove this more formally in Appendix E. For the applications in", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 384, + 506, + 398 + ], + "spans": [ + { + "bbox": [ + 105, + 384, + 147, + 398 + ], + "score": 1.0, + "content": "Section 3,", + "type": "text" + }, + { + "bbox": [ + 148, + 387, + 157, + 395 + ], + "score": 0.68, + "content": "w", + "type": "inline_equation" + }, + { + "bbox": [ + 157, + 384, + 174, + 398 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 174, + 387, + 180, + 395 + ], + "score": 0.6, + "content": "e", + "type": "inline_equation" + }, + { + "bbox": [ + 181, + 384, + 330, + 398 + ], + "score": 1.0, + "content": "are small constants (up to 2), so those", + "type": "text" + }, + { + "bbox": [ + 331, + 385, + 339, + 395 + ], + "score": 0.3, + "content": "\\mathrm { K }", + "type": "inline_equation" + }, + { + "bbox": [ + 339, + 384, + 397, + 398 + ], + "score": 1.0, + "content": "-matrices have", + "type": "text" + }, + { + "bbox": [ + 397, + 385, + 442, + 397 + ], + "score": 0.93, + "content": "O ( n \\log n )", + "type": "inline_equation" + }, + { + "bbox": [ + 442, + 384, + 506, + 398 + ], + "score": 1.0, + "content": "parameters and", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 397, + 143, + 408 + ], + "spans": [ + { + "bbox": [ + 105, + 397, + 143, + 408 + ], + "score": 1.0, + "content": "runtime.", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 18 + }, + { + "type": "title", + "bbox": [ + 107, + 421, + 482, + 432 + ], + "lines": [ + { + "bbox": [ + 106, + 421, + 483, + 433 + ], + "spans": [ + { + "bbox": [ + 106, + 421, + 483, + 433 + ], + "score": 1.0, + "content": "2.3 ALL LOW-DEPTH STRUCTURED MATRICES ARE IN THE KALEIDOSCOPE HIERARCHY", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 22 + }, + { + "type": "text", + "bbox": [ + 107, + 441, + 505, + 507 + ], + "lines": [ + { + "bbox": [ + 106, + 442, + 505, + 453 + ], + "spans": [ + { + "bbox": [ + 106, + 442, + 505, + 453 + ], + "score": 1.0, + "content": "We now present our main theoretical result: the fact that general linear transformations, expressed as", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 452, + 505, + 465 + ], + "spans": [ + { + "bbox": [ + 105, + 452, + 330, + 465 + ], + "score": 1.0, + "content": "low-depth linear arithmetic circuits, are captured in the", + "type": "text" + }, + { + "bbox": [ + 331, + 452, + 350, + 462 + ], + "score": 0.89, + "content": "B B ^ { * }", + "type": "inline_equation" + }, + { + "bbox": [ + 351, + 452, + 505, + 465 + ], + "score": 1.0, + "content": "hierarchy with low width. Arithmetic", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 463, + 506, + 476 + ], + "spans": [ + { + "bbox": [ + 105, + 463, + 506, + 476 + ], + "score": 1.0, + "content": "circuits are commonly used to formalize algebraic algorithmic complexity (Bürgisser et al., 2013);", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 474, + 506, + 487 + ], + "spans": [ + { + "bbox": [ + 105, + 474, + 506, + 487 + ], + "score": 1.0, + "content": "we include a primer on this in Appendix M. The quantities of interest are the total number of gates", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 485, + 505, + 498 + ], + "spans": [ + { + "bbox": [ + 105, + 485, + 505, + 498 + ], + "score": 1.0, + "content": "in the circuit, representing the total number of steps required to perform the algorithm for a serial", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 496, + 507, + 510 + ], + "spans": [ + { + "bbox": [ + 105, + 496, + 507, + 510 + ], + "score": 1.0, + "content": "processor, and the depth, representing the minimum number of steps required for a parallel processor.", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 25.5 + }, + { + "type": "text", + "bbox": [ + 107, + 510, + 504, + 538 + ], + "lines": [ + { + "bbox": [ + 106, + 510, + 506, + 523 + ], + "spans": [ + { + "bbox": [ + 106, + 510, + 210, + 523 + ], + "score": 1.0, + "content": "Theorem 1. Let M be an", + "type": "text" + }, + { + "bbox": [ + 210, + 512, + 234, + 521 + ], + "score": 0.88, + "content": "n \\times n", + "type": "inline_equation" + }, + { + "bbox": [ + 235, + 510, + 506, + 523 + ], + "score": 1.0, + "content": "matrix such that multiplication of M times an arbitrary vector v can", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 103, + 518, + 505, + 542 + ], + "spans": [ + { + "bbox": [ + 103, + 518, + 505, + 542 + ], + "score": 1.0, + "content": "be represented as a linear arithmetic circuit with s total gates and depth d. Then, M ∈ (BB∗)O(d)O( sn ).", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 29.5 + }, + { + "type": "text", + "bbox": [ + 107, + 547, + 505, + 603 + ], + "lines": [ + { + "bbox": [ + 105, + 546, + 506, + 560 + ], + "spans": [ + { + "bbox": [ + 105, + 546, + 247, + 560 + ], + "score": 1.0, + "content": "The representation of such a matrix", + "type": "text" + }, + { + "bbox": [ + 248, + 548, + 260, + 558 + ], + "score": 0.33, + "content": "\\mathbf { M }", + "type": "inline_equation" + }, + { + "bbox": [ + 261, + 546, + 285, + 560 + ], + "score": 1.0, + "content": "in the", + "type": "text" + }, + { + "bbox": [ + 286, + 548, + 305, + 558 + ], + "score": 0.89, + "content": "B B ^ { * }", + "type": "inline_equation" + }, + { + "bbox": [ + 305, + 546, + 361, + 560 + ], + "score": 1.0, + "content": "hierarchy has", + "type": "text" + }, + { + "bbox": [ + 362, + 547, + 409, + 560 + ], + "score": 0.92, + "content": "O ( d s \\log s )", + "type": "inline_equation" + }, + { + "bbox": [ + 409, + 546, + 506, + 560 + ], + "score": 1.0, + "content": "parameters and yields a", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 107, + 558, + 505, + 571 + ], + "spans": [ + { + "bbox": [ + 107, + 559, + 154, + 571 + ], + "score": 0.92, + "content": "O ( d s \\log s )", + "type": "inline_equation" + }, + { + "bbox": [ + 154, + 558, + 328, + 571 + ], + "score": 1.0, + "content": "multiplication algorithm, compared to the", + "type": "text" + }, + { + "bbox": [ + 329, + 559, + 350, + 570 + ], + "score": 0.92, + "content": "O ( s )", + "type": "inline_equation" + }, + { + "bbox": [ + 350, + 558, + 505, + 571 + ], + "score": 1.0, + "content": "parameters and runtime of the circuit", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 104, + 569, + 506, + 582 + ], + "spans": [ + { + "bbox": [ + 104, + 569, + 506, + 582 + ], + "score": 1.0, + "content": "representation. To the best of our knowledge, the most general classes of efficient matrices that have", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 580, + 505, + 593 + ], + "spans": [ + { + "bbox": [ + 105, + 580, + 282, + 593 + ], + "score": 1.0, + "content": "been studied (De Sa et al., 2018) have depth", + "type": "text" + }, + { + "bbox": [ + 283, + 581, + 289, + 590 + ], + "score": 0.8, + "content": "d", + "type": "inline_equation" + }, + { + "bbox": [ + 290, + 580, + 351, + 593 + ], + "score": 1.0, + "content": "on the order of", + "type": "text" + }, + { + "bbox": [ + 351, + 581, + 373, + 592 + ], + "score": 0.85, + "content": "\\log n", + "type": "inline_equation" + }, + { + "bbox": [ + 373, + 580, + 405, + 593 + ], + "score": 1.0, + "content": "or poly", + "type": "text" + }, + { + "bbox": [ + 406, + 581, + 427, + 592 + ], + "score": 0.65, + "content": "\\log n", + "type": "inline_equation" + }, + { + "bbox": [ + 428, + 580, + 505, + 593 + ], + "score": 1.0, + "content": ". In these cases, the", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 591, + 482, + 604 + ], + "spans": [ + { + "bbox": [ + 105, + 591, + 482, + 604 + ], + "score": 1.0, + "content": "representation with K-matrices matches the best known bounds up to polylogarithmic factors.", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 33 + }, + { + "type": "text", + "bbox": [ + 106, + 608, + 505, + 682 + ], + "lines": [ + { + "bbox": [ + 106, + 608, + 505, + 621 + ], + "spans": [ + { + "bbox": [ + 106, + 608, + 505, + 621 + ], + "score": 1.0, + "content": "The crux of the proof of Theorem 1 (shown in Appendix F) is the construction of an almost tight", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 619, + 506, + 632 + ], + "spans": [ + { + "bbox": [ + 105, + 619, + 268, + 632 + ], + "score": 1.0, + "content": "representation of any sparse matrix as a", + "type": "text" + }, + { + "bbox": [ + 268, + 620, + 276, + 629 + ], + "score": 0.34, + "content": "\\mathbf { K }", + "type": "inline_equation" + }, + { + "bbox": [ + 277, + 619, + 506, + 632 + ], + "score": 1.0, + "content": "-matrix (i.e. a product of butterfly matrices): specifically,", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 629, + 507, + 652 + ], + "spans": [ + { + "bbox": [ + 105, + 629, + 177, + 652 + ], + "score": 1.0, + "content": "we show that any", + "type": "text" + }, + { + "bbox": [ + 177, + 637, + 203, + 646 + ], + "score": 0.88, + "content": "n \\times n", + "type": "inline_equation" + }, + { + "bbox": [ + 203, + 629, + 280, + 652 + ], + "score": 1.0, + "content": "sparse matrix with", + "type": "text" + }, + { + "bbox": [ + 280, + 639, + 286, + 646 + ], + "score": 0.66, + "content": "s", + "type": "inline_equation" + }, + { + "bbox": [ + 286, + 629, + 345, + 652 + ], + "score": 1.0, + "content": "nonzeros is in", + "type": "text" + }, + { + "bbox": [ + 345, + 630, + 400, + 651 + ], + "score": 0.94, + "content": "( B B ^ { * } ) _ { O ( 1 ) } ^ { O ( \\left\\lceil \\frac { s } { n } \\right\\rceil ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 399, + 633, + 507, + 649 + ], + "score": 1.0, + "content": "(Theorem 3, Appendix I).", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 106, + 649, + 505, + 661 + ], + "spans": [ + { + "bbox": [ + 106, + 649, + 505, + 661 + ], + "score": 1.0, + "content": "We then leverage the expressivity result of products of sparse matrices to represent all arithmetic", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 106, + 660, + 505, + 672 + ], + "spans": [ + { + "bbox": [ + 106, + 660, + 505, + 672 + ], + "score": 1.0, + "content": "circuits (similar to the sparse product width result of De Sa et al. (2018) referenced in Section 2.1) to", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 106, + 671, + 244, + 683 + ], + "spans": [ + { + "bbox": [ + 106, + 671, + 244, + 683 + ], + "score": 1.0, + "content": "complete the proof of Theorem 1.", + "type": "text" + } + ], + "index": 41 + } + ], + "index": 38.5 + }, + { + "type": "text", + "bbox": [ + 107, + 687, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 687, + 504, + 699 + ], + "spans": [ + { + "bbox": [ + 105, + 687, + 497, + 699 + ], + "score": 1.0, + "content": "This intermediate result is also a novel characterization of sparse matrices. 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This", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 307, + 490, + 320 + ], + "spans": [ + { + "bbox": [ + 105, + 307, + 208, + 320 + ], + "score": 1.0, + "content": "hierarchy generalizes the", + "type": "text" + }, + { + "bbox": [ + 209, + 308, + 224, + 317 + ], + "score": 0.81, + "content": "B \\mathcal { P }", + "type": "inline_equation" + }, + { + "bbox": [ + 225, + 307, + 490, + 320 + ], + "score": 1.0, + "content": "hierarchy proposed by Dao et al. (2019), as shown in Appendix J.", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 12.5, + "bbox_fs": [ + 104, + 273, + 507, + 320 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 330, + 505, + 407 + ], + "lines": [ + { + "bbox": [ + 105, + 329, + 506, + 344 + ], + "spans": [ + { + "bbox": [ + 105, + 329, + 302, + 344 + ], + "score": 1.0, + "content": "Efficiency in space and speed Each matrix in", + "type": "text" + }, + { + "bbox": [ + 302, + 330, + 335, + 342 + ], + "score": 0.92, + "content": "( B B ^ { * } ) _ { e } ^ { w }", + "type": "inline_equation" + }, + { + "bbox": [ + 335, + 329, + 397, + 344 + ], + "score": 1.0, + "content": "is a product of", + "type": "text" + }, + { + "bbox": [ + 397, + 331, + 410, + 340 + ], + "score": 0.83, + "content": "2 w", + "type": "inline_equation" + }, + { + "bbox": [ + 411, + 329, + 506, + 344 + ], + "score": 1.0, + "content": "total butterfly matrices", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 340, + 505, + 353 + ], + "spans": [ + { + "bbox": [ + 105, + 340, + 419, + 353 + ], + "score": 1.0, + "content": "and transposes of butterfly matrices, each of which is in turn a product of", + "type": "text" + }, + { + "bbox": [ + 419, + 341, + 451, + 353 + ], + "score": 0.9, + "content": "\\log ( n e )", + "type": "inline_equation" + }, + { + "bbox": [ + 451, + 340, + 505, + 353 + ], + "score": 1.0, + "content": "factors with", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 351, + 506, + 365 + ], + "spans": [ + { + "bbox": [ + 106, + 352, + 123, + 363 + ], + "score": 0.61, + "content": "2 n e", + "type": "inline_equation" + }, + { + "bbox": [ + 124, + 351, + 325, + 365 + ], + "score": 1.0, + "content": "nonzeros (NNZ) each. 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We prove this more formally in Appendix E. For the applications in", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 384, + 506, + 398 + ], + "spans": [ + { + "bbox": [ + 105, + 384, + 147, + 398 + ], + "score": 1.0, + "content": "Section 3,", + "type": "text" + }, + { + "bbox": [ + 148, + 387, + 157, + 395 + ], + "score": 0.68, + "content": "w", + "type": "inline_equation" + }, + { + "bbox": [ + 157, + 384, + 174, + 398 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 174, + 387, + 180, + 395 + ], + "score": 0.6, + "content": "e", + "type": "inline_equation" + }, + { + "bbox": [ + 181, + 384, + 330, + 398 + ], + "score": 1.0, + "content": "are small constants (up to 2), so those", + "type": "text" + }, + { + "bbox": [ + 331, + 385, + 339, + 395 + ], + "score": 0.3, + "content": "\\mathrm { K }", + "type": "inline_equation" + }, + { + "bbox": [ + 339, + 384, + 397, + 398 + ], + "score": 1.0, + "content": "-matrices have", + "type": "text" + }, + { + "bbox": [ + 397, + 385, + 442, + 397 + ], + "score": 0.93, + "content": "O ( n \\log n )", + "type": "inline_equation" + }, + { + "bbox": [ + 442, + 384, + 506, + 398 + ], + "score": 1.0, + "content": "parameters and", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 397, + 143, + 408 + ], + "spans": [ + { + "bbox": [ + 105, + 397, + 143, + 408 + ], + "score": 1.0, + "content": "runtime.", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 18, + "bbox_fs": [ + 105, + 329, + 506, + 408 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 421, + 482, + 432 + ], + "lines": [ + { + "bbox": [ + 106, + 421, + 483, + 433 + ], + "spans": [ + { + "bbox": [ + 106, + 421, + 483, + 433 + ], + "score": 1.0, + "content": "2.3 ALL LOW-DEPTH STRUCTURED MATRICES ARE IN THE KALEIDOSCOPE HIERARCHY", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 22 + }, + { + "type": "text", + "bbox": [ + 107, + 441, + 505, + 507 + ], + "lines": [ + { + "bbox": [ + 106, + 442, + 505, + 453 + ], + "spans": [ + { + "bbox": [ + 106, + 442, + 505, + 453 + ], + "score": 1.0, + "content": "We now present our main theoretical result: the fact that general linear transformations, expressed as", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 452, + 505, + 465 + ], + "spans": [ + { + "bbox": [ + 105, + 452, + 330, + 465 + ], + "score": 1.0, + "content": "low-depth linear arithmetic circuits, are captured in the", + "type": "text" + }, + { + "bbox": [ + 331, + 452, + 350, + 462 + ], + "score": 0.89, + "content": "B B ^ { * }", + "type": "inline_equation" + }, + { + "bbox": [ + 351, + 452, + 505, + 465 + ], + "score": 1.0, + "content": "hierarchy with low width. Arithmetic", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 463, + 506, + 476 + ], + "spans": [ + { + "bbox": [ + 105, + 463, + 506, + 476 + ], + "score": 1.0, + "content": "circuits are commonly used to formalize algebraic algorithmic complexity (Bürgisser et al., 2013);", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 474, + 506, + 487 + ], + "spans": [ + { + "bbox": [ + 105, + 474, + 506, + 487 + ], + "score": 1.0, + "content": "we include a primer on this in Appendix M. The quantities of interest are the total number of gates", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 485, + 505, + 498 + ], + "spans": [ + { + "bbox": [ + 105, + 485, + 505, + 498 + ], + "score": 1.0, + "content": "in the circuit, representing the total number of steps required to perform the algorithm for a serial", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 496, + 507, + 510 + ], + "spans": [ + { + "bbox": [ + 105, + 496, + 507, + 510 + ], + "score": 1.0, + "content": "processor, and the depth, representing the minimum number of steps required for a parallel processor.", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 25.5, + "bbox_fs": [ + 105, + 442, + 507, + 510 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 510, + 504, + 538 + ], + "lines": [ + { + "bbox": [ + 106, + 510, + 506, + 523 + ], + "spans": [ + { + "bbox": [ + 106, + 510, + 210, + 523 + ], + "score": 1.0, + "content": "Theorem 1. Let M be an", + "type": "text" + }, + { + "bbox": [ + 210, + 512, + 234, + 521 + ], + "score": 0.88, + "content": "n \\times n", + "type": "inline_equation" + }, + { + "bbox": [ + 235, + 510, + 506, + 523 + ], + "score": 1.0, + "content": "matrix such that multiplication of M times an arbitrary vector v can", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 103, + 518, + 505, + 542 + ], + "spans": [ + { + "bbox": [ + 103, + 518, + 505, + 542 + ], + "score": 1.0, + "content": "be represented as a linear arithmetic circuit with s total gates and depth d. Then, M ∈ (BB∗)O(d)O( sn ).", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 29.5, + "bbox_fs": [ + 103, + 510, + 506, + 542 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 547, + 505, + 603 + ], + "lines": [ + { + "bbox": [ + 105, + 546, + 506, + 560 + ], + "spans": [ + { + "bbox": [ + 105, + 546, + 247, + 560 + ], + "score": 1.0, + "content": "The representation of such a matrix", + "type": "text" + }, + { + "bbox": [ + 248, + 548, + 260, + 558 + ], + "score": 0.33, + "content": "\\mathbf { M }", + "type": "inline_equation" + }, + { + "bbox": [ + 261, + 546, + 285, + 560 + ], + "score": 1.0, + "content": "in the", + "type": "text" + }, + { + "bbox": [ + 286, + 548, + 305, + 558 + ], + "score": 0.89, + "content": "B B ^ { * }", + "type": "inline_equation" + }, + { + "bbox": [ + 305, + 546, + 361, + 560 + ], + "score": 1.0, + "content": "hierarchy has", + "type": "text" + }, + { + "bbox": [ + 362, + 547, + 409, + 560 + ], + "score": 0.92, + "content": "O ( d s \\log s )", + "type": "inline_equation" + }, + { + "bbox": [ + 409, + 546, + 506, + 560 + ], + "score": 1.0, + "content": "parameters and yields a", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 107, + 558, + 505, + 571 + ], + "spans": [ + { + "bbox": [ + 107, + 559, + 154, + 571 + ], + "score": 0.92, + "content": "O ( d s \\log s )", + "type": "inline_equation" + }, + { + "bbox": [ + 154, + 558, + 328, + 571 + ], + "score": 1.0, + "content": "multiplication algorithm, compared to the", + "type": "text" + }, + { + "bbox": [ + 329, + 559, + 350, + 570 + ], + "score": 0.92, + "content": "O ( s )", + "type": "inline_equation" + }, + { + "bbox": [ + 350, + 558, + 505, + 571 + ], + "score": 1.0, + "content": "parameters and runtime of the circuit", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 104, + 569, + 506, + 582 + ], + "spans": [ + { + "bbox": [ + 104, + 569, + 506, + 582 + ], + "score": 1.0, + "content": "representation. To the best of our knowledge, the most general classes of efficient matrices that have", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 580, + 505, + 593 + ], + "spans": [ + { + "bbox": [ + 105, + 580, + 282, + 593 + ], + "score": 1.0, + "content": "been studied (De Sa et al., 2018) have depth", + "type": "text" + }, + { + "bbox": [ + 283, + 581, + 289, + 590 + ], + "score": 0.8, + "content": "d", + "type": "inline_equation" + }, + { + "bbox": [ + 290, + 580, + 351, + 593 + ], + "score": 1.0, + "content": "on the order of", + "type": "text" + }, + { + "bbox": [ + 351, + 581, + 373, + 592 + ], + "score": 0.85, + "content": "\\log n", + "type": "inline_equation" + }, + { + "bbox": [ + 373, + 580, + 405, + 593 + ], + "score": 1.0, + "content": "or poly", + "type": "text" + }, + { + "bbox": [ + 406, + 581, + 427, + 592 + ], + "score": 0.65, + "content": "\\log n", + "type": "inline_equation" + }, + { + "bbox": [ + 428, + 580, + 505, + 593 + ], + "score": 1.0, + "content": ". In these cases, the", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 591, + 482, + 604 + ], + "spans": [ + { + "bbox": [ + 105, + 591, + 482, + 604 + ], + "score": 1.0, + "content": "representation with K-matrices matches the best known bounds up to polylogarithmic factors.", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 33, + "bbox_fs": [ + 104, + 546, + 506, + 604 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 608, + 505, + 682 + ], + "lines": [ + { + "bbox": [ + 106, + 608, + 505, + 621 + ], + "spans": [ + { + "bbox": [ + 106, + 608, + 505, + 621 + ], + "score": 1.0, + "content": "The crux of the proof of Theorem 1 (shown in Appendix F) is the construction of an almost tight", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 619, + 506, + 632 + ], + "spans": [ + { + "bbox": [ + 105, + 619, + 268, + 632 + ], + "score": 1.0, + "content": "representation of any sparse matrix as a", + "type": "text" + }, + { + "bbox": [ + 268, + 620, + 276, + 629 + ], + "score": 0.34, + "content": "\\mathbf { K }", + "type": "inline_equation" + }, + { + "bbox": [ + 277, + 619, + 506, + 632 + ], + "score": 1.0, + "content": "-matrix (i.e. a product of butterfly matrices): specifically,", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 629, + 507, + 652 + ], + "spans": [ + { + "bbox": [ + 105, + 629, + 177, + 652 + ], + "score": 1.0, + "content": "we show that any", + "type": "text" + }, + { + "bbox": [ + 177, + 637, + 203, + 646 + ], + "score": 0.88, + "content": "n \\times n", + "type": "inline_equation" + }, + { + "bbox": [ + 203, + 629, + 280, + 652 + ], + "score": 1.0, + "content": "sparse matrix with", + "type": "text" + }, + { + "bbox": [ + 280, + 639, + 286, + 646 + ], + "score": 0.66, + "content": "s", + "type": "inline_equation" + }, + { + "bbox": [ + 286, + 629, + 345, + 652 + ], + "score": 1.0, + "content": "nonzeros is in", + "type": "text" + }, + { + "bbox": [ + 345, + 630, + 400, + 651 + ], + "score": 0.94, + "content": "( B B ^ { * } ) _ { O ( 1 ) } ^ { O ( \\left\\lceil \\frac { s } { n } \\right\\rceil ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 399, + 633, + 507, + 649 + ], + "score": 1.0, + "content": "(Theorem 3, Appendix I).", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 106, + 649, + 505, + 661 + ], + "spans": [ + { + "bbox": [ + 106, + 649, + 505, + 661 + ], + "score": 1.0, + "content": "We then leverage the expressivity result of products of sparse matrices to represent all arithmetic", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 106, + 660, + 505, + 672 + ], + "spans": [ + { + "bbox": [ + 106, + 660, + 505, + 672 + ], + "score": 1.0, + "content": "circuits (similar to the sparse product width result of De Sa et al. (2018) referenced in Section 2.1) to", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 106, + 671, + 244, + 683 + ], + "spans": [ + { + "bbox": [ + 106, + 671, + 244, + 683 + ], + "score": 1.0, + "content": "complete the proof of Theorem 1.", + "type": "text" + } + ], + "index": 41 + } + ], + "index": 38.5, + "bbox_fs": [ + 105, + 608, + 507, + 683 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 687, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 687, + 504, + 699 + ], + "spans": [ + { + "bbox": [ + 105, + 687, + 497, + 699 + ], + "score": 1.0, + "content": "This intermediate result is also a novel characterization of sparse matrices. For a matrix with", + "type": "text" + }, + { + "bbox": [ + 498, + 690, + 504, + 698 + ], + "score": 0.51, + "content": "s", + "type": "inline_equation" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 698, + 506, + 712 + ], + "spans": [ + { + "bbox": [ + 105, + 698, + 274, + 712 + ], + "score": 1.0, + "content": "NNZ, the kaleidoscope representation has", + "type": "text" + }, + { + "bbox": [ + 274, + 699, + 317, + 711 + ], + "score": 0.92, + "content": "O ( s \\log { n } )", + "type": "inline_equation" + }, + { + "bbox": [ + 317, + 698, + 506, + 712 + ], + "score": 1.0, + "content": "parameters and runtime, instead of the optimal", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 107, + 709, + 506, + 722 + ], + "spans": [ + { + "bbox": [ + 107, + 710, + 128, + 722 + ], + "score": 0.89, + "content": "O ( s )", + "type": "inline_equation" + }, + { + "bbox": [ + 128, + 709, + 506, + 722 + ], + "score": 1.0, + "content": "parameters and runtime; so, we trade off an extra logarithmic factor in space and time for full", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 720, + 506, + 733 + ], + "spans": [ + { + "bbox": [ + 105, + 720, + 506, + 733 + ], + "score": 1.0, + "content": "differentiability (thanks to the fixed sparsity patterns in the representation). The intuition behind", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 106, + 82, + 505, + 95 + ], + "spans": [ + { + "bbox": [ + 106, + 83, + 294, + 95 + ], + "score": 1.0, + "content": "the result is as follows: a sparse matrix with", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 295, + 84, + 301, + 92 + ], + "score": 0.54, + "content": "s", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 302, + 83, + 441, + 95 + ], + "score": 1.0, + "content": "NNZ can be written as a sum of", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 441, + 82, + 466, + 95 + ], + "score": 0.92, + "content": "\\lceil s / n \\rceil", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 466, + 83, + 505, + 95 + ], + "score": 1.0, + "content": "matrices", + "type": "text", + "cross_page": true + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 93, + 505, + 106 + ], + "spans": [ + { + "bbox": [ + 105, + 93, + 185, + 106 + ], + "score": 1.0, + "content": "each with at most", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 185, + 95, + 193, + 104 + ], + "score": 0.6, + "content": "n", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 193, + 93, + 243, + 106 + ], + "score": 1.0, + "content": "NNZ. Any", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 243, + 95, + 270, + 104 + ], + "score": 0.89, + "content": "n \\times n", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 271, + 93, + 359, + 106 + ], + "score": 1.0, + "content": "matrix with at most", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 359, + 95, + 367, + 104 + ], + "score": 0.63, + "content": "n", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 367, + 93, + 505, + 106 + ], + "score": 1.0, + "content": "NNZ, up to permuting the rows", + "type": "text", + "cross_page": true + } + ], + "index": 1 + }, + { + "bbox": [ + 106, + 105, + 506, + 117 + ], + "spans": [ + { + "bbox": [ + 106, + 105, + 506, + 117 + ], + "score": 1.0, + "content": "and columns, is a product of two butterfly matrices (Lemma I.1). Sorting networks (Knuth, 1997)", + "type": "text", + "cross_page": true + } + ], + "index": 2 + }, + { + "bbox": [ + 104, + 115, + 506, + 129 + ], + "spans": [ + { + "bbox": [ + 104, + 115, + 261, + 129 + ], + "score": 1.0, + "content": "imply that permutation matrices are in", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 262, + 115, + 317, + 128 + ], + "score": 0.93, + "content": "( \\tilde { B B ^ { * } } ) ^ { O ( \\log n ) }", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 318, + 115, + 506, + 129 + ], + "score": 1.0, + "content": ", but we tighten the result to show that they are", + "type": "text", + "cross_page": true + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 126, + 506, + 140 + ], + "spans": [ + { + "bbox": [ + 105, + 126, + 146, + 140 + ], + "score": 1.0, + "content": "in fact in", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 146, + 128, + 165, + 138 + ], + "score": 0.88, + "content": "B B ^ { * }", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 166, + 126, + 264, + 140 + ], + "score": 1.0, + "content": "(Theorem 2, Appendix", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 264, + 128, + 273, + 138 + ], + "score": 0.3, + "content": "\\mathbf { G }", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 273, + 126, + 506, + 140 + ], + "score": 1.0, + "content": "). We thus obtain a kaleidoscope representation for each", + "type": "text", + "cross_page": true + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 138, + 505, + 152 + ], + "spans": [ + { + "bbox": [ + 105, + 138, + 195, + 152 + ], + "score": 1.0, + "content": "summand matrix with", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 195, + 138, + 240, + 150 + ], + "score": 0.92, + "content": "O ( n \\log n )", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 240, + 138, + 445, + 152 + ], + "score": 1.0, + "content": "parameters. By the addition closure property of the", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 445, + 138, + 464, + 149 + ], + "score": 0.89, + "content": "B B ^ { * }", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 465, + 138, + 505, + 152 + ], + "score": 1.0, + "content": "hierarchy", + "type": "text", + "cross_page": true + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 149, + 505, + 162 + ], + "spans": [ + { + "bbox": [ + 105, + 150, + 259, + 162 + ], + "score": 1.0, + "content": "(Lemma H.5), each sparse matrix with", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 259, + 151, + 265, + 159 + ], + "score": 0.68, + "content": "s", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 266, + 150, + 461, + 162 + ], + "score": 1.0, + "content": "NNZ then has a kaleidoscope representation with", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 461, + 149, + 505, + 162 + ], + "score": 0.95, + "content": "O ( s \\log { n } )", + "type": "inline_equation", + "cross_page": true + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 162, + 155, + 172 + ], + "spans": [ + { + "bbox": [ + 105, + 162, + 155, + 172 + ], + "score": 1.0, + "content": "parameters.", + "type": "text", + "cross_page": true + } + ], + "index": 7 + } + ], + "index": 43.5, + "bbox_fs": [ + 105, + 687, + 506, + 733 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 106, + 82, + 505, + 172 + ], + "lines": [ + { + "bbox": [ + 106, + 82, + 505, + 95 + ], + "spans": [ + { + "bbox": [ + 106, + 83, + 294, + 95 + ], + "score": 1.0, + "content": "the result is as follows: a sparse matrix with", + "type": "text" + }, + { + "bbox": [ + 295, + 84, + 301, + 92 + ], + "score": 0.54, + "content": "s", + "type": "inline_equation" + }, + { + "bbox": [ + 302, + 83, + 441, + 95 + ], + "score": 1.0, + "content": "NNZ can be written as a sum of", + "type": "text" + }, + { + "bbox": [ + 441, + 82, + 466, + 95 + ], + "score": 0.92, + "content": "\\lceil s / n \\rceil", + "type": "inline_equation" + }, + { + "bbox": [ + 466, + 83, + 505, + 95 + ], + "score": 1.0, + "content": "matrices", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 93, + 505, + 106 + ], + "spans": [ + { + "bbox": [ + 105, + 93, + 185, + 106 + ], + "score": 1.0, + "content": "each with at most", + "type": "text" + }, + { + "bbox": [ + 185, + 95, + 193, + 104 + ], + "score": 0.6, + "content": "n", + "type": "inline_equation" + }, + { + "bbox": [ + 193, + 93, + 243, + 106 + ], + "score": 1.0, + "content": "NNZ. 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Sorting networks (Knuth, 1997)", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 104, + 115, + 506, + 129 + ], + "spans": [ + { + "bbox": [ + 104, + 115, + 261, + 129 + ], + "score": 1.0, + "content": "imply that permutation matrices are in", + "type": "text" + }, + { + "bbox": [ + 262, + 115, + 317, + 128 + ], + "score": 0.93, + "content": "( \\tilde { B B ^ { * } } ) ^ { O ( \\log n ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 318, + 115, + 506, + 129 + ], + "score": 1.0, + "content": ", but we tighten the result to show that they are", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 126, + 506, + 140 + ], + "spans": [ + { + "bbox": [ + 105, + 126, + 146, + 140 + ], + "score": 1.0, + "content": "in fact in", + "type": "text" + }, + { + "bbox": [ + 146, + 128, + 165, + 138 + ], + "score": 0.88, + "content": "B B ^ { * }", + "type": "inline_equation" + }, + { + "bbox": [ + 166, + 126, + 264, + 140 + ], + "score": 1.0, + "content": "(Theorem 2, Appendix", + "type": "text" + }, + { + "bbox": [ + 264, + 128, + 273, + 138 + ], + "score": 0.3, + "content": "\\mathbf { G }", + "type": "inline_equation" + }, + { + "bbox": [ + 273, + 126, + 506, + 140 + ], + "score": 1.0, + "content": "). 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There", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 277, + 505, + 288 + ], + "spans": [ + { + "bbox": [ + 106, + 277, + 505, + 288 + ], + "score": 1.0, + "content": "have been other large classes of structured matrices proposed in the machine learning literature, such", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 287, + 505, + 299 + ], + "spans": [ + { + "bbox": [ + 106, + 287, + 505, + 299 + ], + "score": 1.0, + "content": "as Toeplitz-like (Sindhwani et al., 2015) and low displacement rank (LDR) (Thomas et al., 2018), but", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 298, + 505, + 310 + ], + "spans": [ + { + "bbox": [ + 106, + 298, + 505, + 310 + ], + "score": 1.0, + "content": "they are not known to be able to capture these common structures as tightly as K-matrices can. More", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 308, + 267, + 321 + ], + "spans": [ + { + "bbox": [ + 105, + 308, + 267, + 321 + ], + "score": 1.0, + "content": "detailed discussions are in Appendix A.", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 13.5 + }, + { + "type": "title", + "bbox": [ + 108, + 338, + 187, + 349 + ], + "lines": [ + { + "bbox": [ + 105, + 336, + 189, + 351 + ], + "spans": [ + { + "bbox": [ + 105, + 336, + 189, + 351 + ], + "score": 1.0, + "content": "2.4 EXTENSIONS", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 20 + }, + { + "type": "text", + "bbox": [ + 106, + 360, + 505, + 438 + ], + "lines": [ + { + "bbox": [ + 105, + 360, + 505, + 373 + ], + "spans": [ + { + "bbox": [ + 105, + 360, + 505, + 373 + ], + "score": 1.0, + "content": "ReLU networks with low-depth structured weight matrices In Appendix L, we prove that", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 372, + 505, + 383 + ], + "spans": [ + { + "bbox": [ + 106, + 372, + 505, + 383 + ], + "score": 1.0, + "content": "finding an efficient circuit for a ReLU network can be reduced to finding efficient circuits for each", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 382, + 506, + 394 + ], + "spans": [ + { + "bbox": [ + 105, + 382, + 506, + 394 + ], + "score": 1.0, + "content": "of its weight matrices, with at most a constant factor greater size and run-time (i.e. number of", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 393, + 506, + 406 + ], + "spans": [ + { + "bbox": [ + 105, + 393, + 506, + 406 + ], + "score": 1.0, + "content": "gates). We also show that ReLU networks with kaleidoscope weight matrices have near-linear VC", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 403, + 505, + 417 + ], + "spans": [ + { + "bbox": [ + 105, + 403, + 505, + 417 + ], + "score": 1.0, + "content": "dimension in the number of parameters, matching the bound for networks with unconstrained weight", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 415, + 506, + 428 + ], + "spans": [ + { + "bbox": [ + 106, + 415, + 506, + 428 + ], + "score": 1.0, + "content": "matrices (Bartlett et al., 1999; Harvey et al., 2017) and LDR (Thomas et al., 2018). This yields a", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 426, + 274, + 439 + ], + "spans": [ + { + "bbox": [ + 106, + 426, + 274, + 439 + ], + "score": 1.0, + "content": "corresponding sample complexity bound.", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 24 + }, + { + "type": "text", + "bbox": [ + 106, + 453, + 505, + 523 + ], + "lines": [ + { + "bbox": [ + 106, + 452, + 506, + 466 + ], + "spans": [ + { + "bbox": [ + 106, + 452, + 506, + 466 + ], + "score": 1.0, + "content": "Orthogonal kaleidoscope hierarchy Orthogonal butterfly matrices are one commonly used variant", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 464, + 505, + 478 + ], + "spans": [ + { + "bbox": [ + 105, + 464, + 505, + 478 + ], + "score": 1.0, + "content": "due to their improved stability (Parker, 1995), where each butterfly factor is constrained to be", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 475, + 506, + 502 + ], + "spans": [ + { + "bbox": [ + 105, + 480, + 155, + 495 + ], + "score": 1.0, + "content": "orthogonal:", + "type": "text" + }, + { + "bbox": [ + 155, + 475, + 198, + 502 + ], + "score": 0.91, + "content": "\\left[ \\begin{array} { l l } { \\mathbf { C } } & { \\mathbf { s } } \\\\ { - \\mathbf { S } } & { \\mathbf { C } } \\end{array} \\right]", + "type": "inline_equation" + }, + { + "bbox": [ + 199, + 480, + 219, + 497 + ], + "score": 1.0, + "content": "with", + "type": "text" + }, + { + "bbox": [ + 220, + 482, + 240, + 494 + ], + "score": 0.38, + "content": "\\mathbf { C } , \\mathbf { S }", + "type": "inline_equation" + }, + { + "bbox": [ + 241, + 480, + 318, + 497 + ], + "score": 1.0, + "content": "being diagonal and", + "type": "text" + }, + { + "bbox": [ + 319, + 482, + 373, + 493 + ], + "score": 0.92, + "content": "\\mathbf { C } ^ { 2 } + \\mathbf { S } ^ { 2 } = \\mathbf { I }", + "type": "inline_equation" + }, + { + "bbox": [ + 374, + 480, + 433, + 497 + ], + "score": 1.0, + "content": ". 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There", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 277, + 505, + 288 + ], + "spans": [ + { + "bbox": [ + 106, + 277, + 505, + 288 + ], + "score": 1.0, + "content": "have been other large classes of structured matrices proposed in the machine learning literature, such", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 287, + 505, + 299 + ], + "spans": [ + { + "bbox": [ + 106, + 287, + 505, + 299 + ], + "score": 1.0, + "content": "as Toeplitz-like (Sindhwani et al., 2015) and low displacement rank (LDR) (Thomas et al., 2018), but", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 298, + 505, + 310 + ], + "spans": [ + { + "bbox": [ + 106, + 298, + 505, + 310 + ], + "score": 1.0, + "content": "they are not known to be able to capture these common structures as tightly as K-matrices can. 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We also show that ReLU networks with kaleidoscope weight matrices have near-linear VC", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 403, + 505, + 417 + ], + "spans": [ + { + "bbox": [ + 105, + 403, + 505, + 417 + ], + "score": 1.0, + "content": "dimension in the number of parameters, matching the bound for networks with unconstrained weight", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 415, + 506, + 428 + ], + "spans": [ + { + "bbox": [ + 106, + 415, + 506, + 428 + ], + "score": 1.0, + "content": "matrices (Bartlett et al., 1999; Harvey et al., 2017) and LDR (Thomas et al., 2018). This yields a", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 426, + 274, + 439 + ], + "spans": [ + { + "bbox": [ + 106, + 426, + 274, + 439 + ], + "score": 1.0, + "content": "corresponding sample complexity bound.", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 24, + "bbox_fs": [ + 105, + 360, + 506, + 439 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 453, + 505, + 523 + ], + "lines": [ + { + "bbox": [ + 106, + 452, + 506, + 466 + ], + "spans": [ + { + "bbox": [ + 106, + 452, + 506, + 466 + ], + "score": 1.0, + "content": "Orthogonal kaleidoscope hierarchy Orthogonal butterfly matrices are one commonly used variant", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 464, + 505, + 478 + ], + "spans": [ + { + "bbox": [ + 105, + 464, + 505, + 478 + ], + "score": 1.0, + "content": "due to their improved stability (Parker, 1995), where each butterfly factor is constrained to be", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 475, + 506, + 502 + ], + "spans": [ + { + "bbox": [ + 105, + 480, + 155, + 495 + ], + "score": 1.0, + "content": "orthogonal:", + "type": "text" + }, + { + "bbox": [ + 155, + 475, + 198, + 502 + ], + "score": 0.91, + "content": "\\left[ \\begin{array} { l l } { \\mathbf { C } } & { \\mathbf { s } } \\\\ { - \\mathbf { S } } & { \\mathbf { C } } \\end{array} \\right]", + "type": "inline_equation" + }, + { + "bbox": [ + 199, + 480, + 219, + 497 + ], + "score": 1.0, + "content": "with", + "type": "text" + }, + { + "bbox": [ + 220, + 482, + 240, + 494 + ], + "score": 0.38, + "content": "\\mathbf { C } , \\mathbf { S }", + "type": "inline_equation" + }, + { + "bbox": [ + 241, + 480, + 318, + 497 + ], + "score": 1.0, + "content": "being diagonal and", + "type": "text" + }, + { + "bbox": [ + 319, + 482, + 373, + 493 + ], + "score": 0.92, + "content": "\\mathbf { C } ^ { 2 } + \\mathbf { S } ^ { 2 } = \\mathbf { I }", + "type": "inline_equation" + }, + { + "bbox": [ + 374, + 480, + 433, + 497 + ], + "score": 1.0, + "content": ". Similar to the", + "type": "text" + }, + { + "bbox": [ + 433, + 483, + 452, + 493 + ], + "score": 0.88, + "content": "B B ^ { * }", + "type": "inline_equation" + }, + { + "bbox": [ + 453, + 480, + 506, + 497 + ], + "score": 1.0, + "content": "hierarchy, in", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 498, + 506, + 512 + ], + "spans": [ + { + "bbox": [ + 105, + 498, + 506, + 512 + ], + "score": 1.0, + "content": "Appendix K, we define the OBB hierarchy consisting of products of orthogonal butterfly matrices", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 511, + 506, + 524 + ], + "spans": [ + { + "bbox": [ + 105, + 511, + 445, + 524 + ], + "score": 1.0, + "content": "and diagonal matrices, and show that this hierarchy has the same expressiveness as the", + "type": "text" + }, + { + "bbox": [ + 445, + 511, + 464, + 521 + ], + "score": 0.88, + "content": "B B ^ { * }", + "type": "inline_equation" + }, + { + "bbox": [ + 465, + 511, + 506, + 524 + ], + "score": 1.0, + "content": "hierarchy.", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 30, + "bbox_fs": [ + 105, + 452, + 506, + 524 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 542, + 254, + 555 + ], + "lines": [ + { + "bbox": [ + 105, + 541, + 256, + 558 + ], + "spans": [ + { + "bbox": [ + 105, + 541, + 256, + 558 + ], + "score": 1.0, + "content": "3 EMPIRICAL EVALUATION", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 33 + }, + { + "type": "text", + "bbox": [ + 108, + 570, + 504, + 592 + ], + "lines": [ + { + "bbox": [ + 106, + 569, + 505, + 583 + ], + "spans": [ + { + "bbox": [ + 106, + 569, + 505, + 583 + ], + "score": 1.0, + "content": "We validate three claims that suggest that kaleidoscopes are a promising technique to learn different", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 106, + 581, + 277, + 594 + ], + "spans": [ + { + "bbox": [ + 106, + 581, + 277, + 594 + ], + "score": 1.0, + "content": "types of structure in modern architectures.", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 34.5, + "bbox_fs": [ + 106, + 569, + 505, + 594 + ] + }, + { + "type": "list", + "bbox": [ + 114, + 598, + 506, + 732 + ], + "lines": [ + { + "bbox": [ + 114, + 598, + 505, + 610 + ], + "spans": [ + { + "bbox": [ + 114, + 598, + 505, + 610 + ], + "score": 1.0, + "content": "1. Section 3.1: for applications in speech and lightweight computer vision relying on highly", + "type": "text" + } + ], + "index": 36, + "is_list_start_line": true + }, + { + "bbox": [ + 125, + 609, + 505, + 622 + ], + "spans": [ + { + "bbox": [ + 125, + 609, + 505, + 622 + ], + "score": 1.0, + "content": "hand-crafted structured transformations, we show that we can recover—and even improve—the", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 126, + 620, + 505, + 633 + ], + "spans": [ + { + "bbox": [ + 126, + 620, + 505, + 633 + ], + "score": 1.0, + "content": "quality of such architectures by simply replacing existing hand-structured components with", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 126, + 631, + 402, + 643 + ], + "spans": [ + { + "bbox": [ + 126, + 631, + 402, + 643 + ], + "score": 1.0, + "content": "K-matrices, with only a small overhead in memory and computation.", + "type": "text" + } + ], + "index": 39, + "is_list_end_line": true + }, + { + "bbox": [ + 114, + 642, + 506, + 655 + ], + "spans": [ + { + "bbox": [ + 114, + 642, + 506, + 655 + ], + "score": 1.0, + "content": "2. In Section 3.2, for a challenging task with latent structure (Permuted CIFAR-10), a K-matrix-", + "type": "text" + } + ], + "index": 40, + "is_list_start_line": true + }, + { + "bbox": [ + 126, + 654, + 505, + 666 + ], + "spans": [ + { + "bbox": [ + 126, + 654, + 505, + 666 + ], + "score": 1.0, + "content": "based relaxation of permutations is able to learn the right latent permutation, yielding 9 points", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 125, + 664, + 505, + 677 + ], + "spans": [ + { + "bbox": [ + 125, + 664, + 505, + 677 + ], + "score": 1.0, + "content": "better accuracy in a downstream CNN compared to standard RNN and CNN baselines used on", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 126, + 676, + 293, + 688 + ], + "spans": [ + { + "bbox": [ + 126, + 676, + 293, + 688 + ], + "score": 1.0, + "content": "such permuted image classification tasks.", + "type": "text" + } + ], + "index": 43, + "is_list_end_line": true + }, + { + "bbox": [ + 114, + 687, + 505, + 700 + ], + "spans": [ + { + "bbox": [ + 114, + 687, + 505, + 700 + ], + "score": 1.0, + "content": "3. 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In particular, we can entirely replace the complex hand-", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 385, + 505, + 398 + ], + "spans": [ + { + "bbox": [ + 106, + 385, + 505, + 398 + ], + "score": 1.0, + "content": "crafted MFSC featurization commonly used in speech recognition tasks with a fully learnable", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 397, + 506, + 409 + ], + "spans": [ + { + "bbox": [ + 106, + 397, + 227, + 409 + ], + "score": 1.0, + "content": "kaleidoscope layer, with only", + "type": "text" + }, + { + "bbox": [ + 227, + 397, + 250, + 407 + ], + "score": 0.85, + "content": "0 . 4 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 250, + 397, + 506, + 409 + ], + "score": 1.0, + "content": "drop in accuracy on the TIMIT speech recognition benchmark.", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 407, + 505, + 420 + ], + "spans": [ + { + "bbox": [ + 105, + 407, + 505, + 420 + ], + "score": 1.0, + "content": "Results are presented in Table 1. Our approach is competitive with the accuracy of standard models", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 418, + 506, + 432 + ], + "spans": [ + { + "bbox": [ + 105, + 418, + 506, + 432 + ], + "score": 1.0, + "content": "that use hand-crafted features, and significantly outperforms current approaches for learning from", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 430, + 173, + 442 + ], + "spans": [ + { + "bbox": [ + 105, + 430, + 173, + 442 + ], + "score": 1.0, + "content": "raw audio input.", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 17 + }, + { + "type": "table", + "bbox": [ + 159, + 525, + 451, + 582 + ], + "blocks": [ + { + "type": "table_caption", + "bbox": [ + 106, + 451, + 506, + 518 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 105, + 451, + 506, + 464 + ], + "spans": [ + { + "bbox": [ + 105, + 451, + 255, + 464 + ], + "score": 1.0, + "content": "Table 1: TIMIT phoneme error rate", + "type": "text" + }, + { + "bbox": [ + 256, + 452, + 290, + 463 + ], + "score": 0.85, + "content": "( \\mathrm { P E R } \\% )", + "type": "inline_equation" + }, + { + "bbox": [ + 291, + 451, + 506, + 464 + ], + "score": 1.0, + "content": "for different methods. 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It significantly outperforms existing approaches that learn", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 484, + 505, + 497 + ], + "spans": [ + { + "bbox": [ + 105, + 484, + 505, + 497 + ], + "score": 1.0, + "content": "from raw audio, i.e. without handcrafted featurization (e.g. SincNet [row 2], which to our knowledge", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 496, + 506, + 508 + ], + "spans": [ + { + "bbox": [ + 105, + 496, + 410, + 508 + ], + "score": 1.0, + "content": "attains the previous state-of-the-art for learning from raw audio), and is only", + "type": "text" + }, + { + "bbox": [ + 410, + 496, + 432, + 506 + ], + "score": 0.86, + "content": "0 . 8 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 432, + 496, + 506, + 508 + ], + "score": 1.0, + "content": "less accurate than", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 505, + 471, + 520 + ], + "spans": [ + { + "bbox": [ + 105, + 505, + 471, + 520 + ], + "score": 1.0, + "content": "the overall state-of-the-art on TIMIT.3 Additional comparisons are given in Appendix B.1.", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 23.5 + }, + { + "type": "table_body", + "bbox": [ + 159, + 525, + 451, + 582 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 159, + 525, + 451, + 582 + ], + "spans": [ + { + "bbox": [ + 159, + 525, + 451, + 582 + ], + "score": 0.97, + "html": "
MethodTest set PER%Raw audio input
MFSC features +LSTM14.2x//
SincNet (Ravanelli et al., 2019)17.2
Kaleidoscope + LSTM14.6
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Full details for all of the", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 104, + 104, + 425, + 118 + ], + "spans": [ + { + "bbox": [ + 104, + 104, + 405, + 118 + ], + "score": 1.0, + "content": "experiments (precise architectures, hyperparameters, etc.) are in Appendix", + "type": "text" + }, + { + "bbox": [ + 406, + 104, + 420, + 115 + ], + "score": 0.31, + "content": "\\boldsymbol { \\mathrm { ~ B ~ } ^ { 2 } }", + "type": "inline_equation" + }, + { + "bbox": [ + 420, + 104, + 425, + 118 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 2 + } + ], + "index": 1, + "bbox_fs": [ + 104, + 81, + 505, + 118 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 130, + 312, + 141 + ], + "lines": [ + { + "bbox": [ + 106, + 129, + 312, + 142 + ], + "spans": [ + { + "bbox": [ + 106, + 129, + 312, + 142 + ], + "score": 1.0, + "content": "3.1 REPLACING HAND-CRAFTED STRUCTURES", + "type": "text" + } + ], + "index": 3 + } + ], + "index": 3 + }, + { + "type": "text", + "bbox": [ + 107, + 150, + 505, + 206 + ], + "lines": [ + { + "bbox": [ + 106, + 150, + 505, + 164 + ], + "spans": [ + { + "bbox": [ + 106, + 150, + 505, + 164 + ], + "score": 1.0, + "content": "We validate that kaleidoscope matrices can recover or improve on the performance of hand-crafted", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 162, + 505, + 175 + ], + "spans": [ + { + "bbox": [ + 105, + 162, + 505, + 175 + ], + "score": 1.0, + "content": "structure in ML models. 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MethodTest set PER%Raw audio input
MFSC features +LSTM14.2x//
SincNet (Ravanelli et al., 2019)17.2
Kaleidoscope + LSTM14.6
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Instead, we use a standard RNN speech", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 659, + 493, + 673 + ], + "spans": [ + { + "bbox": [ + 105, + 659, + 493, + 673 + ], + "score": 1.0, + "content": "recognition architecture, but use a learnable kaleidoscope layer to replace the featurization steps.", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 33, + "bbox_fs": [ + 105, + 593, + 505, + 673 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 82, + 504, + 149 + ], + "lines": [ + { + "bbox": [ + 105, + 81, + 506, + 95 + ], + "spans": [ + { + "bbox": [ + 105, + 81, + 506, + 95 + ], + "score": 1.0, + "content": "The baseline architecture takes as input filter bank (MFSC) features, which are a popular standard", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 94, + 505, + 106 + ], + "spans": [ + { + "bbox": [ + 106, + 94, + 505, + 106 + ], + "score": 1.0, + "content": "featurization for speech recognition (Paliwal, 1999) and involve several steps hand-crafted specifically", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 106, + 105, + 505, + 116 + ], + "spans": [ + { + "bbox": [ + 106, + 105, + 505, + 116 + ], + "score": 1.0, + "content": "for this domain. These features are extracted from the raw audio waveform, and fed as the input into", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 115, + 505, + 128 + ], + "spans": [ + { + "bbox": [ + 105, + 115, + 505, + 128 + ], + "score": 1.0, + "content": "a Bi-LSTM model. We significantly simplify this pipeline by replacing the featurization step with", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 126, + 506, + 139 + ], + "spans": [ + { + "bbox": [ + 105, + 126, + 506, + 139 + ], + "score": 1.0, + "content": "a trainable kaleidoscope layer that is trained end-to-end together with the Bi-LSTM. The original", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 137, + 398, + 150 + ], + "spans": [ + { + "bbox": [ + 105, + 137, + 398, + 150 + ], + "score": 1.0, + "content": "pipeline and our modified kaleidoscope version are depicted in Figure 2.", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 2.5 + }, + { + "type": "text", + "bbox": [ + 106, + 154, + 505, + 253 + ], + "lines": [ + { + "bbox": [ + 105, + 153, + 506, + 167 + ], + "spans": [ + { + "bbox": [ + 105, + 153, + 506, + 167 + ], + "score": 1.0, + "content": "The computation of MFSC features involves a series of painstakingly hand-designed steps (further", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 166, + 505, + 177 + ], + "spans": [ + { + "bbox": [ + 106, + 166, + 505, + 177 + ], + "score": 1.0, + "content": "described in Appendix B.1), each involving their own hyperparameters: (i) the waveform is framed", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 176, + 506, + 189 + ], + "spans": [ + { + "bbox": [ + 105, + 176, + 506, + 189 + ], + "score": 1.0, + "content": "(split into chunks), (ii) the waveform is dithered (noise is added), (iii) pre-emphasis is applied, (iv)", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 187, + 506, + 200 + ], + "spans": [ + { + "bbox": [ + 105, + 187, + 506, + 200 + ], + "score": 1.0, + "content": "the Hamming window is applied, (v) the FFT is applied and the power spectrum is computed, (vi)", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 198, + 506, + 211 + ], + "spans": [ + { + "bbox": [ + 105, + 198, + 506, + 211 + ], + "score": 1.0, + "content": "the result is mapped to the mel scale (which involves applying a particular linear transformation and", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 209, + 506, + 222 + ], + "spans": [ + { + "bbox": [ + 106, + 209, + 506, + 222 + ], + "score": 1.0, + "content": "then taking the logarithm of the result), (vii) cepstral mean and variance normalization is applied.", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 219, + 506, + 234 + ], + "spans": [ + { + "bbox": [ + 105, + 219, + 506, + 234 + ], + "score": 1.0, + "content": "We replace the last six steps (ii-vii) of this featurization process with a learnable kaleidoscope layer;", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 230, + 506, + 244 + ], + "spans": [ + { + "bbox": [ + 105, + 230, + 506, + 244 + ], + "score": 1.0, + "content": "specifically, after windowing, we multiply the input by a K-matrix, and then compute the logarithm", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 241, + 372, + 254 + ], + "spans": [ + { + "bbox": [ + 105, + 241, + 372, + 254 + ], + "score": 1.0, + "content": "of the power spectrum; the output is fed into the Bi-LSTM model.", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 10 + }, + { + "type": "title", + "bbox": [ + 108, + 267, + 298, + 278 + ], + "lines": [ + { + "bbox": [ + 106, + 267, + 300, + 280 + ], + "spans": [ + { + "bbox": [ + 106, + 267, + 300, + 280 + ], + "score": 1.0, + "content": "3.1.2 REPLACING CNN CHANNEL SHUFFLE", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 15 + }, + { + "type": "text", + "bbox": [ + 107, + 287, + 505, + 353 + ], + "lines": [ + { + "bbox": [ + 106, + 286, + 506, + 299 + ], + "spans": [ + { + "bbox": [ + 106, + 286, + 506, + 299 + ], + "score": 1.0, + "content": "We evaluate how K-matrices can improve the quality of hand-crafted, lightweight architectures for", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 298, + 506, + 311 + ], + "spans": [ + { + "bbox": [ + 106, + 298, + 506, + 311 + ], + "score": 1.0, + "content": "computer vision tasks, without the need for hand-tuning. We select ShuffleNet (Zhang et al., 2018),", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 308, + 506, + 322 + ], + "spans": [ + { + "bbox": [ + 105, + 308, + 506, + 322 + ], + "score": 1.0, + "content": "which is a state-of-the-art lightweight CNN architecture that uses a manually designed “channel", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 320, + 506, + 333 + ], + "spans": [ + { + "bbox": [ + 105, + 320, + 506, + 333 + ], + "score": 1.0, + "content": "shuffle” permutation matrix to improve performance. By replacing this fixed permutation with a", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 331, + 506, + 344 + ], + "spans": [ + { + "bbox": [ + 106, + 331, + 260, + 344 + ], + "score": 1.0, + "content": "learnable K-matrix, we achieve up to", + "type": "text" + }, + { + "bbox": [ + 261, + 331, + 276, + 342 + ], + "score": 0.86, + "content": "5 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 276, + 331, + 506, + 344 + ], + "score": 1.0, + "content": "further improvement in classification accuracy, without", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 342, + 506, + 354 + ], + "spans": [ + { + "bbox": [ + 106, + 342, + 368, + 354 + ], + "score": 1.0, + "content": "hand-tuned components and with a modest space penalty of up to", + "type": "text" + }, + { + "bbox": [ + 369, + 342, + 387, + 352 + ], + "score": 0.86, + "content": "10 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 388, + 342, + 506, + 354 + ], + "score": 1.0, + "content": ". Results are given in Table 2.", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 18.5 + }, + { + "type": "table", + "bbox": [ + 106, + 441, + 515, + 498 + ], + "blocks": [ + { + "type": "table_caption", + "bbox": [ + 106, + 365, + 505, + 432 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 105, + 364, + 506, + 378 + ], + "spans": [ + { + "bbox": [ + 105, + 364, + 506, + 378 + ], + "score": 1.0, + "content": "Table 2: Top-1 classification accuracy of ShuffleNet on ImageNet validation set (parameter counts in", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 376, + 506, + 388 + ], + "spans": [ + { + "bbox": [ + 105, + 376, + 506, + 388 + ], + "score": 1.0, + "content": "parentheses). We compare our approach (col. 3) with our reimplementation of ‘vanilla’ ShuffleNet", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 387, + 506, + 399 + ], + "spans": [ + { + "bbox": [ + 105, + 387, + 506, + 399 + ], + "score": 1.0, + "content": "(col. 1) and a recent approach based on the Hadamard transform (col. 2).4 We report results for", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 396, + 506, + 412 + ], + "spans": [ + { + "bbox": [ + 105, + 396, + 506, + 412 + ], + "score": 1.0, + "content": "different network width multipliers (# channels). The last column shows the differences in accuracy", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 409, + 506, + 422 + ], + "spans": [ + { + "bbox": [ + 106, + 409, + 506, + 422 + ], + "score": 1.0, + "content": "and parameter count between our approach and vanilla ShuffleNet; using a learnable K-matrix in", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 420, + 483, + 433 + ], + "spans": [ + { + "bbox": [ + 105, + 420, + 465, + 433 + ], + "score": 1.0, + "content": "place of each fixed permutation (shuffle) or Hadamard matrix improves accuracy by up to", + "type": "text" + }, + { + "bbox": [ + 466, + 420, + 480, + 430 + ], + "score": 0.85, + "content": "5 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 480, + 420, + 483, + 433 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 24.5 + }, + { + "type": "table_body", + "bbox": [ + 106, + 441, + 515, + 498 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 106, + 441, + 515, + 498 + ], + "spans": [ + { + "bbox": [ + 106, + 441, + 515, + 498 + ], + "score": 0.976, + "html": "
ShuffleHadamardKaleidoscope (K.)K. vs. Shuffle
0.25 ShuffleNet g844.1% (0.46M)43.9% (0.46M)49.2% (0.51M)+5.0% (+0.05M)
0.5 ShuffleNet g857.1% (1.0M)56.2% (1.0M)59.5% (1.1M)+2.4% (+0.1M)
1.0 ShuffleNet g865.3% (2.5M)65.0% (2.5M)66.5% (2.8M)+1.2% (+0.2M)
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To remedy this, ShuffleNet uses a permutation matrix to shuffle the channels", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 543, + 505, + 556 + ], + "spans": [ + { + "bbox": [ + 106, + 543, + 505, + 556 + ], + "score": 1.0, + "content": "after each grouped convolution. Zhao et al. (2019) propose to instead use the Hadamard transform", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 106, + 554, + 505, + 567 + ], + "spans": [ + { + "bbox": [ + 106, + 554, + 505, + 567 + ], + "score": 1.0, + "content": "before and after each grouped convolution to mix the channels. In place of these hand-engineered", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 564, + 506, + 578 + ], + "spans": [ + { + "bbox": [ + 105, + 564, + 506, + 578 + ], + "score": 1.0, + "content": "solutions, we use a K-matrix before and after each grouped convolution, and learn these end-to-end", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 576, + 505, + 588 + ], + "spans": [ + { + "bbox": [ + 105, + 576, + 505, + 588 + ], + "score": 1.0, + "content": "together with the rest of the network. As shown in Table 2, across a range of sizes, replacing the", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 586, + 504, + 600 + ], + "spans": [ + { + "bbox": [ + 105, + 586, + 504, + 600 + ], + "score": 1.0, + "content": "channel shuffles with K-matrices results in improved performance at comparable parameter counts.", + "type": "text" + } + ], + "index": 38 + } + ], + "index": 34.5 + }, + { + "type": "title", + "bbox": [ + 108, + 614, + 284, + 625 + ], + "lines": [ + { + "bbox": [ + 106, + 614, + 286, + 626 + ], + "spans": [ + { + "bbox": [ + 106, + 614, + 286, + 626 + ], + "score": 1.0, + "content": "3.2 LEARNING A LATENT PERMUTATION", + "type": "text" + } + ], + "index": 39 + } + ], + "index": 39 + }, + { + "type": "text", + "bbox": [ + 107, + 635, + 505, + 690 + ], + "lines": [ + { + "bbox": [ + 106, + 634, + 505, + 648 + ], + "spans": [ + { + "bbox": [ + 106, + 634, + 505, + 648 + ], + "score": 1.0, + "content": "We show that K-matrices can be used in a challenging task for which existing classes of structured", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 645, + 505, + 659 + ], + "spans": [ + { + "bbox": [ + 105, + 645, + 505, + 659 + ], + "score": 1.0, + "content": "linear maps have not been found suitable. We investigate the problem of image classification on", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 104, + 656, + 506, + 670 + ], + "spans": [ + { + "bbox": [ + 104, + 656, + 506, + 670 + ], + "score": 1.0, + "content": "a permuted image dataset (Permuted CIFAR-10). This problem is challenging due to the discrete", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 668, + 506, + 680 + ], + "spans": [ + { + "bbox": [ + 105, + 668, + 506, + 680 + ], + "score": 1.0, + "content": "nature of learning the latent permutation of the dataset; we present a differentiable relaxation for this", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 678, + 506, + 691 + ], + "spans": [ + { + "bbox": [ + 105, + 678, + 506, + 691 + ], + "score": 1.0, + "content": "using a K-matrix as a key component. Results are presented in Table 3; compared to methods that do", + "type": "text" + } + ], + "index": 44 + } + ], + "index": 42 + } + ], + "page_idx": 6, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 107, + 701, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 118, + 699, + 506, + 713 + ], + "spans": [ + { + "bbox": [ + 118, + 699, + 506, + 713 + ], + "score": 1.0, + "content": "4Despite our best effort, we were unable to reproduce the original accuracy reported by Zhang et al. (2018), a", + "type": "text" + } + ] + }, + { + "bbox": [ + 105, + 711, + 506, + 723 + ], + "spans": [ + { + "bbox": [ + 105, + 711, + 506, + 723 + ], + "score": 1.0, + "content": "problem similarly faced by Zhao et al. (2019) and Lyu et al. (2019). Zhao et al. (2019) use block Hadamard", + "type": "text" + } + ] + }, + { + "bbox": [ + 106, + 721, + 500, + 733 + ], + "spans": [ + { + "bbox": [ + 106, + 721, + 500, + 733 + ], + "score": 1.0, + "content": "transform and pre-activation ShuffleNet, so their results are not directly comparable with those reported here.", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 108, + 27, + 293, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 26, + 294, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 26, + 294, + 38 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2020", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 302, + 751, + 308, + 759 + ], + "lines": [ + { + "bbox": [ + 302, + 750, + 309, + 762 + ], + "spans": [ + { + "bbox": [ + 302, + 750, + 309, + 762 + ], + "score": 1.0, + "content": "7", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 82, + 504, + 149 + ], + "lines": [ + { + "bbox": [ + 105, + 81, + 506, + 95 + ], + "spans": [ + { + "bbox": [ + 105, + 81, + 506, + 95 + ], + "score": 1.0, + "content": "The baseline architecture takes as input filter bank (MFSC) features, which are a popular standard", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 94, + 505, + 106 + ], + "spans": [ + { + "bbox": [ + 106, + 94, + 505, + 106 + ], + "score": 1.0, + "content": "featurization for speech recognition (Paliwal, 1999) and involve several steps hand-crafted specifically", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 106, + 105, + 505, + 116 + ], + "spans": [ + { + "bbox": [ + 106, + 105, + 505, + 116 + ], + "score": 1.0, + "content": "for this domain. These features are extracted from the raw audio waveform, and fed as the input into", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 115, + 505, + 128 + ], + "spans": [ + { + "bbox": [ + 105, + 115, + 505, + 128 + ], + "score": 1.0, + "content": "a Bi-LSTM model. We significantly simplify this pipeline by replacing the featurization step with", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 126, + 506, + 139 + ], + "spans": [ + { + "bbox": [ + 105, + 126, + 506, + 139 + ], + "score": 1.0, + "content": "a trainable kaleidoscope layer that is trained end-to-end together with the Bi-LSTM. 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We select ShuffleNet (Zhang et al., 2018),", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 308, + 506, + 322 + ], + "spans": [ + { + "bbox": [ + 105, + 308, + 506, + 322 + ], + "score": 1.0, + "content": "which is a state-of-the-art lightweight CNN architecture that uses a manually designed “channel", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 320, + 506, + 333 + ], + "spans": [ + { + "bbox": [ + 105, + 320, + 506, + 333 + ], + "score": 1.0, + "content": "shuffle” permutation matrix to improve performance. By replacing this fixed permutation with a", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 331, + 506, + 344 + ], + "spans": [ + { + "bbox": [ + 106, + 331, + 260, + 344 + ], + "score": 1.0, + "content": "learnable K-matrix, we achieve up to", + "type": "text" + }, + { + "bbox": [ + 261, + 331, + 276, + 342 + ], + "score": 0.86, + "content": "5 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 276, + 331, + 506, + 344 + ], + "score": 1.0, + "content": "further improvement in classification accuracy, without", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 342, + 506, + 354 + ], + "spans": [ + { + "bbox": [ + 106, + 342, + 368, + 354 + ], + "score": 1.0, + "content": "hand-tuned components and with a modest space penalty of up to", + "type": "text" + }, + { + "bbox": [ + 369, + 342, + 387, + 352 + ], + "score": 0.86, + "content": "10 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 388, + 342, + 506, + 354 + ], + "score": 1.0, + "content": ". Results are given in Table 2.", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 18.5, + "bbox_fs": [ + 105, + 286, + 506, + 354 + ] + }, + { + "type": "table", + "bbox": [ + 106, + 441, + 515, + 498 + ], + "blocks": [ + { + "type": "table_caption", + "bbox": [ + 106, + 365, + 505, + 432 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 105, + 364, + 506, + 378 + ], + "spans": [ + { + "bbox": [ + 105, + 364, + 506, + 378 + ], + "score": 1.0, + "content": "Table 2: Top-1 classification accuracy of ShuffleNet on ImageNet validation set (parameter counts in", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 376, + 506, + 388 + ], + "spans": [ + { + "bbox": [ + 105, + 376, + 506, + 388 + ], + "score": 1.0, + "content": "parentheses). 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The last column shows the differences in accuracy", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 409, + 506, + 422 + ], + "spans": [ + { + "bbox": [ + 106, + 409, + 506, + 422 + ], + "score": 1.0, + "content": "and parameter count between our approach and vanilla ShuffleNet; using a learnable K-matrix in", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 420, + 483, + 433 + ], + "spans": [ + { + "bbox": [ + 105, + 420, + 465, + 433 + ], + "score": 1.0, + "content": "place of each fixed permutation (shuffle) or Hadamard matrix improves accuracy by up to", + "type": "text" + }, + { + "bbox": [ + 466, + 420, + 480, + 430 + ], + "score": 0.85, + "content": "5 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 480, + 420, + 483, + 433 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 24.5 + }, + { + "type": "table_body", + "bbox": [ + 106, + 441, + 515, + 498 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 106, + 441, + 515, + 498 + ], + "spans": [ + { + "bbox": [ + 106, + 441, + 515, + 498 + ], + "score": 0.976, + "html": "
ShuffleHadamardKaleidoscope (K.)K. vs. Shuffle
0.25 ShuffleNet g844.1% (0.46M)43.9% (0.46M)49.2% (0.51M)+5.0% (+0.05M)
0.5 ShuffleNet g857.1% (1.0M)56.2% (1.0M)59.5% (1.1M)+2.4% (+0.1M)
1.0 ShuffleNet g865.3% (2.5M)65.0% (2.5M)66.5% (2.8M)+1.2% (+0.2M)
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To remedy this, ShuffleNet uses a permutation matrix to shuffle the channels", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 543, + 505, + 556 + ], + "spans": [ + { + "bbox": [ + 106, + 543, + 505, + 556 + ], + "score": 1.0, + "content": "after each grouped convolution. Zhao et al. (2019) propose to instead use the Hadamard transform", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 106, + 554, + 505, + 567 + ], + "spans": [ + { + "bbox": [ + 106, + 554, + 505, + 567 + ], + "score": 1.0, + "content": "before and after each grouped convolution to mix the channels. In place of these hand-engineered", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 564, + 506, + 578 + ], + "spans": [ + { + "bbox": [ + 105, + 564, + 506, + 578 + ], + "score": 1.0, + "content": "solutions, we use a K-matrix before and after each grouped convolution, and learn these end-to-end", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 576, + 505, + 588 + ], + "spans": [ + { + "bbox": [ + 105, + 576, + 505, + 588 + ], + "score": 1.0, + "content": "together with the rest of the network. As shown in Table 2, across a range of sizes, replacing the", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 586, + 504, + 600 + ], + "spans": [ + { + "bbox": [ + 105, + 586, + 504, + 600 + ], + "score": 1.0, + "content": "channel shuffles with K-matrices results in improved performance at comparable parameter counts.", + "type": "text" + } + ], + "index": 38 + } + ], + "index": 34.5, + "bbox_fs": [ + 105, + 510, + 506, + 600 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 614, + 284, + 625 + ], + "lines": [ + { + "bbox": [ + 106, + 614, + 286, + 626 + ], + "spans": [ + { + "bbox": [ + 106, + 614, + 286, + 626 + ], + "score": 1.0, + "content": "3.2 LEARNING A LATENT PERMUTATION", + "type": "text" + } + ], + "index": 39 + } + ], + "index": 39 + }, + { + "type": "text", + "bbox": [ + 107, + 635, + 505, + 690 + ], + "lines": [ + { + "bbox": [ + 106, + 634, + 505, + 648 + ], + "spans": [ + { + "bbox": [ + 106, + 634, + 505, + 648 + ], + "score": 1.0, + "content": "We show that K-matrices can be used in a challenging task for which existing classes of structured", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 645, + 505, + 659 + ], + "spans": [ + { + "bbox": [ + 105, + 645, + 505, + 659 + ], + "score": 1.0, + "content": "linear maps have not been found suitable. We investigate the problem of image classification on", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 104, + 656, + 506, + 670 + ], + "spans": [ + { + "bbox": [ + 104, + 656, + 506, + 670 + ], + "score": 1.0, + "content": "a permuted image dataset (Permuted CIFAR-10). This problem is challenging due to the discrete", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 668, + 506, + 680 + ], + "spans": [ + { + "bbox": [ + 105, + 668, + 506, + 680 + ], + "score": 1.0, + "content": "nature of learning the latent permutation of the dataset; we present a differentiable relaxation for this", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 678, + 506, + 691 + ], + "spans": [ + { + "bbox": [ + 105, + 678, + 506, + 691 + ], + "score": 1.0, + "content": "using a K-matrix as a key component. Results are presented in Table 3; compared to methods that do", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 82, + 506, + 95 + ], + "spans": [ + { + "bbox": [ + 105, + 82, + 433, + 95 + ], + "score": 1.0, + "content": "not have a permutation learning step, our approach gets 9 points higher accuracy", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 433, + 83, + 461, + 93 + ], + "score": 0.83, + "content": "8 4 . 4 \\%", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 461, + 82, + 472, + 95 + ], + "score": 1.0, + "content": "to", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 472, + 82, + 500, + 93 + ], + "score": 0.85, + "content": "9 3 . 6 \\%", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 500, + 82, + 506, + 95 + ], + "score": 1.0, + "content": "),", + "type": "text", + "cross_page": true + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 94, + 412, + 106 + ], + "spans": [ + { + "bbox": [ + 106, + 94, + 375, + 106 + ], + "score": 1.0, + "content": "coming within 2 points of the accuracy on the un-permuted dataset", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 376, + 94, + 408, + 105 + ], + "score": 0.86, + "content": "( 9 4 . 9 \\% )", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 408, + 94, + 412, + 106 + ], + "score": 1.0, + "content": ".", + "type": "text", + "cross_page": true + } + ], + "index": 1 + } + ], + "index": 42, + "bbox_fs": [ + 104, + 634, + 506, + 691 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 105, + 82, + 504, + 105 + ], + "lines": [ + { + "bbox": [ + 105, + 82, + 506, + 95 + ], + "spans": [ + { + "bbox": [ + 105, + 82, + 433, + 95 + ], + "score": 1.0, + "content": "not have a permutation learning step, our approach gets 9 points higher accuracy", + "type": "text" + }, + { + "bbox": [ + 433, + 83, + 461, + 93 + ], + "score": 0.83, + "content": "8 4 . 4 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 461, + 82, + 472, + 95 + ], + "score": 1.0, + "content": "to", + "type": "text" + }, + { + "bbox": [ + 472, + 82, + 500, + 93 + ], + "score": 0.85, + "content": "9 3 . 6 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 500, + 82, + 506, + 95 + ], + "score": 1.0, + "content": "),", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 94, + 412, + 106 + ], + "spans": [ + { + "bbox": [ + 106, + 94, + 375, + 106 + ], + "score": 1.0, + "content": "coming within 2 points of the accuracy on the un-permuted dataset", + "type": "text" + }, + { + "bbox": [ + 376, + 94, + 408, + 105 + ], + "score": 0.86, + "content": "( 9 4 . 9 \\% )", + "type": "inline_equation" + }, + { + "bbox": [ + 408, + 94, + 412, + 106 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 1 + } + ], + "index": 0.5 + }, + { + "type": "table", + "bbox": [ + 143, + 153, + 465, + 187 + ], + "blocks": [ + { + "type": "table_caption", + "bbox": [ + 107, + 114, + 505, + 147 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 105, + 113, + 506, + 127 + ], + "spans": [ + { + "bbox": [ + 105, + 113, + 379, + 127 + ], + "score": 1.0, + "content": "Table 3: Permuted CIFAR-10 validation set classification accuracy", + "type": "text" + }, + { + "bbox": [ + 379, + 115, + 394, + 125 + ], + "score": 0.75, + "content": "( \\% )", + "type": "inline_equation" + }, + { + "bbox": [ + 395, + 113, + 506, + 127 + ], + "score": 1.0, + "content": ". Our kaleidoscope layer is", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 106, + 125, + 505, + 137 + ], + "spans": [ + { + "bbox": [ + 106, + 125, + 505, + 137 + ], + "score": 1.0, + "content": "able to nearly perfectly recover the latent structure, allowing a downstream CNN to approach the", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 136, + 405, + 148 + ], + "spans": [ + { + "bbox": [ + 105, + 136, + 405, + 148 + ], + "score": 1.0, + "content": "accuracy of a standard ResNet18 on the unpermuted dataset (last column).", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 3 + }, + { + "type": "table_body", + "bbox": [ + 143, + 153, + 465, + 187 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 143, + 153, + 465, + 187 + ], + "spans": [ + { + "bbox": [ + 143, + 153, + 465, + 187 + ], + "score": 0.973, + "html": "
ModelFCRNNCNNDense +CNNK +CNN Unpermuted
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Large learnable classes of structured", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 226, + 506, + 239 + ], + "spans": [ + { + "bbox": [ + 105, + 226, + 506, + 239 + ], + "score": 1.0, + "content": "matrices such as Toeplitz-like matrices (Sindhwani et al., 2015) and low-displacement rank (LDR)", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 237, + 506, + 250 + ], + "spans": [ + { + "bbox": [ + 105, + 237, + 506, + 250 + ], + "score": 1.0, + "content": "matrices (Thomas et al., 2018) have been used for model compression. However, despite their", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 248, + 505, + 261 + ], + "spans": [ + { + "bbox": [ + 105, + 248, + 505, + 261 + ], + "score": 1.0, + "content": "theoretical speedup, these structured matrix classes lack efficient implementations, especially on", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 258, + 506, + 272 + ], + "spans": [ + { + "bbox": [ + 105, + 258, + 506, + 272 + ], + "score": 1.0, + "content": "GPUs. Therefore, their use has largely been confined to small models (e.g. single hidden layer neural", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 271, + 273, + 282 + ], + "spans": [ + { + "bbox": [ + 106, + 271, + 273, + 282 + ], + "score": 1.0, + "content": "nets) and small datasets (e.g. CIFAR-10).", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 8.5 + }, + { + "type": "text", + "bbox": [ + 107, + 287, + 505, + 386 + ], + "lines": [ + { + "bbox": [ + 105, + 286, + 505, + 300 + ], + "spans": [ + { + "bbox": [ + 105, + 286, + 505, + 300 + ], + "score": 1.0, + "content": "Butterfly matrices encode the recursive divide-and-conquer structure of the fast Fourier transform", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 297, + 506, + 311 + ], + "spans": [ + { + "bbox": [ + 106, + 297, + 506, + 311 + ], + "score": 1.0, + "content": "(FFT) algorithm. They were first used in numerical linear algebra for fast preconditioning (Parker,", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 308, + 506, + 322 + ], + "spans": [ + { + "bbox": [ + 105, + 308, + 506, + 322 + ], + "score": 1.0, + "content": "1995). The butterfly factorization is then generalized to encompass complementary low-rank matrices", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 320, + 506, + 333 + ], + "spans": [ + { + "bbox": [ + 105, + 320, + 506, + 333 + ], + "score": 1.0, + "content": "commonly encountered in solving differential and integral equations (Rokhlin & Tygert, 2006; Tygert,", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 331, + 506, + 344 + ], + "spans": [ + { + "bbox": [ + 106, + 331, + 506, + 344 + ], + "score": 1.0, + "content": "2008; 2010b;a; Li et al., 2015; 2018). In machine learning, butterfly matrices have been use to", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 341, + 506, + 354 + ], + "spans": [ + { + "bbox": [ + 105, + 341, + 506, + 354 + ], + "score": 1.0, + "content": "approximate the Hessian for fast optimization (Mathieu & LeCun, 2014), and to perform fast random", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 353, + 506, + 365 + ], + "spans": [ + { + "bbox": [ + 105, + 353, + 506, + 365 + ], + "score": 1.0, + "content": "projection (Jing et al., 2017; Munkhoeva et al., 2018; Choromanski et al., 2019). Dao et al. (2019)", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 364, + 505, + 376 + ], + "spans": [ + { + "bbox": [ + 106, + 364, + 505, + 376 + ], + "score": 1.0, + "content": "show that butterfly matrices can be used to learn fast algorithms for discrete transforms such as the", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 375, + 428, + 387 + ], + "spans": [ + { + "bbox": [ + 106, + 375, + 428, + 387 + ], + "score": 1.0, + "content": "Fourier transform, cosine/sine transform, Hadamard transform, and convolution.", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 20 + }, + { + "type": "title", + "bbox": [ + 108, + 401, + 212, + 412 + ], + "lines": [ + { + "bbox": [ + 106, + 399, + 214, + 413 + ], + "spans": [ + { + "bbox": [ + 106, + 399, + 214, + 413 + ], + "score": 1.0, + "content": "A.2 SPARSE MATRICES", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 25 + }, + { + "type": "text", + "bbox": [ + 107, + 422, + 505, + 597 + ], + "lines": [ + { + "bbox": [ + 105, + 421, + 506, + 434 + ], + "spans": [ + { + "bbox": [ + 105, + 421, + 506, + 434 + ], + "score": 1.0, + "content": "Several classes of structured linear transforms are ubiquitous in modern deep learning architectures;", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 432, + 505, + 446 + ], + "spans": [ + { + "bbox": [ + 105, + 432, + 505, + 446 + ], + "score": 1.0, + "content": "particularly widespread examples include convolution and multiheaded attention. Recently, attempts", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 444, + 505, + 456 + ], + "spans": [ + { + "bbox": [ + 105, + 444, + 505, + 456 + ], + "score": 1.0, + "content": "to impose sparsity on the neural network weights have been gaining traction. State-of-the art", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 104, + 453, + 506, + 469 + ], + "spans": [ + { + "bbox": [ + 104, + 453, + 506, + 469 + ], + "score": 1.0, + "content": "approaches of this type typically accomplish this by pruning small weights (either gradually during", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 465, + 506, + 479 + ], + "spans": [ + { + "bbox": [ + 105, + 465, + 506, + 479 + ], + "score": 1.0, + "content": "training (Zhu & Gupta, 2017), or post-training (Han et al., 2016)) or by training a dense network", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 476, + 505, + 489 + ], + "spans": [ + { + "bbox": [ + 105, + 476, + 505, + 489 + ], + "score": 1.0, + "content": "and then identifying “winning lottery tickets”—sparse subnetworks which may then be retrained", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 488, + 505, + 500 + ], + "spans": [ + { + "bbox": [ + 105, + 488, + 505, + 500 + ], + "score": 1.0, + "content": "from scratch with appropriate initialization (Frankle & Carbin, 2019). Importantly, these approaches", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 499, + 505, + 510 + ], + "spans": [ + { + "bbox": [ + 106, + 499, + 505, + 510 + ], + "score": 1.0, + "content": "start from a dense network, and therefore training is expensive. There is also a more nascent line of", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 510, + 505, + 522 + ], + "spans": [ + { + "bbox": [ + 105, + 510, + 505, + 522 + ], + "score": 1.0, + "content": "work that aims to train unstructured sparse neural networks directly (Mocanu et al., 2018; Mostafa", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 520, + 506, + 533 + ], + "spans": [ + { + "bbox": [ + 105, + 520, + 506, + 533 + ], + "score": 1.0, + "content": "& Wang, 2019; Dettmers & Zettlemoyer, 2019; Evci et al., 2019). These approaches maintain a", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 531, + 505, + 544 + ], + "spans": [ + { + "bbox": [ + 105, + 531, + 505, + 544 + ], + "score": 1.0, + "content": "constant network sparsity level throughout training, and use heuristics to evolve the sparsity pattern", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 542, + 505, + 555 + ], + "spans": [ + { + "bbox": [ + 105, + 542, + 505, + 555 + ], + "score": 1.0, + "content": "during training. One drawback is that the indices of the nonzero entries need to be stored in addition", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 553, + 505, + 565 + ], + "spans": [ + { + "bbox": [ + 105, + 553, + 505, + 565 + ], + "score": 1.0, + "content": "to the entry values themselves, which increases the memory required to store the sparse weight", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 564, + 505, + 577 + ], + "spans": [ + { + "bbox": [ + 105, + 564, + 505, + 577 + ], + "score": 1.0, + "content": "tensors. Another drawback is that these approaches to learn the sparsity pattern are based on intricate", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 574, + 505, + 588 + ], + "spans": [ + { + "bbox": [ + 105, + 574, + 505, + 588 + ], + "score": 1.0, + "content": "heuristics, which can be brittle. We note that these heuristic sparsification techniques could potentially", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 586, + 397, + 599 + ], + "spans": [ + { + "bbox": [ + 105, + 586, + 397, + 599 + ], + "score": 1.0, + "content": "be combined with our approach, to further sparsify the K-matrix factors.", + "type": "text" + } + ], + "index": 41 + } + ], + "index": 33.5 + }, + { + "type": "title", + "bbox": [ + 107, + 613, + 309, + 623 + ], + "lines": [ + { + "bbox": [ + 105, + 612, + 310, + 624 + ], + "spans": [ + { + "bbox": [ + 105, + 612, + 310, + 624 + ], + "score": 1.0, + "content": "A.3 SPEECH RECOGNITION FROM RAW AUDIO", + "type": "text" + } + ], + "index": 42 + } + ], + "index": 42 + }, + { + "type": "text", + "bbox": [ + 107, + 633, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 633, + 505, + 645 + ], + "spans": [ + { + "bbox": [ + 106, + 633, + 505, + 645 + ], + "score": 1.0, + "content": "Numerous works focus on the problem of speech recognition from raw audio input, i.e. without", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 643, + 506, + 656 + ], + "spans": [ + { + "bbox": [ + 105, + 643, + 506, + 656 + ], + "score": 1.0, + "content": "manual featurization. SincNet (Ravanelli & Bengio, 2018) is a CNN-based architecture parameterized", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 654, + 506, + 668 + ], + "spans": [ + { + "bbox": [ + 105, + 654, + 506, + 668 + ], + "score": 1.0, + "content": "with sinc functions, designed so that the first convolutional layer imitates a band-pass filter. Zeghidour", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 666, + 506, + 678 + ], + "spans": [ + { + "bbox": [ + 105, + 666, + 506, + 678 + ], + "score": 1.0, + "content": "et al. (2018) formulate a learnable version of a filter bank featurization; their filters are initialized as", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 677, + 506, + 689 + ], + "spans": [ + { + "bbox": [ + 105, + 677, + 506, + 689 + ], + "score": 1.0, + "content": "an approximation of MFSC features and then fine-tuned jointly with the rest of the model. Sainath", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 104, + 686, + 506, + 702 + ], + "spans": [ + { + "bbox": [ + 104, + 686, + 506, + 702 + ], + "score": 1.0, + "content": "et al. (2015) proposed a powerful combined convolutional LSTM (CLDNN)-based model for learning", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 105, + 699, + 506, + 712 + ], + "spans": [ + { + "bbox": [ + 105, + 699, + 506, + 712 + ], + "score": 1.0, + "content": "from raw audio, using a large amount of training data. The WaveNet generative architecture (van den", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 106, + 710, + 504, + 722 + ], + "spans": [ + { + "bbox": [ + 106, + 710, + 504, + 722 + ], + "score": 1.0, + "content": "Oord et al., 2016), based on dilated convolutions, has been adapted to speech recognition and can be", + "type": "text" + } + ], + "index": 50 + }, + { + "bbox": [ + 105, + 720, + 506, + 733 + ], + "spans": [ + { + "bbox": [ + 105, + 720, + 506, + 733 + ], + "score": 1.0, + "content": "trained on raw audio. 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Large learnable classes of structured", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 226, + 506, + 239 + ], + "spans": [ + { + "bbox": [ + 105, + 226, + 506, + 239 + ], + "score": 1.0, + "content": "matrices such as Toeplitz-like matrices (Sindhwani et al., 2015) and low-displacement rank (LDR)", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 237, + 506, + 250 + ], + "spans": [ + { + "bbox": [ + 105, + 237, + 506, + 250 + ], + "score": 1.0, + "content": "matrices (Thomas et al., 2018) have been used for model compression. However, despite their", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 248, + 505, + 261 + ], + "spans": [ + { + "bbox": [ + 105, + 248, + 505, + 261 + ], + "score": 1.0, + "content": "theoretical speedup, these structured matrix classes lack efficient implementations, especially on", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 258, + 506, + 272 + ], + "spans": [ + { + "bbox": [ + 105, + 258, + 506, + 272 + ], + "score": 1.0, + "content": "GPUs. Therefore, their use has largely been confined to small models (e.g. single hidden layer neural", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 271, + 273, + 282 + ], + "spans": [ + { + "bbox": [ + 106, + 271, + 273, + 282 + ], + "score": 1.0, + "content": "nets) and small datasets (e.g. CIFAR-10).", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 8.5, + "bbox_fs": [ + 105, + 127, + 506, + 282 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 287, + 505, + 386 + ], + "lines": [ + { + "bbox": [ + 105, + 286, + 505, + 300 + ], + "spans": [ + { + "bbox": [ + 105, + 286, + 505, + 300 + ], + "score": 1.0, + "content": "Butterfly matrices encode the recursive divide-and-conquer structure of the fast Fourier transform", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 297, + 506, + 311 + ], + "spans": [ + { + "bbox": [ + 106, + 297, + 506, + 311 + ], + "score": 1.0, + "content": "(FFT) algorithm. They were first used in numerical linear algebra for fast preconditioning (Parker,", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 308, + 506, + 322 + ], + "spans": [ + { + "bbox": [ + 105, + 308, + 506, + 322 + ], + "score": 1.0, + "content": "1995). The butterfly factorization is then generalized to encompass complementary low-rank matrices", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 320, + 506, + 333 + ], + "spans": [ + { + "bbox": [ + 105, + 320, + 506, + 333 + ], + "score": 1.0, + "content": "commonly encountered in solving differential and integral equations (Rokhlin & Tygert, 2006; Tygert,", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 331, + 506, + 344 + ], + "spans": [ + { + "bbox": [ + 106, + 331, + 506, + 344 + ], + "score": 1.0, + "content": "2008; 2010b;a; Li et al., 2015; 2018). In machine learning, butterfly matrices have been use to", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 341, + 506, + 354 + ], + "spans": [ + { + "bbox": [ + 105, + 341, + 506, + 354 + ], + "score": 1.0, + "content": "approximate the Hessian for fast optimization (Mathieu & LeCun, 2014), and to perform fast random", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 353, + 506, + 365 + ], + "spans": [ + { + "bbox": [ + 105, + 353, + 506, + 365 + ], + "score": 1.0, + "content": "projection (Jing et al., 2017; Munkhoeva et al., 2018; Choromanski et al., 2019). Dao et al. (2019)", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 364, + 505, + 376 + ], + "spans": [ + { + "bbox": [ + 106, + 364, + 505, + 376 + ], + "score": 1.0, + "content": "show that butterfly matrices can be used to learn fast algorithms for discrete transforms such as the", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 375, + 428, + 387 + ], + "spans": [ + { + "bbox": [ + 106, + 375, + 428, + 387 + ], + "score": 1.0, + "content": "Fourier transform, cosine/sine transform, Hadamard transform, and convolution.", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 20, + "bbox_fs": [ + 105, + 286, + 506, + 387 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 401, + 212, + 412 + ], + "lines": [ + { + "bbox": [ + 106, + 399, + 214, + 413 + ], + "spans": [ + { + "bbox": [ + 106, + 399, + 214, + 413 + ], + "score": 1.0, + "content": "A.2 SPARSE MATRICES", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 25 + }, + { + "type": "text", + "bbox": [ + 107, + 422, + 505, + 597 + ], + "lines": [ + { + "bbox": [ + 105, + 421, + 506, + 434 + ], + "spans": [ + { + "bbox": [ + 105, + 421, + 506, + 434 + ], + "score": 1.0, + "content": "Several classes of structured linear transforms are ubiquitous in modern deep learning architectures;", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 432, + 505, + 446 + ], + "spans": [ + { + "bbox": [ + 105, + 432, + 505, + 446 + ], + "score": 1.0, + "content": "particularly widespread examples include convolution and multiheaded attention. Recently, attempts", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 444, + 505, + 456 + ], + "spans": [ + { + "bbox": [ + 105, + 444, + 505, + 456 + ], + "score": 1.0, + "content": "to impose sparsity on the neural network weights have been gaining traction. State-of-the art", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 104, + 453, + 506, + 469 + ], + "spans": [ + { + "bbox": [ + 104, + 453, + 506, + 469 + ], + "score": 1.0, + "content": "approaches of this type typically accomplish this by pruning small weights (either gradually during", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 465, + 506, + 479 + ], + "spans": [ + { + "bbox": [ + 105, + 465, + 506, + 479 + ], + "score": 1.0, + "content": "training (Zhu & Gupta, 2017), or post-training (Han et al., 2016)) or by training a dense network", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 476, + 505, + 489 + ], + "spans": [ + { + "bbox": [ + 105, + 476, + 505, + 489 + ], + "score": 1.0, + "content": "and then identifying “winning lottery tickets”—sparse subnetworks which may then be retrained", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 488, + 505, + 500 + ], + "spans": [ + { + "bbox": [ + 105, + 488, + 505, + 500 + ], + "score": 1.0, + "content": "from scratch with appropriate initialization (Frankle & Carbin, 2019). Importantly, these approaches", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 499, + 505, + 510 + ], + "spans": [ + { + "bbox": [ + 106, + 499, + 505, + 510 + ], + "score": 1.0, + "content": "start from a dense network, and therefore training is expensive. There is also a more nascent line of", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 510, + 505, + 522 + ], + "spans": [ + { + "bbox": [ + 105, + 510, + 505, + 522 + ], + "score": 1.0, + "content": "work that aims to train unstructured sparse neural networks directly (Mocanu et al., 2018; Mostafa", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 520, + 506, + 533 + ], + "spans": [ + { + "bbox": [ + 105, + 520, + 506, + 533 + ], + "score": 1.0, + "content": "& Wang, 2019; Dettmers & Zettlemoyer, 2019; Evci et al., 2019). These approaches maintain a", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 531, + 505, + 544 + ], + "spans": [ + { + "bbox": [ + 105, + 531, + 505, + 544 + ], + "score": 1.0, + "content": "constant network sparsity level throughout training, and use heuristics to evolve the sparsity pattern", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 542, + 505, + 555 + ], + "spans": [ + { + "bbox": [ + 105, + 542, + 505, + 555 + ], + "score": 1.0, + "content": "during training. One drawback is that the indices of the nonzero entries need to be stored in addition", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 553, + 505, + 565 + ], + "spans": [ + { + "bbox": [ + 105, + 553, + 505, + 565 + ], + "score": 1.0, + "content": "to the entry values themselves, which increases the memory required to store the sparse weight", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 564, + 505, + 577 + ], + "spans": [ + { + "bbox": [ + 105, + 564, + 505, + 577 + ], + "score": 1.0, + "content": "tensors. Another drawback is that these approaches to learn the sparsity pattern are based on intricate", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 574, + 505, + 588 + ], + "spans": [ + { + "bbox": [ + 105, + 574, + 505, + 588 + ], + "score": 1.0, + "content": "heuristics, which can be brittle. We note that these heuristic sparsification techniques could potentially", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 586, + 397, + 599 + ], + "spans": [ + { + "bbox": [ + 105, + 586, + 397, + 599 + ], + "score": 1.0, + "content": "be combined with our approach, to further sparsify the K-matrix factors.", + "type": "text" + } + ], + "index": 41 + } + ], + "index": 33.5, + "bbox_fs": [ + 104, + 421, + 506, + 599 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 613, + 309, + 623 + ], + "lines": [ + { + "bbox": [ + 105, + 612, + 310, + 624 + ], + "spans": [ + { + "bbox": [ + 105, + 612, + 310, + 624 + ], + "score": 1.0, + "content": "A.3 SPEECH RECOGNITION FROM RAW AUDIO", + "type": "text" + } + ], + "index": 42 + } + ], + "index": 42 + }, + { + "type": "text", + "bbox": [ + 107, + 633, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 633, + 505, + 645 + ], + "spans": [ + { + "bbox": [ + 106, + 633, + 505, + 645 + ], + "score": 1.0, + "content": "Numerous works focus on the problem of speech recognition from raw audio input, i.e. without", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 643, + 506, + 656 + ], + "spans": [ + { + "bbox": [ + 105, + 643, + 506, + 656 + ], + "score": 1.0, + "content": "manual featurization. SincNet (Ravanelli & Bengio, 2018) is a CNN-based architecture parameterized", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 654, + 506, + 668 + ], + "spans": [ + { + "bbox": [ + 105, + 654, + 506, + 668 + ], + "score": 1.0, + "content": "with sinc functions, designed so that the first convolutional layer imitates a band-pass filter. Zeghidour", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 666, + 506, + 678 + ], + "spans": [ + { + "bbox": [ + 105, + 666, + 506, + 678 + ], + "score": 1.0, + "content": "et al. (2018) formulate a learnable version of a filter bank featurization; their filters are initialized as", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 677, + 506, + 689 + ], + "spans": [ + { + "bbox": [ + 105, + 677, + 506, + 689 + ], + "score": 1.0, + "content": "an approximation of MFSC features and then fine-tuned jointly with the rest of the model. Sainath", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 104, + 686, + 506, + 702 + ], + "spans": [ + { + "bbox": [ + 104, + 686, + 506, + 702 + ], + "score": 1.0, + "content": "et al. (2015) proposed a powerful combined convolutional LSTM (CLDNN)-based model for learning", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 105, + 699, + 506, + 712 + ], + "spans": [ + { + "bbox": [ + 105, + 699, + 506, + 712 + ], + "score": 1.0, + "content": "from raw audio, using a large amount of training data. The WaveNet generative architecture (van den", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 106, + 710, + 504, + 722 + ], + "spans": [ + { + "bbox": [ + 106, + 710, + 504, + 722 + ], + "score": 1.0, + "content": "Oord et al., 2016), based on dilated convolutions, has been adapted to speech recognition and can be", + "type": "text" + } + ], + "index": 50 + }, + { + "bbox": [ + 105, + 720, + 506, + 733 + ], + "spans": [ + { + "bbox": [ + 105, + 720, + 506, + 733 + ], + "score": 1.0, + "content": "trained on raw audio. Other approaches that can learn from raw audio can be found in (Palaz et al.,", + "type": "text" + } + ], + "index": 51 + } + ], + "index": 47, + "bbox_fs": [ + 104, + 633, + 506, + 733 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 82, + 504, + 115 + ], + "lines": [ + { + "bbox": [ + 106, + 82, + 505, + 94 + ], + "spans": [ + { + "bbox": [ + 106, + 82, + 418, + 94 + ], + "score": 1.0, + "content": "2013; Collobert et al., 2016; Ghahremani et al., 2016). To our knowledge, the", + "type": "text" + }, + { + "bbox": [ + 419, + 82, + 446, + 93 + ], + "score": 0.85, + "content": "1 4 . 6 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 446, + 82, + 505, + 94 + ], + "score": 1.0, + "content": "PER achieved", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 93, + 505, + 106 + ], + "spans": [ + { + "bbox": [ + 105, + 93, + 186, + 106 + ], + "score": 1.0, + "content": "by our kaleidoscope", + "type": "text" + }, + { + "bbox": [ + 187, + 94, + 223, + 104 + ], + "score": 0.38, + "content": "+ \\mathrm { L S T M }", + "type": "inline_equation" + }, + { + "bbox": [ + 223, + 93, + 505, + 106 + ], + "score": 1.0, + "content": "model on the TIMIT test set is the lowest error rate obtained by a model", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 106, + 105, + 240, + 116 + ], + "spans": [ + { + "bbox": [ + 106, + 105, + 240, + 116 + ], + "score": 1.0, + "content": "trained directly on the raw audio.", + "type": "text" + } + ], + "index": 2 + } + ], + "index": 1 + }, + { + "type": "title", + "bbox": [ + 108, + 129, + 248, + 140 + ], + "lines": [ + { + "bbox": [ + 107, + 129, + 249, + 141 + ], + "spans": [ + { + "bbox": [ + 107, + 129, + 249, + 141 + ], + "score": 1.0, + "content": "A.4 LEARNING PERMUTATIONS", + "type": "text" + } + ], + "index": 3 + } + ], + "index": 3 + }, + { + "type": "text", + "bbox": [ + 107, + 150, + 505, + 193 + ], + "lines": [ + { + "bbox": [ + 105, + 149, + 505, + 162 + ], + "spans": [ + { + "bbox": [ + 105, + 149, + 505, + 162 + ], + "score": 1.0, + "content": "Permutation matrices find use in tasks such as matching and sorting (among many others). Techniques", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 160, + 506, + 173 + ], + "spans": [ + { + "bbox": [ + 105, + 160, + 506, + 173 + ], + "score": 1.0, + "content": "to obtain posterior distributions over permutations have been developed, such as the exponential", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 172, + 506, + 184 + ], + "spans": [ + { + "bbox": [ + 105, + 172, + 506, + 184 + ], + "score": 1.0, + "content": "weights algorithm (Helmbold & Warmuth, 2009) and the Gumbel-Sinkhorn network (Mena et al.,", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 104, + 180, + 136, + 196 + ], + "spans": [ + { + "bbox": [ + 104, + 180, + 136, + 196 + ], + "score": 1.0, + "content": "2018).", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 5.5 + }, + { + "type": "text", + "bbox": [ + 107, + 199, + 505, + 276 + ], + "lines": [ + { + "bbox": [ + 105, + 199, + 505, + 212 + ], + "spans": [ + { + "bbox": [ + 105, + 199, + 505, + 212 + ], + "score": 1.0, + "content": "Classifying images with permuted pixels is a standard task to benchmark the ability of RNNs to learn", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 210, + 506, + 224 + ], + "spans": [ + { + "bbox": [ + 105, + 210, + 506, + 224 + ], + "score": 1.0, + "content": "long range dependencies. Le et al. (2015) propose the Permuted MNIST task, in which the model has", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 221, + 506, + 235 + ], + "spans": [ + { + "bbox": [ + 105, + 221, + 506, + 235 + ], + "score": 1.0, + "content": "to classify digit images with all the pixels permuted. Many new RNN architectures, with unitary or", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 233, + 506, + 244 + ], + "spans": [ + { + "bbox": [ + 105, + 233, + 506, + 244 + ], + "score": 1.0, + "content": "orthogonal weight matrices to avoid gradient explosion or vanishing, have been proposed and tested", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 243, + 506, + 256 + ], + "spans": [ + { + "bbox": [ + 105, + 243, + 506, + 256 + ], + "score": 1.0, + "content": "on this task (Le et al., 2015; Arjovsky et al., 2016; Wisdom et al., 2016; Mhammedi et al., 2017;", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 254, + 506, + 266 + ], + "spans": [ + { + "bbox": [ + 105, + 254, + 506, + 266 + ], + "score": 1.0, + "content": "Trinh et al., 2018). Standard gated RNN architectures such as LSTM and GRU have also been found", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 266, + 437, + 277 + ], + "spans": [ + { + "bbox": [ + 106, + 266, + 437, + 277 + ], + "score": 1.0, + "content": "to be competitive with these new RNN architectures on this task (Bai et al., 2018).", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 11 + }, + { + "type": "title", + "bbox": [ + 108, + 293, + 327, + 305 + ], + "lines": [ + { + "bbox": [ + 105, + 291, + 328, + 307 + ], + "spans": [ + { + "bbox": [ + 105, + 291, + 328, + 307 + ], + "score": 1.0, + "content": "B ADDITIONAL EXPERIMENTAL DETAILS", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 15 + }, + { + "type": "title", + "bbox": [ + 108, + 317, + 239, + 328 + ], + "lines": [ + { + "bbox": [ + 106, + 316, + 240, + 329 + ], + "spans": [ + { + "bbox": [ + 106, + 316, + 240, + 329 + ], + "score": 1.0, + "content": "B.1 SPEECH PREPROCESSING", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 16 + }, + { + "type": "text", + "bbox": [ + 106, + 338, + 504, + 360 + ], + "lines": [ + { + "bbox": [ + 105, + 337, + 505, + 351 + ], + "spans": [ + { + "bbox": [ + 105, + 337, + 505, + 351 + ], + "score": 1.0, + "content": "In this section, we fully describe our settings and procedures for the speech preprocessing experiments", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 349, + 389, + 361 + ], + "spans": [ + { + "bbox": [ + 106, + 349, + 389, + 361 + ], + "score": 1.0, + "content": "in Section 3.1.1, and present additional auxiliary baselines and results.", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 17.5 + }, + { + "type": "title", + "bbox": [ + 108, + 372, + 239, + 383 + ], + "lines": [ + { + "bbox": [ + 106, + 372, + 239, + 384 + ], + "spans": [ + { + "bbox": [ + 106, + 372, + 239, + 384 + ], + "score": 1.0, + "content": "B.1.1 EXPERIMENTAL SETUP", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 19 + }, + { + "type": "text", + "bbox": [ + 107, + 391, + 505, + 468 + ], + "lines": [ + { + "bbox": [ + 105, + 391, + 506, + 405 + ], + "spans": [ + { + "bbox": [ + 105, + 391, + 506, + 405 + ], + "score": 1.0, + "content": "We evaluate our speech recognition models on the TIMIT speech corpus (Garofolo et al., 1993), a", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 403, + 505, + 415 + ], + "spans": [ + { + "bbox": [ + 105, + 403, + 387, + 415 + ], + "score": 1.0, + "content": "standard benchmark for speech recognition. The input is audio (16-bit,", + "type": "text" + }, + { + "bbox": [ + 388, + 403, + 417, + 414 + ], + "score": 0.32, + "content": "1 6 \\mathrm { k H z }", + "type": "inline_equation" + }, + { + "bbox": [ + 418, + 403, + 505, + 415 + ], + "score": 1.0, + "content": ".wav format), and the", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 414, + 505, + 425 + ], + "spans": [ + { + "bbox": [ + 105, + 414, + 505, + 425 + ], + "score": 1.0, + "content": "target is the transcription into a sequence of phonemes (units of spoken sound). 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We use PyTorch (Paszke et al., 2017), the Kaldi speech recognition toolkit", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 447, + 505, + 459 + ], + "spans": [ + { + "bbox": [ + 105, + 447, + 505, + 459 + ], + "score": 1.0, + "content": "(Povey et al., 2011), and the PyTorch-Kaldi toolkit (Ravanelli et al., 2019) for developing PyTorch", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 457, + 376, + 470 + ], + "spans": [ + { + "bbox": [ + 105, + 457, + 376, + 470 + ], + "score": 1.0, + "content": "speech recognition models for all our experiments and evaluations.", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 23, + "bbox_fs": [ + 105, + 391, + 506, + 470 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 480, + 253, + 492 + ], + "lines": [ + { + "bbox": [ + 105, + 480, + 254, + 493 + ], + "spans": [ + { + "bbox": [ + 105, + 480, + 254, + 493 + ], + "score": 1.0, + "content": "B.1.2 MODEL AND EVALUATION", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 27 + }, + { + "type": "text", + "bbox": [ + 107, + 500, + 505, + 653 + ], + "lines": [ + { + "bbox": [ + 104, + 497, + 506, + 514 + ], + "spans": [ + { + "bbox": [ + 104, + 497, + 506, + 514 + ], + "score": 1.0, + "content": "Our baseline Bi-LSTM architecture is taken from the PyTorch-Kaldi repository.6 This is a strong", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 511, + 505, + 524 + ], + "spans": [ + { + "bbox": [ + 105, + 511, + 505, + 524 + ], + "score": 1.0, + "content": "baseline model that, to the best of our knowledge, matches state-of-the-art performance for models", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 521, + 505, + 533 + ], + "spans": [ + { + "bbox": [ + 105, + 521, + 505, + 533 + ], + "score": 1.0, + "content": "that use a single type of input featurization (Ravanelli et al., 2019). The original Bi-LSTM model", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 533, + 505, + 545 + ], + "spans": [ + { + "bbox": [ + 106, + 533, + 505, + 545 + ], + "score": 1.0, + "content": "takes as input filter bank features. These are computed as follows: (i) the waveform is framed (split", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 543, + 505, + 556 + ], + "spans": [ + { + "bbox": [ + 105, + 543, + 168, + 556 + ], + "score": 1.0, + "content": "into chunks of", + "type": "text" + }, + { + "bbox": [ + 168, + 544, + 195, + 554 + ], + "score": 0.43, + "content": "2 5 ~ \\mathrm { m s }", + "type": "inline_equation" + }, + { + "bbox": [ + 195, + 543, + 281, + 556 + ], + "score": 1.0, + "content": "each that overlap by", + "type": "text" + }, + { + "bbox": [ + 281, + 544, + 308, + 555 + ], + "score": 0.46, + "content": "1 0 \\mathrm { m s }", + "type": "inline_equation" + }, + { + "bbox": [ + 308, + 543, + 505, + 556 + ], + "score": 1.0, + "content": "each), (ii) the waveform is dithered (zero-mean", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 555, + 505, + 568 + ], + "spans": [ + { + "bbox": [ + 105, + 555, + 505, + 568 + ], + "score": 1.0, + "content": "Gaussian random noise is added), (iii) pre-emphasis is applied to amplify high frequencies, (iv) the", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 565, + 505, + 578 + ], + "spans": [ + { + "bbox": [ + 105, + 565, + 505, + 578 + ], + "score": 1.0, + "content": "Hamming window function (Harris, 1978) is applied, (v) the FFT is applied, and the power spectrum", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 577, + 505, + 589 + ], + "spans": [ + { + "bbox": [ + 105, + 577, + 505, + 589 + ], + "score": 1.0, + "content": "of the resulting (complex-valued) output is computed, (vi) the power spectrum (which has dimension", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 587, + 505, + 600 + ], + "spans": [ + { + "bbox": [ + 105, + 587, + 505, + 600 + ], + "score": 1.0, + "content": "512) is mapped to the “mel scale” (which is a scale intended to mimic human auditory perception", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 598, + 506, + 611 + ], + "spans": [ + { + "bbox": [ + 105, + 598, + 445, + 611 + ], + "score": 1.0, + "content": "(Stevens et al., 1937)) by multiplication with a specific banded matrix of dimension", + "type": "text" + }, + { + "bbox": [ + 446, + 599, + 484, + 609 + ], + "score": 0.9, + "content": "5 1 2 \\times 2 3", + "type": "inline_equation" + }, + { + "bbox": [ + 484, + 598, + 506, + 611 + ], + "score": 1.0, + "content": ", and", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 609, + 505, + 622 + ], + "spans": [ + { + "bbox": [ + 105, + 609, + 505, + 622 + ], + "score": 1.0, + "content": "the entrywise logarithm of the output is taken (the 23 outputs are called the filters), and (vii) cepstral", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 620, + 505, + 633 + ], + "spans": [ + { + "bbox": [ + 105, + 620, + 505, + 633 + ], + "score": 1.0, + "content": "mean and variance normalization (Liu et al., 1993) is applied. Numerical hyperparameters of this", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 632, + 505, + 644 + ], + "spans": [ + { + "bbox": [ + 105, + 632, + 505, + 644 + ], + "score": 1.0, + "content": "procedure include the dither noise scale, the pre-emphasis coefficient, the Hamming window size, the", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 642, + 493, + 655 + ], + "spans": [ + { + "bbox": [ + 105, + 642, + 493, + 655 + ], + "score": 1.0, + "content": "number of mel filters, and more; we kept all these the same as the Kaldi/PyTorch-Kaldi defaults.", + "type": "text" + } + ], + "index": 41 + } + ], + "index": 34.5, + "bbox_fs": [ + 104, + 497, + 506, + 655 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 659, + 505, + 703 + ], + "lines": [ + { + "bbox": [ + 106, + 659, + 505, + 671 + ], + "spans": [ + { + "bbox": [ + 106, + 659, + 505, + 671 + ], + "score": 1.0, + "content": "In contrast, our “K-matrix version” of the model takes as input the raw waveform, split into chunks", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 106, + 671, + 505, + 682 + ], + "spans": [ + { + "bbox": [ + 106, + 671, + 505, + 682 + ], + "score": 1.0, + "content": "the same way as before but with no normalization, dithering, or other preprocessing, which is then fed", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 680, + 506, + 694 + ], + "spans": [ + { + "bbox": [ + 105, + 680, + 252, + 694 + ], + "score": 1.0, + "content": "into a complex-valued kaleidoscope", + "type": "text" + }, + { + "bbox": [ + 253, + 681, + 290, + 693 + ], + "score": 0.91, + "content": "[ ( B B ^ { * } ) ^ { 2 } ]", + "type": "inline_equation" + }, + { + "bbox": [ + 291, + 680, + 506, + 694 + ], + "score": 1.0, + "content": "matrix. Similarly to the nonlinear steps in computing", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 692, + 506, + 705 + ], + "spans": [ + { + "bbox": [ + 105, + 692, + 506, + 705 + ], + "score": 1.0, + "content": "filter bank features, the logarithm of the power spectrum of the output (which has dimension 512)", + "type": "text" + } + ], + "index": 45 + } + ], + "index": 43.5, + "bbox_fs": [ + 105, + 659, + 506, + 705 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 82, + 505, + 182 + ], + "lines": [ + { + "bbox": [ + 105, + 83, + 505, + 95 + ], + "spans": [ + { + "bbox": [ + 105, + 83, + 505, + 95 + ], + "score": 1.0, + "content": "is then computed. This output is fed into the Bi-LSTM; the Bi-LSTM and kaleidoscope layer are", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 93, + 505, + 105 + ], + "spans": [ + { + "bbox": [ + 105, + 93, + 505, + 105 + ], + "score": 1.0, + "content": "trained together in standard end-to-end fashion. The Bi-LSTM architecture is not modified aside", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 106, + 105, + 505, + 117 + ], + "spans": [ + { + "bbox": [ + 106, + 105, + 400, + 117 + ], + "score": 1.0, + "content": "from changing the input dimension from 23 to 512; this (along with the", + "type": "text" + }, + { + "bbox": [ + 401, + 105, + 431, + 115 + ], + "score": 0.81, + "content": "\\approx 7 5 \\mathrm { K }", + "type": "inline_equation" + }, + { + "bbox": [ + 432, + 105, + 505, + 117 + ], + "score": 1.0, + "content": "parameters in the", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 114, + 506, + 128 + ], + "spans": [ + { + "bbox": [ + 105, + 114, + 506, + 128 + ], + "score": 1.0, + "content": "kaleidoscope layer itself) results in approximately a 1.1M increase in the total number of parameters", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 126, + 505, + 140 + ], + "spans": [ + { + "bbox": [ + 105, + 126, + 358, + 140 + ], + "score": 1.0, + "content": "compared to the model that takes in MFSC features (a modest", + "type": "text" + }, + { + "bbox": [ + 358, + 127, + 373, + 137 + ], + "score": 0.85, + "content": "8 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 373, + 126, + 505, + 140 + ], + "score": 1.0, + "content": "relative increase). Total training", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 136, + 506, + 150 + ], + "spans": [ + { + "bbox": [ + 105, + 136, + 294, + 150 + ], + "score": 1.0, + "content": "time for our kaleidoscope-based architecture is", + "type": "text" + }, + { + "bbox": [ + 294, + 137, + 309, + 147 + ], + "score": 0.85, + "content": "7 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 309, + 136, + 506, + 150 + ], + "score": 1.0, + "content": "greater than that required for the model that uses", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 148, + 506, + 162 + ], + "spans": [ + { + "bbox": [ + 105, + 148, + 506, + 162 + ], + "score": 1.0, + "content": "MFSC features, not counting the time required to precompute the MFSC features; the FLOPs for", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 159, + 505, + 172 + ], + "spans": [ + { + "bbox": [ + 105, + 159, + 240, + 172 + ], + "score": 1.0, + "content": "inference-time are approximately", + "type": "text" + }, + { + "bbox": [ + 240, + 159, + 259, + 170 + ], + "score": 0.86, + "content": "15 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 260, + 159, + 505, + 172 + ], + "score": 1.0, + "content": "greater (mostly due to the larger dimension of the input to the", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 170, + 437, + 183 + ], + "spans": [ + { + "bbox": [ + 105, + 170, + 331, + 183 + ], + "score": 1.0, + "content": "Bi-LSTM; the kaleidoscope layer accounts for less than", + "type": "text" + }, + { + "bbox": [ + 331, + 171, + 353, + 181 + ], + "score": 0.85, + "content": "0 . 5 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 354, + 170, + 437, + 183 + ], + "score": 1.0, + "content": "of the total FLOPs).", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 4 + }, + { + "type": "text", + "bbox": [ + 107, + 187, + 505, + 264 + ], + "lines": [ + { + "bbox": [ + 105, + 187, + 506, + 200 + ], + "spans": [ + { + "bbox": [ + 105, + 187, + 506, + 200 + ], + "score": 1.0, + "content": "As baselines, we also compare to inserting other types of linear transformations before the Bi-LSTM:", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 197, + 507, + 210 + ], + "spans": [ + { + "bbox": [ + 105, + 197, + 507, + 210 + ], + "score": 1.0, + "content": "fixed linear transformations (such as the fixed FFT, or no transform at all [i.e. the identity]), other train-", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 209, + 505, + 222 + ], + "spans": [ + { + "bbox": [ + 105, + 209, + 505, + 222 + ], + "score": 1.0, + "content": "able structured layers (low-rank, circulant, and sparse [using the sparse training algorithm of Dettmers", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 220, + 506, + 233 + ], + "spans": [ + { + "bbox": [ + 105, + 220, + 506, + 233 + ], + "score": 1.0, + "content": "& Zettlemoyer (2019)]), and a trainable unstructured (dense) linear layer. The kaleidoscope layer", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 230, + 506, + 244 + ], + "spans": [ + { + "bbox": [ + 105, + 230, + 506, + 244 + ], + "score": 1.0, + "content": "performs the best out of all such approaches. The fact that it outperforms even a dense linear layer", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 242, + 505, + 254 + ], + "spans": [ + { + "bbox": [ + 105, + 242, + 505, + 254 + ], + "score": 1.0, + "content": "with more parameters is particularly notable, as it suggests that the structural bias imposed by the", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 253, + 504, + 265 + ], + "spans": [ + { + "bbox": [ + 106, + 253, + 504, + 265 + ], + "score": 1.0, + "content": "K-matrix representation is beneficial for performance on this task. Full results are given in Table 5.", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 12 + }, + { + "type": "table", + "bbox": [ + 167, + 294, + 443, + 433 + ], + "blocks": [ + { + "type": "table_caption", + "bbox": [ + 122, + 276, + 485, + 288 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 124, + 275, + 486, + 288 + ], + "spans": [ + { + "bbox": [ + 124, + 275, + 270, + 288 + ], + "score": 1.0, + "content": "Table 5: TIMIT phoneme error rate", + "type": "text" + }, + { + "bbox": [ + 271, + 276, + 299, + 286 + ], + "score": 0.73, + "content": "( \\mathrm { P E R } \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 299, + 275, + 302, + 288 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 302, + 276, + 313, + 286 + ], + "score": 0.51, + "content": "\\pm", + "type": "inline_equation" + }, + { + "bbox": [ + 313, + 275, + 486, + 288 + ], + "score": 1.0, + "content": "standard deviation across 5 random seeds).", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 16 + }, + { + "type": "table_body", + "bbox": [ + 167, + 294, + 443, + 433 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 167, + 294, + 443, + 433 + ], + "spans": [ + { + "bbox": [ + 167, + 294, + 443, + 433 + ], + "score": 0.983, + "html": "
ModelTest set PER%#Parameters
Low rank + LSTM23.6 ± 0.915.5M
Sparse +LSTM21.7 ± 0.915.5M
Circulant +LSTM23.9 ± 0.915.4M
Dense +LSTM15.4 ± 0.615.9M
FFT+LSTM15.7 ± 0.115.4M
Identity +LSTM20.7 ± 0.315.4M
Kaleidoscope + LSTM14.6 ± 0.315.4M
MFSC features + LSTM14.2 ± 0.214.3M
SincNet (Ravanelli et al., 2019)17.210.0M
LiGRU (Ravanelli et al., 2018)13.812.3M
", + "type": "table", + "image_path": "339a0f9dd315c6fd892eac7a9f09786548540de0105e3cba345bef5d6336e941.jpg" + } + ] + } + ], + "index": 18, + "virtual_lines": [ + { + "bbox": [ + 167, + 294, + 443, + 340.3333333333333 + ], + "spans": [], + "index": 17 + }, + { + "bbox": [ + 167, + 340.3333333333333, + 443, + 386.66666666666663 + ], + "spans": [], + "index": 18 + }, + { + "bbox": [ + 167, + 386.66666666666663, + 443, + 432.99999999999994 + ], + "spans": [], + "index": 19 + } + ] + } + ], + "index": 17.0 + }, + { + "type": "text", + "bbox": [ + 106, + 445, + 505, + 588 + ], + "lines": [ + { + "bbox": [ + 106, + 446, + 505, + 457 + ], + "spans": [ + { + "bbox": [ + 106, + 446, + 505, + 457 + ], + "score": 1.0, + "content": "In our experiments, we grid search the initial learning rate for the “preprocessing layer” (if applicable)", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 456, + 506, + 469 + ], + "spans": [ + { + "bbox": [ + 105, + 456, + 506, + 469 + ], + "score": 1.0, + "content": "in {5e-5, 1e-4, 2e-4, 4e-4, 8e-4, 1.6e-3}, and fix all other hyperparameters (including the initial", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 104, + 466, + 507, + 481 + ], + "spans": [ + { + "bbox": [ + 104, + 466, + 507, + 481 + ], + "score": 1.0, + "content": "learning rates for the other parts of the network) to their default values in the PyTorch-Kaldi repository.", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 478, + 506, + 491 + ], + "spans": [ + { + "bbox": [ + 106, + 478, + 506, + 491 + ], + "score": 1.0, + "content": "The model and any preprocessing layers are trained end-to-end with the RMSProp optimizer for 24", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 489, + 506, + 502 + ], + "spans": [ + { + "bbox": [ + 105, + 489, + 506, + 502 + ], + "score": 1.0, + "content": "epochs (as per the defaults in PyTorch-Kaldi). For each model, we use the validation set to select", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 500, + 506, + 513 + ], + "spans": [ + { + "bbox": [ + 105, + 500, + 506, + 513 + ], + "score": 1.0, + "content": "the best preprocessing learning rate, while the final error rates are reported on the separate held-out", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 511, + 506, + 524 + ], + "spans": [ + { + "bbox": [ + 105, + 511, + 423, + 524 + ], + "score": 1.0, + "content": "test set. 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The final test", + "type": "text" + }, + { + "bbox": [ + 428, + 688, + 456, + 698 + ], + "score": 0.85, + "content": "\\mathrm { P E R } \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 457, + 687, + 505, + 700 + ], + "score": 1.0, + "content": "attained by", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 698, + 505, + 711 + ], + "spans": [ + { + "bbox": [ + 105, + 698, + 197, + 711 + ], + "score": 1.0, + "content": "this “hybrid” model is", + "type": "text" + }, + { + "bbox": [ + 198, + 699, + 241, + 709 + ], + "score": 0.87, + "content": "{ \\bf 1 4 . 0 \\pm 0 . 3 }", + "type": "inline_equation" + }, + { + "bbox": [ + 241, + 698, + 505, + 711 + ], + "score": 1.0, + "content": "; the model has 14.4M parameters—a negligible increase over the", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 106, + 709, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 106, + 709, + 134, + 720 + ], + "score": 0.36, + "content": "1 4 . 3 \\mathbf { M }", + "type": "inline_equation" + }, + { + "bbox": [ + 134, + 709, + 505, + 722 + ], + "score": 1.0, + "content": "in the original architecture. 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Total training", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 136, + 506, + 150 + ], + "spans": [ + { + "bbox": [ + 105, + 136, + 294, + 150 + ], + "score": 1.0, + "content": "time for our kaleidoscope-based architecture is", + "type": "text" + }, + { + "bbox": [ + 294, + 137, + 309, + 147 + ], + "score": 0.85, + "content": "7 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 309, + 136, + 506, + 150 + ], + "score": 1.0, + "content": "greater than that required for the model that uses", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 148, + 506, + 162 + ], + "spans": [ + { + "bbox": [ + 105, + 148, + 506, + 162 + ], + "score": 1.0, + "content": "MFSC features, not counting the time required to precompute the MFSC features; the FLOPs for", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 159, + 505, + 172 + ], + "spans": [ + { + "bbox": [ + 105, + 159, + 240, + 172 + ], + "score": 1.0, + "content": "inference-time are approximately", + "type": "text" + }, + { + "bbox": [ + 240, + 159, + 259, + 170 + ], + "score": 0.86, + "content": "15 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 260, + 159, + 505, + 172 + ], + "score": 1.0, + "content": "greater (mostly due to the larger dimension of the input to the", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 170, + 437, + 183 + ], + "spans": [ + { + "bbox": [ + 105, + 170, + 331, + 183 + ], + "score": 1.0, + "content": "Bi-LSTM; the kaleidoscope layer accounts for less than", + "type": "text" + }, + { + "bbox": [ + 331, + 171, + 353, + 181 + ], + "score": 0.85, + "content": "0 . 5 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 354, + 170, + 437, + 183 + ], + "score": 1.0, + "content": "of the total FLOPs).", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 4, + "bbox_fs": [ + 105, + 83, + 506, + 183 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 187, + 505, + 264 + ], + "lines": [ + { + "bbox": [ + 105, + 187, + 506, + 200 + ], + "spans": [ + { + "bbox": [ + 105, + 187, + 506, + 200 + ], + "score": 1.0, + "content": "As baselines, we also compare to inserting other types of linear transformations before the Bi-LSTM:", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 197, + 507, + 210 + ], + "spans": [ + { + "bbox": [ + 105, + 197, + 507, + 210 + ], + "score": 1.0, + "content": "fixed linear transformations (such as the fixed FFT, or no transform at all [i.e. the identity]), other train-", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 209, + 505, + 222 + ], + "spans": [ + { + "bbox": [ + 105, + 209, + 505, + 222 + ], + "score": 1.0, + "content": "able structured layers (low-rank, circulant, and sparse [using the sparse training algorithm of Dettmers", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 220, + 506, + 233 + ], + "spans": [ + { + "bbox": [ + 105, + 220, + 506, + 233 + ], + "score": 1.0, + "content": "& Zettlemoyer (2019)]), and a trainable unstructured (dense) linear layer. The kaleidoscope layer", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 230, + 506, + 244 + ], + "spans": [ + { + "bbox": [ + 105, + 230, + 506, + 244 + ], + "score": 1.0, + "content": "performs the best out of all such approaches. The fact that it outperforms even a dense linear layer", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 242, + 505, + 254 + ], + "spans": [ + { + "bbox": [ + 105, + 242, + 505, + 254 + ], + "score": 1.0, + "content": "with more parameters is particularly notable, as it suggests that the structural bias imposed by the", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 253, + 504, + 265 + ], + "spans": [ + { + "bbox": [ + 106, + 253, + 504, + 265 + ], + "score": 1.0, + "content": "K-matrix representation is beneficial for performance on this task. Full results are given in Table 5.", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 12, + "bbox_fs": [ + 105, + 187, + 507, + 265 + ] + }, + { + "type": "table", + "bbox": [ + 167, + 294, + 443, + 433 + ], + "blocks": [ + { + "type": "table_caption", + "bbox": [ + 122, + 276, + 485, + 288 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 124, + 275, + 486, + 288 + ], + "spans": [ + { + "bbox": [ + 124, + 275, + 270, + 288 + ], + "score": 1.0, + "content": "Table 5: TIMIT phoneme error rate", + "type": "text" + }, + { + "bbox": [ + 271, + 276, + 299, + 286 + ], + "score": 0.73, + "content": "( \\mathrm { P E R } \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 299, + 275, + 302, + 288 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 302, + 276, + 313, + 286 + ], + "score": 0.51, + "content": "\\pm", + "type": "inline_equation" + }, + { + "bbox": [ + 313, + 275, + 486, + 288 + ], + "score": 1.0, + "content": "standard deviation across 5 random seeds).", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 16 + }, + { + "type": "table_body", + "bbox": [ + 167, + 294, + 443, + 433 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 167, + 294, + 443, + 433 + ], + "spans": [ + { + "bbox": [ + 167, + 294, + 443, + 433 + ], + "score": 0.983, + "html": "
ModelTest set PER%#Parameters
Low rank + LSTM23.6 ± 0.915.5M
Sparse +LSTM21.7 ± 0.915.5M
Circulant +LSTM23.9 ± 0.915.4M
Dense +LSTM15.4 ± 0.615.9M
FFT+LSTM15.7 ± 0.115.4M
Identity +LSTM20.7 ± 0.315.4M
Kaleidoscope + LSTM14.6 ± 0.315.4M
MFSC features + LSTM14.2 ± 0.214.3M
SincNet (Ravanelli et al., 2019)17.210.0M
LiGRU (Ravanelli et al., 2018)13.812.3M
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For each model, we use the validation set to select", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 500, + 506, + 513 + ], + "spans": [ + { + "bbox": [ + 105, + 500, + 506, + 513 + ], + "score": 1.0, + "content": "the best preprocessing learning rate, while the final error rates are reported on the separate held-out", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 511, + 506, + 524 + ], + "spans": [ + { + "bbox": [ + 105, + 511, + 423, + 524 + ], + "score": 1.0, + "content": "test set. 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The architecture for each residual block", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 228, + 505, + 244 + ], + "spans": [ + { + "bbox": [ + 105, + 228, + 240, + 244 + ], + "score": 1.0, + "content": "in ShuffleNet is: 1x1 group conv", + "type": "text" + }, + { + "bbox": [ + 241, + 230, + 253, + 240 + ], + "score": 0.66, + "content": "", + "type": "inline_equation" + }, + { + "bbox": [ + 253, + 228, + 332, + 244 + ], + "score": 1.0, + "content": "Batch norm, ReLU", + "type": "text" + }, + { + "bbox": [ + 333, + 230, + 345, + 240 + ], + "score": 0.81, + "content": "", + "type": "inline_equation" + }, + { + "bbox": [ + 346, + 228, + 397, + 244 + ], + "score": 1.0, + "content": "Permutation", + "type": "text" + }, + { + "bbox": [ + 397, + 230, + 426, + 240 + ], + "score": 0.86, + "content": "\\to 3 { \\bf x } 3", + "type": "inline_equation" + }, + { + "bbox": [ + 426, + 228, + 492, + 244 + ], + "score": 1.0, + "content": "depthwise conv", + "type": "text" + }, + { + "bbox": [ + 492, + 231, + 505, + 240 + ], + "score": 0.78, + "content": "", + "type": "inline_equation" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 241, + 337, + 253 + ], + "spans": [ + { + "bbox": [ + 106, + 241, + 155, + 253 + ], + "score": 1.0, + "content": "Batch norm", + "type": "text" + }, + { + "bbox": [ + 155, + 241, + 184, + 251 + ], + "score": 0.79, + "content": " 1 \\mathrm { x } 1", + "type": "inline_equation" + }, + { + "bbox": [ + 185, + 241, + 337, + 253 + ], + "score": 1.0, + "content": "group conv. 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The K-matrices", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 384, + 385, + 396 + ], + "spans": [ + { + "bbox": [ + 105, + 384, + 385, + 396 + ], + "score": 1.0, + "content": "are trained along with the rest of the network, rather than being fixed.", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 20.5 + }, + { + "type": "title", + "bbox": [ + 107, + 410, + 239, + 421 + ], + "lines": [ + { + "bbox": [ + 106, + 410, + 240, + 422 + ], + "spans": [ + { + "bbox": [ + 106, + 410, + 240, + 422 + ], + "score": 1.0, + "content": "B.2.2 EXPERIMENTAL SETUP", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 24 + }, + { + "type": "text", + "bbox": [ + 106, + 430, + 505, + 519 + ], + "lines": [ + { + "bbox": [ + 106, + 430, + 506, + 442 + ], + "spans": [ + { + "bbox": [ + 106, + 430, + 506, + 442 + ], + "score": 1.0, + "content": "We evaluate the CNN architectures on the image classification task of the standard ImageNet", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 442, + 505, + 453 + ], + "spans": [ + { + "bbox": [ + 106, + 442, + 505, + 453 + ], + "score": 1.0, + "content": "dataset (Russakovsky et al., 2015). We use the standard data augmentation, training, and evaluation", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 453, + 505, + 465 + ], + "spans": [ + { + "bbox": [ + 105, + 453, + 505, + 465 + ], + "score": 1.0, + "content": "pipeline as in (Xie et al., 2017). We train with SGD on 8 GPUs for 90 epochs, with a total batch size", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 464, + 505, + 475 + ], + "spans": [ + { + "bbox": [ + 106, + 464, + 505, + 475 + ], + "score": 1.0, + "content": "of 2048 and initial learning rate 0.8. For the 1.0 ShuffleNet g8 architecture, we reduce the total batch", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 474, + 506, + 488 + ], + "spans": [ + { + "bbox": [ + 105, + 474, + 506, + 488 + ], + "score": 1.0, + "content": "size to 1792 to fit into GPU memory, and correspondingly linearly scale the initial learning rate to", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 104, + 484, + 506, + 498 + ], + "spans": [ + { + "bbox": [ + 104, + 484, + 506, + 498 + ], + "score": 1.0, + "content": "0.7. Other hyperparameters (e.g. learning rate schedule, weight decay, etc.) are kept the same as in", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 496, + 506, + 510 + ], + "spans": [ + { + "bbox": [ + 105, + 496, + 506, + 510 + ], + "score": 1.0, + "content": "the ShuffleNet paper (Zhang et al., 2018). 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ShuffleHadamardKaleidoscope (K.)|K. vs. Shuffle
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ShuffleHadamardKaleidoscope (K.)|K. vs. Shuffle
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Fully connected (FC): This is a 3-layer MLP, with hidden size 1024 and ReLU nonlinearity", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 141, + 400, + 297, + 412 + ], + "spans": [ + { + "bbox": [ + 141, + 400, + 297, + 412 + ], + "score": 1.0, + "content": "in-between the fully connected layers.", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 129, + 414, + 506, + 426 + ], + "spans": [ + { + "bbox": [ + 129, + 414, + 506, + 426 + ], + "score": 1.0, + "content": "2. Recurrent neural network (RNN): We use a gated recurrent unit (GRU) model (Cho et al.,", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 141, + 425, + 506, + 438 + ], + "spans": [ + { + "bbox": [ + 141, + 425, + 506, + 438 + ], + "score": 1.0, + "content": "2014), with hidden size 1024. 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Standard gated architectures such as LSTM and GRU have shown competitive", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 141, + 457, + 506, + 471 + ], + "spans": [ + { + "bbox": [ + 141, + 457, + 506, + 471 + ], + "score": 1.0, + "content": "performance on the Permuted MNIST dataset, and we choose GRU as a baseline since it has", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 142, + 469, + 390, + 481 + ], + "spans": [ + { + "bbox": [ + 142, + 469, + 390, + 481 + ], + "score": 1.0, + "content": "been reported to slightly outperform LSTM (Bai et al., 2018).", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 129, + 482, + 505, + 496 + ], + "spans": [ + { + "bbox": [ + 129, + 482, + 505, + 496 + ], + "score": 1.0, + "content": "3. CNN: We use the standard ResNet18 architecture, adapted to smaller image size of the", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 142, + 493, + 505, + 507 + ], + "spans": [ + { + "bbox": [ + 142, + 493, + 505, + 507 + ], + "score": 1.0, + "content": "CIFAR-10 dataset (changing stride from 2 to 1 of the first convolutional layer, and removing", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 141, + 505, + 273, + 517 + ], + "spans": [ + { + "bbox": [ + 141, + 505, + 273, + 517 + ], + "score": 1.0, + "content": "max-pooling layer that follows).", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 130, + 519, + 505, + 531 + ], + "spans": [ + { + "bbox": [ + 130, + 519, + 169, + 531 + ], + "score": 1.0, + "content": "4. 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We also remark that other classes of structured linear maps such as low-rank, circulant, and", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 336, + 505, + 349 + ], + "spans": [ + { + "bbox": [ + 105, + 336, + 505, + 349 + ], + "score": 1.0, + "content": "so on, are even less suited to this task than dense matrices, as they are incapable of representing all", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 348, + 163, + 360 + ], + "spans": [ + { + "bbox": [ + 105, + 348, + 163, + 360 + ], + "score": 1.0, + "content": "permutations.", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 18, + "bbox_fs": [ + 105, + 280, + 506, + 360 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 370, + 201, + 381 + ], + "lines": [ + { + "bbox": [ + 106, + 369, + 203, + 383 + ], + "spans": [ + { + "bbox": [ + 106, + 369, + 203, + 383 + ], + "score": 1.0, + "content": "Baseline architectures", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 22 + }, + { + "type": "list", + "bbox": [ + 129, + 388, + 505, + 576 + ], + "lines": [ + { + "bbox": [ + 130, + 389, + 505, + 402 + ], + "spans": [ + { + "bbox": [ + 130, + 389, + 505, + 402 + ], + "score": 1.0, + "content": "1. 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Standard gated architectures such as LSTM and GRU have shown competitive", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 141, + 457, + 506, + 471 + ], + "spans": [ + { + "bbox": [ + 141, + 457, + 506, + 471 + ], + "score": 1.0, + "content": "performance on the Permuted MNIST dataset, and we choose GRU as a baseline since it has", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 142, + 469, + 390, + 481 + ], + "spans": [ + { + "bbox": [ + 142, + 469, + 390, + 481 + ], + "score": 1.0, + "content": "been reported to slightly outperform LSTM (Bai et al., 2018).", + "type": "text" + } + ], + "index": 30, + "is_list_end_line": true + }, + { + "bbox": [ + 129, + 482, + 505, + 496 + ], + "spans": [ + { + "bbox": [ + 129, + 482, + 505, + 496 + ], + "score": 1.0, + "content": "3. 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Baseline CNN (unpermuted): We use the standard ResNet18 architecture applied to the", + "type": "text" + } + ], + "index": 37, + "is_list_start_line": true + }, + { + "bbox": [ + 142, + 567, + 269, + 578 + ], + "spans": [ + { + "bbox": [ + 142, + 567, + 269, + 578 + ], + "score": 1.0, + "content": "unpermuted CIFAR-10 dataset.", + "type": "text" + } + ], + "index": 38, + "is_list_end_line": true + } + ], + "index": 30.5, + "bbox_fs": [ + 128, + 389, + 506, + 578 + ] + }, + { + "type": "text", + "bbox": [ + 108, + 585, + 505, + 618 + ], + "lines": [ + { + "bbox": [ + 105, + 583, + 505, + 599 + ], + "spans": [ + { + "bbox": [ + 105, + 583, + 505, + 599 + ], + "score": 1.0, + "content": "All models are trained for 200 total epochs, with the Adam optimizer. We use the standard learning", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 595, + 505, + 609 + ], + "spans": [ + { + "bbox": [ + 105, + 595, + 505, + 609 + ], + "score": 1.0, + "content": "rate schedule and weight decay from Mostafa & Wang (2019). We use Hyperband (Li et al., 2017) to", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 106, + 607, + 454, + 620 + ], + "spans": [ + { + "bbox": [ + 106, + 607, + 454, + 620 + ], + "score": 1.0, + "content": "tune other hyperparameters such as the initial learning rate and annealing temperature.", + "type": "text" + } + ], + "index": 41 + } + ], + "index": 40, + "bbox_fs": [ + 105, + 583, + 505, + 620 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 630, + 321, + 642 + ], + "lines": [ + { + "bbox": [ + 105, + 630, + 322, + 643 + ], + "spans": [ + { + "bbox": [ + 105, + 630, + 322, + 643 + ], + "score": 1.0, + "content": "B.4 SPEEDING UP DYNAMICCONV’S INFERENCE", + "type": "text" + } + ], + "index": 42 + } + ], + "index": 42 + }, + { + "type": "title", + "bbox": [ + 108, + 651, + 244, + 663 + ], + "lines": [ + { + "bbox": [ + 106, + 651, + 245, + 663 + ], + "spans": [ + { + "bbox": [ + 106, + 651, + 245, + 663 + ], + "score": 1.0, + "content": "B.4.1 MODEL ARCHITECTURE", + "type": "text" + } + ], + "index": 43 + } + ], + "index": 43 + }, + { + "type": "text", + "bbox": [ + 107, + 671, + 506, + 715 + ], + "lines": [ + { + "bbox": [ + 106, + 671, + 505, + 682 + ], + "spans": [ + { + "bbox": [ + 106, + 671, + 505, + 682 + ], + "score": 1.0, + "content": "We start with the DynamicConv Transformer architecture (Wu et al., 2019), which is a variant of the", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 106, + 682, + 505, + 694 + ], + "spans": [ + { + "bbox": [ + 106, + 682, + 505, + 694 + ], + "score": 1.0, + "content": "Transformer architecture (Vaswani et al., 2017) where the self-attention in each layer is replaced with", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 692, + 507, + 705 + ], + "spans": [ + { + "bbox": [ + 105, + 692, + 507, + 705 + ], + "score": 1.0, + "content": "a light-weight DynamicConv module. We use the implementation from the Fairseq library(Ott et al.,", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 702, + 244, + 716 + ], + "spans": [ + { + "bbox": [ + 105, + 702, + 244, + 716 + ], + "score": 1.0, + "content": "2019),9 with PyTorch version 1.2.", + "type": "text" + } + ], + "index": 47 + } + ], + "index": 45.5, + "bbox_fs": [ + 105, + 671, + 507, + 716 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 82, + 505, + 126 + ], + "lines": [ + { + "bbox": [ + 106, + 82, + 505, + 94 + ], + "spans": [ + { + "bbox": [ + 106, + 82, + 329, + 94 + ], + "score": 1.0, + "content": "The architecture of each layer of the decoder is: Linear", + "type": "text" + }, + { + "bbox": [ + 329, + 83, + 342, + 93 + ], + "score": 0.8, + "content": "", + "type": "inline_equation" + }, + { + "bbox": [ + 342, + 82, + 401, + 94 + ], + "score": 1.0, + "content": "DynamicConv", + "type": "text" + }, + { + "bbox": [ + 402, + 83, + 415, + 93 + ], 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"content": "In every layer of the decoder, we replace the dense weight matrix in each of the four Linear layers", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 115, + 335, + 127 + ], + "spans": [ + { + "bbox": [ + 105, + 115, + 133, + 127 + ], + "score": 1.0, + "content": "with a", + "type": "text" + }, + { + "bbox": [ + 133, + 116, + 142, + 125 + ], + "score": 0.27, + "content": "\\mathrm { K }", + "type": "inline_equation" + }, + { + "bbox": [ + 142, + 115, + 208, + 127 + ], + "score": 1.0, + "content": "-matrix from the", + "type": "text" + }, + { + "bbox": [ + 209, + 116, + 217, + 126 + ], + "score": 0.75, + "content": "\\boldsymbol { B }", + "type": "inline_equation" + }, + { + "bbox": [ + 218, + 115, + 335, + 127 + ], + "score": 1.0, + "content": "class (i.e. a butterfly matrix).", + "type": "text" + } + ], + "index": 3 + } + ], + "index": 1.5 + }, + { + "type": "title", + "bbox": [ + 108, + 139, + 263, + 150 + ], + "lines": [ + { + "bbox": [ + 105, + 138, 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(Wu et al., 2019). We note that the DynamicConv model with K-matrices in the decoder trains", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 191, + 506, + 204 + ], + "spans": [ + { + "bbox": [ + 106, + 191, + 506, + 204 + ], + "score": 1.0, + "content": "slightly faster than the default DynamicConv model (both models are trained for 50,000 updates,", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 202, + 488, + 215 + ], + "spans": [ + { + "bbox": [ + 105, + 202, + 227, + 215 + ], + "score": 1.0, + "content": "which requires approximately", + "type": "text" + }, + { + "bbox": [ + 227, + 203, + 242, + 213 + ], + "score": 0.82, + "content": "7 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 243, + 202, + 488, + 215 + ], + "score": 1.0, + "content": "less time for the K-matrix model than for the default model).", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 7 + }, + { + "type": "text", + "bbox": [ + 107, + 219, + 505, + 263 + ], + "lines": [ + { + "bbox": [ + 105, + 219, + 507, + 232 + ], + "spans": [ + { + "bbox": [ + 105, + 219, + 507, + 232 + ], + "score": 1.0, + "content": "To evaluate inference speed, we run the decoding script on the IWSLT-14 De-En test set in single-", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 230, + 507, + 242 + ], + "spans": [ + { + "bbox": [ + 106, + 230, + 507, + 242 + ], + "score": 1.0, + "content": "threaded mode on a server Intel Xeon CPU E5-2690 v4 at 2.60GHz, and measure wall-clock time.", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 241, + 505, + 253 + ], + "spans": [ + { + "bbox": [ + 105, + 241, + 505, + 253 + ], + "score": 1.0, + "content": "The test set contains 6750 sentences, with 149241 tokens. Following Wu et al. (2019), we set the", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 252, + 320, + 264 + ], + "spans": [ + { + "bbox": [ + 105, + 252, + 320, + 264 + ], + "score": 1.0, + "content": "batch size to 1 and beam size to 1 for this evaluation.", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 11.5 + }, + { + "type": "title", + "bbox": [ + 108, + 276, + 418, + 287 + ], + "lines": [ + { + "bbox": [ + 106, + 276, + 419, + 288 + ], + "spans": [ + { + "bbox": [ + 106, + 276, + 419, + 288 + ], + "score": 1.0, + "content": "B.4.3 ADDITIONAL COMPARISON WITH OTHER STRUCTURED MATRICES", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 14 + }, + { + "type": "text", + "bbox": [ + 107, + 295, + 505, + 384 + ], + "lines": [ + { + "bbox": [ + 106, + 296, + 505, + 308 + ], + "spans": [ + { + "bbox": [ + 106, + 296, + 505, + 308 + ], + "score": 1.0, + "content": "We additionally compare the speed-quality tradeoff of K-matrices with other classes of structured", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 307, + 505, + 318 + ], + "spans": [ + { + "bbox": [ + 105, + 307, + 505, + 318 + ], + "score": 1.0, + "content": "matrices, when used to replace the fully-connected layers of DynamicConv’s decoder. We consider", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 318, + 504, + 329 + ], + "spans": [ + { + "bbox": [ + 106, + 318, + 504, + 329 + ], + "score": 1.0, + "content": "the following additional classes of structured matrices: low-rank, circulant, Toeplitz-like (Sindhwani", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 329, + 505, + 340 + ], + "spans": [ + { + "bbox": [ + 105, + 329, + 505, + 340 + ], + "score": 1.0, + "content": "et al., 2015), ACDC (Moczulski et al., 2016), Fastfood (Le et al., 2013), and sparse. For classes with", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 340, + 505, + 352 + ], + "spans": [ + { + "bbox": [ + 105, + 340, + 505, + 352 + ], + "score": 1.0, + "content": "a variable number of parameters (e.g. low-rank, sparse), we set the number of parameters to match", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 351, + 506, + 363 + ], + "spans": [ + { + "bbox": [ + 105, + 351, + 506, + 363 + ], + "score": 1.0, + "content": "that of K-matrices. For sparse matrices, besides the result for an ensemble of 10 models (the default", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 362, + 506, + 374 + ], + "spans": [ + { + "bbox": [ + 105, + 362, + 506, + 374 + ], + "score": 1.0, + "content": "setting in the Fairseq repository), we also report the result for a single model, as that could have faster", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 372, + 470, + 385 + ], + "spans": [ + { + "bbox": [ + 105, + 372, + 470, + 385 + ], + "score": 1.0, + "content": "inference time (since ensembling/averaging sparse matrices produces a less sparse matrix).", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 18.5 + }, + { + "type": "text", + "bbox": [ + 107, + 389, + 505, + 466 + ], + "lines": [ + { + "bbox": [ + 105, + 389, + 505, + 402 + ], + "spans": [ + { + "bbox": [ + 105, + 389, + 505, + 402 + ], + "score": 1.0, + "content": "In Figure 5, we plot the tradeoff between translation quality (measured by BLEU score) and inference", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 401, + 504, + 413 + ], + "spans": [ + { + "bbox": [ + 105, + 401, + 504, + 413 + ], + "score": 1.0, + "content": "speed (sentences per second). Most classes of structured matrices produce similar translation quality", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 411, + 505, + 423 + ], + "spans": [ + { + "bbox": [ + 105, + 411, + 489, + 423 + ], + "score": 1.0, + "content": "(between 34.1 and 34.4 BLEU score). K-matrices have the second fastest inference time, only", + "type": "text" + }, + { + "bbox": [ + 489, + 411, + 505, + 421 + ], + "score": 0.85, + "content": "7 \\%", + "type": "inline_equation" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 423, + 504, + 433 + ], + "spans": [ + { + "bbox": [ + 106, + 423, + 504, + 433 + ], + "score": 1.0, + "content": "slower than low-rank matrices. We note that low-rank matrices benefit from very well-tuned BLAS", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 433, + 505, + 446 + ], + "spans": [ + { + "bbox": [ + 105, + 433, + 505, + 446 + ], + "score": 1.0, + "content": "routines (matrix-matrix multiplication). Even though our implementation of K-matrix multiplication", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 444, + 506, + 456 + ], + "spans": [ + { + "bbox": [ + 105, + 444, + 506, + 456 + ], + "score": 1.0, + "content": "is not yet highly optimized, it is already quite close to the speed of low-rank matrix multiplication at", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 456, + 231, + 469 + ], + "spans": [ + { + "bbox": [ + 106, + 456, + 231, + 469 + ], + "score": 1.0, + "content": "an equivalent parameter count.", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 26 + }, + { + "type": "title", + "bbox": [ + 107, + 484, + 398, + 496 + ], + "lines": [ + { + "bbox": [ + 104, + 481, + 400, + 499 + ], + "spans": [ + { + "bbox": [ + 104, + 481, + 400, + 499 + ], + "score": 1.0, + "content": "C SPEED BENCHMARK AND IMPLEMENTATION DETAILS", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 30 + }, + { + "type": "text", + "bbox": [ + 107, + 507, + 505, + 629 + ], + "lines": [ + { + "bbox": [ + 105, + 507, + 505, + 521 + ], + "spans": [ + { + "bbox": [ + 105, + 507, + 339, + 521 + ], + "score": 1.0, + "content": "Each K-matrix (for fixed width and expansion), has an", + "type": "text" + }, + { + "bbox": [ + 340, + 508, + 384, + 520 + ], + "score": 0.92, + "content": "O ( n \\log n )", + "type": "inline_equation" + }, + { + "bbox": [ + 384, + 507, + 505, + 521 + ], + "score": 1.0, + "content": "matrix-vector multiplication", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 519, + 506, + 532 + ], + "spans": [ + { + "bbox": [ + 105, + 519, + 506, + 532 + ], + "score": 1.0, + "content": "algorithm: sequentially multiply the input vector with each of the sparse factors. Our implementation", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 528, + 507, + 545 + ], + "spans": [ + { + "bbox": [ + 105, + 528, + 507, + 545 + ], + "score": 1.0, + "content": "of this simple algorithm is surprisingly competitive with optimized subroutines, both on GPU (e.g.", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 540, + 505, + 555 + ], + "spans": [ + { + "bbox": [ + 105, + 540, + 505, + 555 + ], + "score": 1.0, + "content": "for training) and on CPU (e.g. for inference). In Figure 6, we compare the speed of multiplying by", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 551, + 506, + 565 + ], + "spans": [ + { + "bbox": [ + 105, + 551, + 185, + 565 + ], + "score": 1.0, + "content": "a K-matrix in class", + "type": "text" + }, + { + "bbox": [ + 186, + 552, + 194, + 562 + ], + "score": 0.76, + "content": "\\boldsymbol { B }", + "type": "inline_equation" + }, + { + "bbox": [ + 195, + 551, + 506, + 565 + ], + "score": 1.0, + "content": "(i.e. a butterfly matrix) against a specialized implementation of the FFT. We", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 563, + 506, + 576 + ], + "spans": [ + { + "bbox": [ + 105, + 563, + 506, + 576 + ], + "score": 1.0, + "content": "normalize the speed by the speed of dense matrix-matrix multiply (on GPU) or dense matrix-vector", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 574, + 505, + 587 + ], + "spans": [ + { + "bbox": [ + 105, + 574, + 300, + 587 + ], + "score": 1.0, + "content": "multiply (on CPU). On GPU, with input sizes", + "type": "text" + }, + { + "bbox": [ + 300, + 574, + 344, + 584 + ], + "score": 0.89, + "content": "n = 1 0 2 4", + "type": "inline_equation" + }, + { + "bbox": [ + 344, + 574, + 505, + 587 + ], + "score": 1.0, + "content": "and batch size 2048, the training time", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 106, + 585, + 505, + 596 + ], + "spans": [ + { + "bbox": [ + 106, + 585, + 322, + 596 + ], + "score": 1.0, + "content": "(forward and backward) of K-matrices matrix is about", + "type": "text" + }, + { + "bbox": [ + 323, + 586, + 334, + 595 + ], + "score": 0.52, + "content": "3 \\mathbf { x }", + "type": "inline_equation" + }, + { + "bbox": [ + 335, + 585, + 505, + 596 + ], + "score": 1.0, + "content": "faster than dense matrix multiply (GEMM", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 595, + 505, + 608 + ], + "spans": [ + { + "bbox": [ + 105, + 595, + 505, + 608 + ], + "score": 1.0, + "content": "from cuBLAS). For inference on CPU, the kaleidoscope fast multiplication can be one or two orders", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 106, + 607, + 505, + 618 + ], + "spans": [ + { + "bbox": [ + 106, + 607, + 505, + 618 + ], + "score": 1.0, + "content": "of magnitude faster than GEMV. Over a range of matrix sizes, our implementation is within a factor", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 106, + 618, + 417, + 630 + ], + "spans": [ + { + "bbox": [ + 106, + 618, + 117, + 630 + ], + "score": 1.0, + "content": "of", + "type": "text" + }, + { + "bbox": [ + 117, + 618, + 137, + 628 + ], + "score": 0.38, + "content": "2 { - } 4 \\mathbf { x }", + "type": "inline_equation" + }, + { + "bbox": [ + 137, + 618, + 417, + 630 + ], + "score": 1.0, + "content": "of specialized implementations of the FFT, a highly optimized kernel.", + "type": "text" + } + ], + "index": 41 + } + ], + "index": 36 + }, + { + "type": "text", + "bbox": [ + 107, + 635, + 505, + 668 + ], + "lines": [ + { + "bbox": [ + 105, + 633, + 507, + 648 + ], + "spans": [ + { + "bbox": [ + 105, + 633, + 410, + 648 + ], + "score": 1.0, + "content": "Our implementation is also memory-efficient. In the forward pass through the", + "type": "text" + }, + { + "bbox": [ + 410, + 635, + 447, + 646 + ], + "score": 0.92, + "content": "O ( \\log n )", + "type": "inline_equation" + }, + { + "bbox": [ + 447, + 633, + 507, + 648 + ], + "score": 1.0, + "content": "sparse factors,", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 106, + 646, + 505, + 658 + ], + "spans": [ + { + "bbox": [ + 106, + 646, + 505, + 658 + ], + "score": 1.0, + "content": "we do not store the intermediate results, but recompute them during the backward pass. Therefore the", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 106, + 657, + 367, + 669 + ], + "spans": [ + { + "bbox": [ + 106, + 657, + 228, + 669 + ], + "score": 1.0, + "content": "activation memory required is", + "type": "text" + }, + { + "bbox": [ + 228, + 657, + 255, + 669 + ], + "score": 0.92, + "content": "O ( b n )", + "type": "inline_equation" + }, + { + "bbox": [ + 255, + 657, + 358, + 669 + ], + "score": 1.0, + "content": "for an input batch size of", + "type": "text" + }, + { + "bbox": [ + 359, + 657, + 364, + 667 + ], + "score": 0.76, + "content": "b", + "type": "inline_equation" + }, + { + "bbox": [ + 364, + 657, + 367, + 669 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 44 + } + ], + "index": 43 + }, + { + "type": "title", + "bbox": [ + 108, + 684, + 289, + 697 + ], + "lines": [ + { + "bbox": [ + 106, + 684, + 291, + 698 + ], + "spans": [ + { + "bbox": [ + 106, + 684, + 291, + 698 + ], + "score": 1.0, + "content": "D SYNTHETIC MATRIX RECOVERY", + "type": "text" + } + ], + "index": 45 + } + ], + "index": 45 + }, + { + "type": "text", + "bbox": [ + 107, + 709, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 708, + 506, + 723 + ], + "spans": [ + { + "bbox": [ + 105, + 708, + 506, + 723 + ], + "score": 1.0, + "content": "We directly validate Theorem 1 on well-known types of structured matrices used in machine learning.", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 720, + 505, + 733 + ], + "spans": [ + { + "bbox": [ + 105, + 720, + 210, + 733 + ], + "score": 1.0, + "content": "Given a structured matrix", + "type": "text" + }, + { + "bbox": [ + 211, + 721, + 223, + 731 + ], + "score": 0.4, + "content": "\\mathbf { M }", + "type": "inline_equation" + }, + { + "bbox": [ + 223, + 720, + 322, + 733 + ], + "score": 1.0, + "content": ", we attempt to represent", + "type": "text" + }, + { + "bbox": [ + 322, + 721, + 335, + 730 + ], + "score": 0.42, + "content": "\\mathbf { M }", + "type": "inline_equation" + }, + { + "bbox": [ + 335, + 720, + 447, + 733 + ], + "score": 1.0, + "content": "as closely as possible using", + "type": "text" + }, + { + "bbox": [ + 447, + 721, + 456, + 731 + ], + "score": 0.27, + "content": "\\mathbf { K }", + "type": "inline_equation" + }, + { + "bbox": [ + 456, + 720, + 505, + 733 + ], + "score": 1.0, + "content": "-matrices as", + "type": "text" + } + ], + "index": 47 + } + ], + "index": 46.5 + } + ], + "page_idx": 20, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 108, + 27, + 293, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 26, + 294, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 26, + 294, + 38 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2020", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 300, + 752, + 310, + 760 + ], + "lines": [ + { + "bbox": [ + 299, + 750, + 312, + 764 + ], + "spans": [ + { + 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}, + { + "type": "text", + "bbox": [ + 107, + 159, + 505, + 213 + ], + "lines": [ + { + "bbox": [ + 105, + 158, + 505, + 171 + ], + "spans": [ + { + "bbox": [ + 105, + 158, + 505, + 171 + ], + "score": 1.0, + "content": "The models are trained from scratch using the training script from the Fairseq repository, with the", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 170, + 505, + 182 + ], + "spans": [ + { + "bbox": [ + 106, + 170, + 505, + 182 + ], + "score": 1.0, + "content": "same hyperparameters (optimizer, learning rate, number of updates, etc.) used in the DynamicConv", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 180, + 506, + 194 + ], + "spans": [ + { + "bbox": [ + 105, + 180, + 506, + 194 + ], + "score": 1.0, + "content": "paper (Wu et al., 2019). We note that the DynamicConv model with K-matrices in the decoder trains", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 191, + 506, + 204 + ], + "spans": [ + { + "bbox": [ + 106, + 191, + 506, + 204 + ], + "score": 1.0, + "content": "slightly faster than the default DynamicConv model (both models are trained for 50,000 updates,", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 202, + 488, + 215 + ], + "spans": [ + { + "bbox": [ + 105, + 202, + 227, + 215 + ], + "score": 1.0, + "content": "which requires approximately", + "type": "text" + }, + { + "bbox": [ + 227, + 203, + 242, + 213 + ], + "score": 0.82, + "content": "7 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 243, + 202, + 488, + 215 + ], + "score": 1.0, + "content": "less time for the K-matrix model than for the default model).", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 7, + "bbox_fs": [ + 105, + 158, + 506, + 215 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 219, + 505, + 263 + ], + "lines": [ + { + "bbox": [ + 105, + 219, + 507, + 232 + ], + "spans": [ + { + "bbox": [ + 105, + 219, + 507, + 232 + ], + "score": 1.0, + "content": "To evaluate inference speed, we run the decoding script on the IWSLT-14 De-En test set in single-", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 230, + 507, + 242 + ], + "spans": [ + { + "bbox": [ + 106, + 230, + 507, + 242 + ], + "score": 1.0, + "content": "threaded mode on a server Intel Xeon CPU E5-2690 v4 at 2.60GHz, and measure wall-clock time.", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 241, + 505, + 253 + ], + "spans": [ + { + "bbox": [ + 105, + 241, + 505, + 253 + ], + "score": 1.0, + "content": "The test set contains 6750 sentences, with 149241 tokens. Following Wu et al. (2019), we set the", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 252, + 320, + 264 + ], + "spans": [ + { + "bbox": [ + 105, + 252, + 320, + 264 + ], + "score": 1.0, + "content": "batch size to 1 and beam size to 1 for this evaluation.", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 11.5, + "bbox_fs": [ + 105, + 219, + 507, + 264 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 276, + 418, + 287 + ], + "lines": [ + { + "bbox": [ + 106, + 276, + 419, + 288 + ], + "spans": [ + { + "bbox": [ + 106, + 276, + 419, + 288 + ], + "score": 1.0, + "content": "B.4.3 ADDITIONAL COMPARISON WITH OTHER STRUCTURED MATRICES", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 14 + }, + { + "type": "text", + "bbox": [ + 107, + 295, + 505, + 384 + ], + "lines": [ + { + "bbox": [ + 106, + 296, + 505, + 308 + ], + "spans": [ + { + "bbox": [ + 106, + 296, + 505, + 308 + ], + "score": 1.0, + "content": "We additionally compare the speed-quality tradeoff of K-matrices with other classes of structured", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 307, + 505, + 318 + ], + "spans": [ + { + "bbox": [ + 105, + 307, + 505, + 318 + ], + "score": 1.0, + "content": "matrices, when used to replace the fully-connected layers of DynamicConv’s decoder. We consider", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 318, + 504, + 329 + ], + "spans": [ + { + "bbox": [ + 106, + 318, + 504, + 329 + ], + "score": 1.0, + "content": "the following additional classes of structured matrices: low-rank, circulant, Toeplitz-like (Sindhwani", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 329, + 505, + 340 + ], + "spans": [ + { + "bbox": [ + 105, + 329, + 505, + 340 + ], + "score": 1.0, + "content": "et al., 2015), ACDC (Moczulski et al., 2016), Fastfood (Le et al., 2013), and sparse. For classes with", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 340, + 505, + 352 + ], + "spans": [ + { + "bbox": [ + 105, + 340, + 505, + 352 + ], + "score": 1.0, + "content": "a variable number of parameters (e.g. low-rank, sparse), we set the number of parameters to match", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 351, + 506, + 363 + ], + "spans": [ + { + "bbox": [ + 105, + 351, + 506, + 363 + ], + "score": 1.0, + "content": "that of K-matrices. For sparse matrices, besides the result for an ensemble of 10 models (the default", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 362, + 506, + 374 + ], + "spans": [ + { + "bbox": [ + 105, + 362, + 506, + 374 + ], + "score": 1.0, + "content": "setting in the Fairseq repository), we also report the result for a single model, as that could have faster", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 372, + 470, + 385 + ], + "spans": [ + { + "bbox": [ + 105, + 372, + 470, + 385 + ], + "score": 1.0, + "content": "inference time (since ensembling/averaging sparse matrices produces a less sparse matrix).", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 18.5, + "bbox_fs": [ + 105, + 296, + 506, + 385 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 389, + 505, + 466 + ], + "lines": [ + { + "bbox": [ + 105, + 389, + 505, + 402 + ], + "spans": [ + { + "bbox": [ + 105, + 389, + 505, + 402 + ], + "score": 1.0, + "content": "In Figure 5, we plot the tradeoff between translation quality (measured by BLEU score) and inference", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 401, + 504, + 413 + ], + "spans": [ + { + "bbox": [ + 105, + 401, + 504, + 413 + ], + "score": 1.0, + "content": "speed (sentences per second). Most classes of structured matrices produce similar translation quality", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 411, + 505, + 423 + ], + "spans": [ + { + "bbox": [ + 105, + 411, + 489, + 423 + ], + "score": 1.0, + "content": "(between 34.1 and 34.4 BLEU score). K-matrices have the second fastest inference time, only", + "type": "text" + }, + { + "bbox": [ + 489, + 411, + 505, + 421 + ], + "score": 0.85, + "content": "7 \\%", + "type": "inline_equation" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 423, + 504, + 433 + ], + "spans": [ + { + "bbox": [ + 106, + 423, + 504, + 433 + ], + "score": 1.0, + "content": "slower than low-rank matrices. We note that low-rank matrices benefit from very well-tuned BLAS", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 433, + 505, + 446 + ], + "spans": [ + { + "bbox": [ + 105, + 433, + 505, + 446 + ], + "score": 1.0, + "content": "routines (matrix-matrix multiplication). Even though our implementation of K-matrix multiplication", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 444, + 506, + 456 + ], + "spans": [ + { + "bbox": [ + 105, + 444, + 506, + 456 + ], + "score": 1.0, + "content": "is not yet highly optimized, it is already quite close to the speed of low-rank matrix multiplication at", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 456, + 231, + 469 + ], + "spans": [ + { + "bbox": [ + 106, + 456, + 231, + 469 + ], + "score": 1.0, + "content": "an equivalent parameter count.", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 26, + "bbox_fs": [ + 105, + 389, + 506, + 469 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 484, + 398, + 496 + ], + "lines": [ + { + "bbox": [ + 104, + 481, + 400, + 499 + ], + "spans": [ + { + "bbox": [ + 104, + 481, + 400, + 499 + ], + "score": 1.0, + "content": "C SPEED BENCHMARK AND IMPLEMENTATION DETAILS", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 30 + }, + { + "type": "text", + "bbox": [ + 107, + 507, + 505, + 629 + ], + "lines": [ + { + "bbox": [ + 105, + 507, + 505, + 521 + ], + "spans": [ + { + "bbox": [ + 105, + 507, + 339, + 521 + ], + "score": 1.0, + "content": "Each K-matrix (for fixed width and expansion), has an", + "type": "text" + }, + { + "bbox": [ + 340, + 508, + 384, + 520 + ], + "score": 0.92, + "content": "O ( n \\log n )", + "type": "inline_equation" + }, + { + "bbox": [ + 384, + 507, + 505, + 521 + ], + "score": 1.0, + "content": "matrix-vector multiplication", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 519, + 506, + 532 + ], + "spans": [ + { + "bbox": [ + 105, + 519, + 506, + 532 + ], + "score": 1.0, + "content": "algorithm: sequentially multiply the input vector with each of the sparse factors. Our implementation", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 528, + 507, + 545 + ], + "spans": [ + { + "bbox": [ + 105, + 528, + 507, + 545 + ], + "score": 1.0, + "content": "of this simple algorithm is surprisingly competitive with optimized subroutines, both on GPU (e.g.", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 540, + 505, + 555 + ], + "spans": [ + { + "bbox": [ + 105, + 540, + 505, + 555 + ], + "score": 1.0, + "content": "for training) and on CPU (e.g. for inference). In Figure 6, we compare the speed of multiplying by", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 551, + 506, + 565 + ], + "spans": [ + { + "bbox": [ + 105, + 551, + 185, + 565 + ], + "score": 1.0, + "content": "a K-matrix in class", + "type": "text" + }, + { + "bbox": [ + 186, + 552, + 194, + 562 + ], + "score": 0.76, + "content": "\\boldsymbol { B }", + "type": "inline_equation" + }, + { + "bbox": [ + 195, + 551, + 506, + 565 + ], + "score": 1.0, + "content": "(i.e. a butterfly matrix) against a specialized implementation of the FFT. We", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 563, + 506, + 576 + ], + "spans": [ + { + "bbox": [ + 105, + 563, + 506, + 576 + ], + "score": 1.0, + "content": "normalize the speed by the speed of dense matrix-matrix multiply (on GPU) or dense matrix-vector", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 574, + 505, + 587 + ], + "spans": [ + { + "bbox": [ + 105, + 574, + 300, + 587 + ], + "score": 1.0, + "content": "multiply (on CPU). On GPU, with input sizes", + "type": "text" + }, + { + "bbox": [ + 300, + 574, + 344, + 584 + ], + "score": 0.89, + "content": "n = 1 0 2 4", + "type": "inline_equation" + }, + { + "bbox": [ + 344, + 574, + 505, + 587 + ], + "score": 1.0, + "content": "and batch size 2048, the training time", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 106, + 585, + 505, + 596 + ], + "spans": [ + { + "bbox": [ + 106, + 585, + 322, + 596 + ], + "score": 1.0, + "content": "(forward and backward) of K-matrices matrix is about", + "type": "text" + }, + { + "bbox": [ + 323, + 586, + 334, + 595 + ], + "score": 0.52, + "content": "3 \\mathbf { x }", + "type": "inline_equation" + }, + { + "bbox": [ + 335, + 585, + 505, + 596 + ], + "score": 1.0, + "content": "faster than dense matrix multiply (GEMM", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 595, + 505, + 608 + ], + "spans": [ + { + "bbox": [ + 105, + 595, + 505, + 608 + ], + "score": 1.0, + "content": "from cuBLAS). For inference on CPU, the kaleidoscope fast multiplication can be one or two orders", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 106, + 607, + 505, + 618 + ], + "spans": [ + { + "bbox": [ + 106, + 607, + 505, + 618 + ], + "score": 1.0, + "content": "of magnitude faster than GEMV. 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Therefore the", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 106, + 657, + 367, + 669 + ], + "spans": [ + { + "bbox": [ + 106, + 657, + 228, + 669 + ], + "score": 1.0, + "content": "activation memory required is", + "type": "text" + }, + { + "bbox": [ + 228, + 657, + 255, + 669 + ], + "score": 0.92, + "content": "O ( b n )", + "type": "inline_equation" + }, + { + "bbox": [ + 255, + 657, + 358, + 669 + ], + "score": 1.0, + "content": "for an input batch size of", + "type": "text" + }, + { + "bbox": [ + 359, + 657, + 364, + 667 + ], + "score": 0.76, + "content": "b", + "type": "inline_equation" + }, + { + "bbox": [ + 364, + 657, + 367, + 669 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 44 + } + ], + "index": 43, + "bbox_fs": [ + 105, + 633, + 507, + 669 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 684, + 289, + 697 + ], + "lines": [ + { + "bbox": [ + 106, + 684, + 291, + 698 + ], + "spans": [ + { + "bbox": [ + 106, + 684, + 291, + 698 + ], + "score": 1.0, + "content": "D SYNTHETIC MATRIX RECOVERY", + "type": "text" + } + ], + "index": 45 + } + ], + "index": 45 + }, + { + "type": "text", + "bbox": [ + 107, + 709, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 708, + 506, + 723 + ], + "spans": [ + { + "bbox": [ + 105, + 708, + 506, + 723 + ], + "score": 1.0, + "content": "We directly validate Theorem 1 on well-known types of structured matrices used in machine learning.", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 720, + 505, + 733 + ], + "spans": [ + { + "bbox": [ + 105, + 720, + 210, + 733 + ], + "score": 1.0, + "content": "Given a structured matrix", + "type": "text" + }, + { + "bbox": [ + 211, + 721, + 223, + 731 + ], + "score": 0.4, + "content": "\\mathbf { M }", + "type": "inline_equation" + }, + { + "bbox": [ + 223, + 720, + 322, + 733 + ], + "score": 1.0, + "content": ", we attempt to represent", + "type": "text" + }, + { + "bbox": [ + 322, + 721, + 335, + 730 + ], + "score": 0.42, + "content": "\\mathbf { M }", + "type": "inline_equation" + }, + { + "bbox": [ + 335, + 720, + 447, + 733 + ], + "score": 1.0, + "content": "as closely as possible using", + "type": "text" + }, + { + "bbox": [ + 447, + 721, + 456, + 731 + ], + "score": 0.27, + "content": "\\mathbf { K }", + "type": "inline_equation" + }, + { + "bbox": [ + 456, + 720, + 505, + 733 + ], + "score": 1.0, + "content": "-matrices as", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 106, + 594, + 505, + 606 + ], + "spans": [ + { + "bbox": [ + 106, + 594, + 505, + 606 + ], + "score": 1.0, + "content": "well as the standard classes of structured matrices: sparse and low-rank. In Table 7, we quantify the", + "type": "text", + "cross_page": true + } + ], + "index": 11 + }, + { + "bbox": [ + 104, + 604, + 506, + 618 + ], + "spans": [ + { + "bbox": [ + 104, + 604, + 506, + 618 + ], + "score": 1.0, + "content": "expressivity of each of these three methods, as measured by their ability to approximate a range of", + "type": "text", + "cross_page": true + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 615, + 505, + 629 + ], + "spans": [ + { + "bbox": [ + 106, + 615, + 505, + 629 + ], + "score": 1.0, + "content": "different structures. Results for “global minimum” of kaleidoscope matrices are obtained from the", + "type": "text", + "cross_page": true + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 626, + 505, + 640 + ], + "spans": [ + { + "bbox": [ + 105, + 626, + 505, + 640 + ], + "score": 1.0, + "content": "theoretical expressiveness results in Section I and Section J. 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All target matrices M were randomly initialized such that", + "type": "text" + }, + { + "bbox": [ + 383, + 720, + 442, + 732 + ], + "score": 0.94, + "content": "\\mathbb { E } [ \\mathbf { M } ^ { T } \\mathbf { M } ] ^ { - \\sum } \\mathbf { I }", + "type": "inline_equation" + }, + { + "bbox": [ + 442, + 720, + 446, + 733 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 21, + "bbox_fs": [ + 105, + 698, + 507, + 733 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 106, + 80, + 505, + 169 + ], + "lines": [ + { + "bbox": [ + 105, + 79, + 506, + 93 + ], + "spans": [ + { + "bbox": [ + 105, + 79, + 506, + 93 + ], + "score": 1.0, + "content": "Table 7: Expressiveness of different classes of structured matrices: Frobenius error of represent-", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 91, + 505, + 104 + ], + "spans": [ + { + "bbox": [ + 106, + 91, + 505, + 104 + ], + "score": 1.0, + "content": "ing common structured matrices (columns) of dimension 256 using three structured representations", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 106, + 102, + 505, + 115 + ], + "spans": [ + { + "bbox": [ + 106, + 102, + 505, + 115 + ], + "score": 1.0, + "content": "of matrices with adjustable numbers of parameters. (Left group: Target matrices in the same class", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 113, + 505, + 125 + ], + "spans": [ + { + "bbox": [ + 105, + 113, + 505, + 125 + ], + "score": 1.0, + "content": "as the methods. Middle group: Target matrices with fixed number of parameters. Right: Random", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 124, + 506, + 136 + ], + "spans": [ + { + "bbox": [ + 105, + 124, + 506, + 136 + ], + "score": 1.0, + "content": "matrix to show typical scale of error.) Each method is allotted the same number of parameters, equal", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 106, + 135, + 505, + 147 + ], + "spans": [ + { + "bbox": [ + 106, + 135, + 124, + 147 + ], + "score": 1.0, + "content": "to a", + "type": "text" + }, + { + "bbox": [ + 124, + 136, + 147, + 146 + ], + "score": 0.82, + "content": "\\log n", + "type": "inline_equation" + }, + { + "bbox": [ + 147, + 135, + 505, + 147 + ], + "score": 1.0, + "content": "factor more than that of the target matrix. Low-rank and sparse matrices are unable to", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 146, + 506, + 158 + ], + "spans": [ + { + "bbox": [ + 105, + 146, + 506, + 158 + ], + "score": 1.0, + "content": "capture any structure outside their own class, while the minima for kaleidoscope matrices found via", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 157, + 488, + 169 + ], + "spans": [ + { + "bbox": [ + 106, + 157, + 488, + 169 + ], + "score": 1.0, + "content": "optimization better capture the actual structure for out-of-class targets better than the baselines.", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 3.5 + }, + { + "type": "table", + "bbox": [ + 106, + 178, + 542, + 257 + ], + "blocks": [ + { + "type": "table_body", + "bbox": [ + 106, + 178, + 542, + 257 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 106, + 178, + 542, + 257 + ], + "spans": [ + { + "bbox": [ + 106, + 178, + 542, + 257 + ], + "score": 0.981, + "html": "
Target MethodKaleidoscopeLow-rankSparseConvolutionFastfoodRandom
Global Min.Kaleidoscope0.00.00.00.00.0
Low-rank14.90.010.814.611.615.5
Sparse11.712.20.013.17.114.1
With SGDKaleidoscope0.00.018.00.05.114.5
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We first make some basic", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 352, + 270, + 365 + ], + "spans": [ + { + "bbox": [ + 106, + 352, + 270, + 365 + ], + "score": 1.0, + "content": "observations about the parameterization.", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 14.5 + }, + { + "type": "text", + "bbox": [ + 108, + 367, + 397, + 379 + ], + "lines": [ + { + "bbox": [ + 106, + 366, + 397, + 381 + ], + "spans": [ + { + "bbox": [ + 106, + 366, + 196, + 381 + ], + "score": 1.0, + "content": "Observation E.1. An", + "type": "text" + }, + { + "bbox": [ + 197, + 369, + 222, + 378 + ], + "score": 0.83, + "content": "n \\times n", + "type": "inline_equation" + }, + { + "bbox": [ + 222, + 366, + 251, + 381 + ], + "score": 1.0, + "content": "matrix", + "type": "text" + }, + { + "bbox": [ + 252, + 367, + 294, + 378 + ], + "score": 0.9, + "content": "\\mathbf { M } \\in B B ^ { * }", + "type": "inline_equation" + }, + { + "bbox": [ + 294, + 366, + 312, + 381 + ], + "score": 1.0, + "content": "has", + "type": "text" + }, + { + "bbox": [ + 312, + 368, + 346, + 379 + ], + "score": 0.58, + "content": "4 n \\log n", + "type": "inline_equation" + }, + { + "bbox": [ + 347, + 366, + 397, + 381 + ], + "score": 1.0, + "content": "parameters.", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 16 + }, + { + "type": "text", + "bbox": [ + 107, + 391, + 505, + 425 + ], + "lines": [ + { + "bbox": [ + 105, + 391, + 506, + 403 + ], + "spans": [ + { + "bbox": [ + 105, + 391, + 283, + 403 + ], + "score": 1.0, + "content": "Proof. M can be expressed as a product of", + "type": "text" + }, + { + "bbox": [ + 283, + 392, + 312, + 403 + ], + "score": 0.89, + "content": "2 \\log n", + "type": "inline_equation" + }, + { + "bbox": [ + 312, + 391, + 441, + 403 + ], + "score": 1.0, + "content": "butterfly factor matrices of size", + "type": "text" + }, + { + "bbox": [ + 442, + 393, + 467, + 402 + ], + "score": 0.88, + "content": "n \\times n", + "type": "inline_equation" + }, + { + "bbox": [ + 467, + 391, + 506, + 403 + ], + "score": 1.0, + "content": ". Each of", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 402, + 505, + 415 + ], + "spans": [ + { + "bbox": [ + 105, + 402, + 353, + 415 + ], + "score": 1.0, + "content": "these factor matrices has 2 parameters per row, for a total of", + "type": "text" + }, + { + "bbox": [ + 353, + 403, + 365, + 413 + ], + "score": 0.84, + "content": "2 n", + "type": "inline_equation" + }, + { + "bbox": [ + 366, + 402, + 505, + 415 + ], + "score": 1.0, + "content": "parameters each. Hence, the total", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 413, + 506, + 426 + ], + "spans": [ + { + "bbox": [ + 105, + 413, + 205, + 426 + ], + "score": 1.0, + "content": "number of parameters is", + "type": "text" + }, + { + "bbox": [ + 206, + 414, + 240, + 425 + ], + "score": 0.9, + "content": "4 n \\log n", + "type": "inline_equation" + }, + { + "bbox": [ + 240, + 413, + 244, + 426 + ], + "score": 1.0, + "content": ".", + "type": "text" + }, + { + "bbox": [ + 493, + 413, + 506, + 425 + ], + "score": 0.998, + "content": "□", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 18 + }, + { + "type": "text", + "bbox": [ + 108, + 431, + 504, + 455 + ], + "lines": [ + { + "bbox": [ + 105, + 429, + 505, + 445 + ], + "spans": [ + { + "bbox": [ + 105, + 429, + 234, + 445 + ], + "score": 1.0, + "content": "Observation E.2. Let M be an", + "type": "text" + }, + { + "bbox": [ + 234, + 433, + 258, + 442 + ], + "score": 0.87, + "content": "n \\times n", + "type": "inline_equation" + }, + { + "bbox": [ + 258, + 429, + 298, + 445 + ], + "score": 1.0, + "content": "matrix in", + "type": "text" + }, + { + "bbox": [ + 298, + 432, + 330, + 444 + ], + "score": 0.92, + "content": "( B B ^ { * } ) _ { e } ^ { w }", + "type": "inline_equation" + }, + { + "bbox": [ + 330, + 429, + 505, + 445 + ], + "score": 1.0, + "content": ". Then, given an arbitrary vector v of length", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 107, + 443, + 359, + 456 + ], + "spans": [ + { + "bbox": [ + 107, + 446, + 114, + 453 + ], + "score": 0.53, + "content": "n", + "type": "inline_equation" + }, + { + "bbox": [ + 114, + 443, + 223, + 456 + ], + "score": 1.0, + "content": ", we can compute Mv with", + "type": "text" + }, + { + "bbox": [ + 224, + 443, + 290, + 455 + ], + "score": 0.89, + "content": "O ( w n e \\log ( n e ) )", + "type": "inline_equation" + }, + { + "bbox": [ + 290, + 443, + 359, + 456 + ], + "score": 1.0, + "content": "field operations.", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 20.5 + }, + { + "type": "text", + "bbox": [ + 107, + 466, + 505, + 500 + ], + "lines": [ + { + "bbox": [ + 105, + 466, + 505, + 479 + ], + "spans": [ + { + "bbox": [ + 105, + 466, + 162, + 479 + ], + "score": 1.0, + "content": "Proof. Since", + "type": "text" + }, + { + "bbox": [ + 162, + 467, + 222, + 479 + ], + "score": 0.92, + "content": "\\mathbf { M } \\in ( B B ^ { * } ) _ { e } ^ { w }", + "type": "inline_equation" + }, + { + "bbox": [ + 222, + 466, + 330, + 479 + ], + "score": 1.0, + "content": ", we can decompose it as", + "type": "text" + }, + { + "bbox": [ + 331, + 466, + 403, + 478 + ], + "score": 0.91, + "content": "\\mathbf { S } \\mathbf { E } _ { 1 } \\mathbf { E } _ { 2 } \\ldots \\mathbf { E } _ { w } \\mathbf { S } ^ { T }", + "type": "inline_equation" + }, + { + "bbox": [ + 403, + 466, + 437, + 479 + ], + "score": 1.0, + "content": ", where", + "type": "text" + }, + { + "bbox": [ + 437, + 467, + 445, + 477 + ], + "score": 0.29, + "content": "\\mathbf { S }", + "type": "inline_equation" + }, + { + "bbox": [ + 446, + 466, + 505, + 479 + ], + "score": 1.0, + "content": "is as given in", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 477, + 505, + 491 + ], + "spans": [ + { + "bbox": [ + 105, + 477, + 207, + 491 + ], + "score": 1.0, + "content": "Definition 2.4, and each", + "type": "text" + }, + { + "bbox": [ + 207, + 479, + 219, + 489 + ], + "score": 0.87, + "content": "\\mathbf { E } _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 219, + 477, + 241, + 491 + ], + "score": 1.0, + "content": "is an", + "type": "text" + }, + { + "bbox": [ + 242, + 479, + 277, + 488 + ], + "score": 0.89, + "content": "e n \\times e n", + "type": "inline_equation" + }, + { + "bbox": [ + 277, + 477, + 317, + 491 + ], + "score": 1.0, + "content": "matrix in", + "type": "text" + }, + { + "bbox": [ + 318, + 478, + 337, + 488 + ], + "score": 0.87, + "content": "B B ^ { * }", + "type": "inline_equation" + }, + { + "bbox": [ + 337, + 477, + 505, + 491 + ], + "score": 1.0, + "content": ". Therefore, to compute Mv, we can use", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 489, + 483, + 500 + ], + "spans": [ + { + "bbox": [ + 105, + 489, + 483, + 500 + ], + "score": 1.0, + "content": "associativity of matrix multiplication to multiply the vector by one of these matrices at a time.", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 23 + }, + { + "type": "text", + "bbox": [ + 107, + 505, + 505, + 583 + ], + "lines": [ + { + "bbox": [ + 106, + 504, + 505, + 518 + ], + "spans": [ + { + "bbox": [ + 106, + 504, + 505, + 518 + ], + "score": 1.0, + "content": "Since all of these factors are sparse, we use the naïve sparse matrix-vector multiplication algorithm", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 516, + 506, + 529 + ], + "spans": [ + { + "bbox": [ + 105, + 516, + 506, + 529 + ], + "score": 1.0, + "content": "(begin with a 0-vector and perform the corresponding multiplication and addition for each nonzero", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 526, + 504, + 540 + ], + "spans": [ + { + "bbox": [ + 105, + 526, + 217, + 540 + ], + "score": 1.0, + "content": "matrix entry). S (and thus", + "type": "text" + }, + { + "bbox": [ + 217, + 527, + 231, + 538 + ], + "score": 0.85, + "content": "\\mathbf { \\vec { S } } ^ { T }", + "type": "inline_equation" + }, + { + "bbox": [ + 231, + 526, + 257, + 540 + ], + "score": 1.0, + "content": ") have", + "type": "text" + }, + { + "bbox": [ + 258, + 529, + 265, + 537 + ], + "score": 0.61, + "content": "n", + "type": "inline_equation" + }, + { + "bbox": [ + 266, + 526, + 489, + 540 + ], + "score": 1.0, + "content": "NNZ. Therefore, matrix-vector multiplication by S or", + "type": "text" + }, + { + "bbox": [ + 490, + 527, + 504, + 538 + ], + "score": 0.87, + "content": "\\mathbf { S } ^ { T }", + "type": "inline_equation" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 538, + 505, + 551 + ], + "spans": [ + { + "bbox": [ + 105, + 538, + 142, + 551 + ], + "score": 1.0, + "content": "requires", + "type": "text" + }, + { + "bbox": [ + 142, + 539, + 164, + 550 + ], + "score": 0.92, + "content": "O ( n )", + "type": "inline_equation" + }, + { + "bbox": [ + 165, + 538, + 505, + 551 + ], + "score": 1.0, + "content": "operations, which is dominated by the butterfly matrix-vector multiplication. Each", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 548, + 505, + 563 + ], + "spans": [ + { + "bbox": [ + 106, + 550, + 118, + 560 + ], + "score": 0.86, + "content": "\\mathbf { E } _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 119, + 548, + 250, + 563 + ], + "score": 1.0, + "content": "can be further decomposed into", + "type": "text" + }, + { + "bbox": [ + 251, + 550, + 290, + 561 + ], + "score": 0.89, + "content": "2 \\log ( n e )", + "type": "inline_equation" + }, + { + "bbox": [ + 290, + 548, + 381, + 563 + ], + "score": 1.0, + "content": "matrices with at most", + "type": "text" + }, + { + "bbox": [ + 381, + 550, + 398, + 560 + ], + "score": 0.49, + "content": "2 n e", + "type": "inline_equation" + }, + { + "bbox": [ + 399, + 548, + 505, + 563 + ], + "score": 1.0, + "content": "non-zero entries each (by", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 560, + 506, + 574 + ], + "spans": [ + { + "bbox": [ + 105, + 560, + 368, + 574 + ], + "score": 1.0, + "content": "Observation E.1). Therefore, matrix vector multiplication by each", + "type": "text" + }, + { + "bbox": [ + 369, + 561, + 380, + 572 + ], + "score": 0.89, + "content": "\\mathbf { E } _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 381, + 560, + 416, + 574 + ], + "score": 1.0, + "content": "requires", + "type": "text" + }, + { + "bbox": [ + 416, + 560, + 476, + 573 + ], + "score": 0.94, + "content": "O ( n e \\log ( n e ) )", + "type": "inline_equation" + }, + { + "bbox": [ + 476, + 560, + 506, + 574 + ], + "score": 1.0, + "content": ". Since", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 571, + 505, + 584 + ], + "spans": [ + { + "bbox": [ + 106, + 571, + 143, + 584 + ], + "score": 1.0, + "content": "there are", + "type": "text" + }, + { + "bbox": [ + 144, + 573, + 152, + 581 + ], + "score": 0.78, + "content": "w", + "type": "inline_equation" + }, + { + "bbox": [ + 153, + 571, + 174, + 584 + ], + "score": 1.0, + "content": "such", + "type": "text" + }, + { + "bbox": [ + 174, + 572, + 186, + 582 + ], + "score": 0.88, + "content": "\\mathbf { E } _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 186, + 571, + 273, + 584 + ], + "score": 1.0, + "content": ", we require a total of", + "type": "text" + }, + { + "bbox": [ + 273, + 571, + 340, + 583 + ], + "score": 0.91, + "content": "O ( w n \\bar { e } \\log ( n e ) )", + "type": "inline_equation" + }, + { + "bbox": [ + 340, + 571, + 388, + 584 + ], + "score": 1.0, + "content": "operations.", + "type": "text" + }, + { + "bbox": [ + 494, + 572, + 505, + 582 + ], + "score": 1.0, + "content": "□", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 28 + }, + { + "type": "text", + "bbox": [ + 106, + 595, + 470, + 607 + ], + "lines": [ + { + "bbox": [ + 105, + 595, + 471, + 609 + ], + "spans": [ + { + "bbox": [ + 105, + 595, + 332, + 609 + ], + "score": 1.0, + "content": "Now, we are ready to show that our definition of classes", + "type": "text" + }, + { + "bbox": [ + 333, + 595, + 365, + 608 + ], + "score": 0.92, + "content": "( B B ^ { * } ) _ { e } ^ { w }", + "type": "inline_equation" + }, + { + "bbox": [ + 365, + 595, + 471, + 609 + ], + "score": 1.0, + "content": "forms a natural hierarchy.", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 32 + }, + { + "type": "text", + "bbox": [ + 108, + 612, + 396, + 623 + ], + "lines": [ + { + "bbox": [ + 105, + 610, + 397, + 627 + ], + "spans": [ + { + "bbox": [ + 105, + 610, + 397, + 627 + ], + "score": 1.0, + "content": "First, we must argue that all matrices are contained within the hierarchy.", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 33 + }, + { + "type": "text", + "bbox": [ + 107, + 628, + 414, + 640 + ], + "lines": [ + { + "bbox": [ + 104, + 626, + 411, + 642 + ], + "spans": [ + { + "bbox": [ + 104, + 626, + 255, + 642 + ], + "score": 1.0, + "content": "Lemma E.3. Let M be an arbitrary", + "type": "text" + }, + { + "bbox": [ + 255, + 630, + 281, + 639 + ], + "score": 0.88, + "content": "n \\times n", + "type": "inline_equation" + }, + { + "bbox": [ + 281, + 626, + 335, + 642 + ], + "score": 1.0, + "content": "matrix. Then", + "type": "text" + }, + { + "bbox": [ + 336, + 627, + 411, + 641 + ], + "score": 0.91, + "content": "\\mathbf { M } \\in ( B B ^ { * } ) ^ { ( 2 n - 2 ) }", + "type": "inline_equation" + } + ], + "index": 34 + } + ], + "index": 34 + }, + { + "type": "text", + "bbox": [ + 107, + 653, + 505, + 708 + ], + "lines": [ + { + "bbox": [ + 106, + 653, + 505, + 664 + ], + "spans": [ + { + "bbox": [ + 106, + 653, + 252, + 664 + ], + "score": 1.0, + "content": "Proof. Corollary E.3 in Appendix", + "type": "text" + }, + { + "bbox": [ + 253, + 653, + 262, + 663 + ], + "score": 0.38, + "content": "\\mathbf { K }", + "type": "inline_equation" + }, + { + "bbox": [ + 262, + 653, + 330, + 664 + ], + "score": 1.0, + "content": "shows that any", + "type": "text" + }, + { + "bbox": [ + 330, + 654, + 358, + 663 + ], + "score": 0.89, + "content": "n \\times n", + "type": "inline_equation" + }, + { + "bbox": [ + 358, + 653, + 505, + 664 + ], + "score": 1.0, + "content": "matrix can be written in the form", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 107, + 661, + 505, + 677 + ], + "spans": [ + { + "bbox": [ + 107, + 663, + 340, + 676 + ], + "score": 0.76, + "content": "\\mathbf { M _ { 1 } } \\mathbf { \\bar { M } ^ { \\prime } } _ { 1 } ^ { * } \\dots \\mathbf { M } _ { n - 1 } \\mathbf { M ^ { \\prime } } _ { n - 1 } ^ { * } \\mathbf { \\bar { M } M _ { n } } \\mathbf { M ^ { \\prime } } _ { n } ^ { * } \\dots \\mathbf { M } _ { 2 n - 2 } \\mathbf { M ^ { \\prime } } _ { n - 2 } ^ { * } \\mathbf { \\bar { M } M ^ { \\prime } } _ { n } ^ { * } \\dots \\mathbf { M } _ { n } ^ { * } \\dots \\mathbf { M ^ { \\prime } } _ { n } ^ { * }", + "type": "inline_equation" + }, + { + "bbox": [ + 340, + 661, + 371, + 677 + ], + "score": 1.0, + "content": ", where", + "type": "text" + }, + { + "bbox": [ + 371, + 664, + 406, + 675 + ], + "score": 0.64, + "content": "\\mathbf { M } _ { i } , \\mathbf { M } _ { i } ^ { \\prime }", + "type": "inline_equation" + }, + { + "bbox": [ + 406, + 661, + 505, + 677 + ], + "score": 1.0, + "content": "are orthogonal butterfly", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 673, + 505, + 688 + ], + "spans": [ + { + "bbox": [ + 105, + 673, + 344, + 688 + ], + "score": 1.0, + "content": "matrices and M is a diagonal matrix. We can combine", + "type": "text" + }, + { + "bbox": [ + 344, + 675, + 354, + 685 + ], + "score": 0.82, + "content": "D", + "type": "inline_equation" + }, + { + "bbox": [ + 354, + 673, + 378, + 688 + ], + "score": 1.0, + "content": "with", + "type": "text" + }, + { + "bbox": [ + 378, + 676, + 394, + 686 + ], + "score": 0.9, + "content": "M _ { n }", + "type": "inline_equation" + }, + { + "bbox": [ + 394, + 673, + 505, + 688 + ], + "score": 1.0, + "content": "to form another (possibly", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 106, + 686, + 505, + 698 + ], + "spans": [ + { + "bbox": [ + 106, + 686, + 374, + 698 + ], + "score": 1.0, + "content": "not orthogonal) butterfly matrix. This yields a decomposition of", + "type": "text" + }, + { + "bbox": [ + 374, + 686, + 387, + 696 + ], + "score": 0.35, + "content": "\\mathbf { M }", + "type": "inline_equation" + }, + { + "bbox": [ + 387, + 686, + 505, + 698 + ], + "score": 1.0, + "content": "as products of (possibly not", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 696, + 504, + 709 + ], + "spans": [ + { + "bbox": [ + 105, + 696, + 448, + 709 + ], + "score": 1.0, + "content": "orthogonal) butterfly matrices and their (conjugate) transposes, completing the proof.", + "type": "text" + }, + { + "bbox": [ + 496, + 699, + 504, + 706 + ], + "score": 0.998, + "content": "□", + "type": "text" + } + ], + "index": 39 + } + ], + "index": 37 + }, + { + "type": "text", + "bbox": [ + 107, + 720, + 364, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 719, + 365, + 734 + ], + "spans": [ + { + "bbox": [ + 105, + 719, + 365, + 734 + ], + "score": 1.0, + "content": "Next, we argue that, up to a certain point, this hierarchy is strict.", + "type": "text" + } + ], + "index": 40 + } + ], + "index": 40 + } + ], + "page_idx": 22, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 107, + 27, + 293, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 26, + 294, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 26, + 294, + 38 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2020", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 300, + 751, + 311, + 760 + ], + "lines": [ + { + "bbox": [ + 298, + 750, + 313, + 764 + ], + "spans": [ + { + "bbox": [ + 298, + 750, + 313, + 764 + ], + "score": 1.0, + "content": "", + "type": "text", + "height": 14, + "width": 15 + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 106, + 80, + 505, + 169 + ], + "lines": [ + { + "bbox": [ + 105, + 79, + 506, + 93 + ], + "spans": [ + { + "bbox": [ + 105, + 79, + 506, + 93 + ], + "score": 1.0, + "content": "Table 7: Expressiveness of different classes of structured matrices: Frobenius error of represent-", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 91, + 505, + 104 + ], + "spans": [ + { + "bbox": [ + 106, + 91, + 505, + 104 + ], + "score": 1.0, + "content": "ing common structured matrices (columns) of dimension 256 using three structured representations", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 106, + 102, + 505, + 115 + ], + "spans": [ + { + "bbox": [ + 106, + 102, + 505, + 115 + ], + "score": 1.0, + "content": "of matrices with adjustable numbers of parameters. (Left group: Target matrices in the same class", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 113, + 505, + 125 + ], + "spans": [ + { + "bbox": [ + 105, + 113, + 505, + 125 + ], + "score": 1.0, + "content": "as the methods. Middle group: Target matrices with fixed number of parameters. Right: Random", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 124, + 506, + 136 + ], + "spans": [ + { + "bbox": [ + 105, + 124, + 506, + 136 + ], + "score": 1.0, + "content": "matrix to show typical scale of error.) Each method is allotted the same number of parameters, equal", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 106, + 135, + 505, + 147 + ], + "spans": [ + { + "bbox": [ + 106, + 135, + 124, + 147 + ], + "score": 1.0, + "content": "to a", + "type": "text" + }, + { + "bbox": [ + 124, + 136, + 147, + 146 + ], + "score": 0.82, + "content": "\\log n", + "type": "inline_equation" + }, + { + "bbox": [ + 147, + 135, + 505, + 147 + ], + "score": 1.0, + "content": "factor more than that of the target matrix. Low-rank and sparse matrices are unable to", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 146, + 506, + 158 + ], + "spans": [ + { + "bbox": [ + 105, + 146, + 506, + 158 + ], + "score": 1.0, + "content": "capture any structure outside their own class, while the minima for kaleidoscope matrices found via", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 157, + 488, + 169 + ], + "spans": [ + { + "bbox": [ + 106, + 157, + 488, + 169 + ], + "score": 1.0, + "content": "optimization better capture the actual structure for out-of-class targets better than the baselines.", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 3.5, + "bbox_fs": [ + 105, + 79, + 506, + 169 + ] + }, + { + "type": "table", + "bbox": [ + 106, + 178, + 542, + 257 + ], + "blocks": [ + { + "type": "table_body", + "bbox": [ + 106, + 178, + 542, + 257 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 106, + 178, + 542, + 257 + ], + "spans": [ + { + "bbox": [ + 106, + 178, + 542, + 257 + ], + "score": 0.981, + "html": "
Target MethodKaleidoscopeLow-rankSparseConvolutionFastfoodRandom
Global Min.Kaleidoscope0.00.00.00.00.0
Low-rank14.90.010.814.611.615.5
Sparse11.712.20.013.17.114.1
With SGDKaleidoscope0.00.018.00.05.114.5
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We first make some basic", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 352, + 270, + 365 + ], + "spans": [ + { + "bbox": [ + 106, + 352, + 270, + 365 + ], + "score": 1.0, + "content": "observations about the parameterization.", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 14.5, + "bbox_fs": [ + 105, + 340, + 506, + 365 + ] + }, + { + "type": "text", + "bbox": [ + 108, + 367, + 397, + 379 + ], + "lines": [ + { + "bbox": [ + 106, + 366, + 397, + 381 + ], + "spans": [ + { + "bbox": [ + 106, + 366, + 196, + 381 + ], + "score": 1.0, + "content": "Observation E.1. An", + "type": "text" + }, + { + "bbox": [ + 197, + 369, + 222, + 378 + ], + "score": 0.83, + "content": "n \\times n", + "type": "inline_equation" + }, + { + "bbox": [ + 222, + 366, + 251, + 381 + ], + "score": 1.0, + "content": "matrix", + "type": "text" + }, + { + "bbox": [ + 252, + 367, + 294, + 378 + ], + "score": 0.9, + "content": "\\mathbf { M } \\in B B ^ { * }", + "type": "inline_equation" + }, + { + "bbox": [ + 294, + 366, + 312, + 381 + ], + "score": 1.0, + "content": "has", + "type": "text" + }, + { + "bbox": [ + 312, + 368, + 346, + 379 + ], + "score": 0.58, + "content": "4 n \\log n", + "type": "inline_equation" + }, + { + "bbox": [ + 347, + 366, + 397, + 381 + ], + "score": 1.0, + "content": "parameters.", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 16, + "bbox_fs": [ + 106, + 366, + 397, + 381 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 391, + 505, + 425 + ], + "lines": [ + { + "bbox": [ + 105, + 391, + 506, + 403 + ], + "spans": [ + { + "bbox": [ + 105, + 391, + 283, + 403 + ], + "score": 1.0, + "content": "Proof. M can be expressed as a product of", + "type": "text" + }, + { + "bbox": [ + 283, + 392, + 312, + 403 + ], + "score": 0.89, + "content": "2 \\log n", + "type": "inline_equation" + }, + { + "bbox": [ + 312, + 391, + 441, + 403 + ], + "score": 1.0, + "content": "butterfly factor matrices of size", + "type": "text" + }, + { + "bbox": [ + 442, + 393, + 467, + 402 + ], + "score": 0.88, + "content": "n \\times n", + "type": "inline_equation" + }, + { + "bbox": [ + 467, + 391, + 506, + 403 + ], + "score": 1.0, + "content": ". Each of", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 402, + 505, + 415 + ], + "spans": [ + { + "bbox": [ + 105, + 402, + 353, + 415 + ], + "score": 1.0, + "content": "these factor matrices has 2 parameters per row, for a total of", + "type": "text" + }, + { + "bbox": [ + 353, + 403, + 365, + 413 + ], + "score": 0.84, + "content": "2 n", + "type": "inline_equation" + }, + { + "bbox": [ + 366, + 402, + 505, + 415 + ], + "score": 1.0, + "content": "parameters each. Hence, the total", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 413, + 506, + 426 + ], + "spans": [ + { + "bbox": [ + 105, + 413, + 205, + 426 + ], + "score": 1.0, + "content": "number of parameters is", + "type": "text" + }, + { + "bbox": [ + 206, + 414, + 240, + 425 + ], + "score": 0.9, + "content": "4 n \\log n", + "type": "inline_equation" + }, + { + "bbox": [ + 240, + 413, + 244, + 426 + ], + "score": 1.0, + "content": ".", + "type": "text" + }, + { + "bbox": [ + 493, + 413, + 506, + 425 + ], + "score": 0.998, + "content": "□", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 18, + "bbox_fs": [ + 105, + 391, + 506, + 426 + ] + }, + { + "type": "text", + "bbox": [ + 108, + 431, + 504, + 455 + ], + "lines": [ + { + "bbox": [ + 105, + 429, + 505, + 445 + ], + "spans": [ + { + "bbox": [ + 105, + 429, + 234, + 445 + ], + "score": 1.0, + "content": "Observation E.2. Let M be an", + "type": "text" + }, + { + "bbox": [ + 234, + 433, + 258, + 442 + ], + "score": 0.87, + "content": "n \\times n", + "type": "inline_equation" + }, + { + "bbox": [ + 258, + 429, + 298, + 445 + ], + "score": 1.0, + "content": "matrix in", + "type": "text" + }, + { + "bbox": [ + 298, + 432, + 330, + 444 + ], + "score": 0.92, + "content": "( B B ^ { * } ) _ { e } ^ { w }", + "type": "inline_equation" + }, + { + "bbox": [ + 330, + 429, + 505, + 445 + ], + "score": 1.0, + "content": ". Then, given an arbitrary vector v of length", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 107, + 443, + 359, + 456 + ], + "spans": [ + { + "bbox": [ + 107, + 446, + 114, + 453 + ], + "score": 0.53, + "content": "n", + "type": "inline_equation" + }, + { + "bbox": [ + 114, + 443, + 223, + 456 + ], + "score": 1.0, + "content": ", we can compute Mv with", + "type": "text" + }, + { + "bbox": [ + 224, + 443, + 290, + 455 + ], + "score": 0.89, + "content": "O ( w n e \\log ( n e ) )", + "type": "inline_equation" + }, + { + "bbox": [ + 290, + 443, + 359, + 456 + ], + "score": 1.0, + "content": "field operations.", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 20.5, + "bbox_fs": [ + 105, + 429, + 505, + 456 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 466, + 505, + 500 + ], + "lines": [ + { + "bbox": [ + 105, + 466, + 505, + 479 + ], + "spans": [ + { + "bbox": [ + 105, + 466, + 162, + 479 + ], + "score": 1.0, + "content": "Proof. Since", + "type": "text" + }, + { + "bbox": [ + 162, + 467, + 222, + 479 + ], + "score": 0.92, + "content": "\\mathbf { M } \\in ( B B ^ { * } ) _ { e } ^ { w }", + "type": "inline_equation" + }, + { + "bbox": [ + 222, + 466, + 330, + 479 + ], + "score": 1.0, + "content": ", we can decompose it as", + "type": "text" + }, + { + "bbox": [ + 331, + 466, + 403, + 478 + ], + "score": 0.91, + "content": "\\mathbf { S } \\mathbf { E } _ { 1 } \\mathbf { E } _ { 2 } \\ldots \\mathbf { E } _ { w } \\mathbf { S } ^ { T }", + "type": "inline_equation" + }, + { + "bbox": [ + 403, + 466, + 437, + 479 + ], + "score": 1.0, + "content": ", where", + "type": "text" + }, + { + "bbox": [ + 437, + 467, + 445, + 477 + ], + "score": 0.29, + "content": "\\mathbf { S }", + "type": "inline_equation" + }, + { + "bbox": [ + 446, + 466, + 505, + 479 + ], + "score": 1.0, + "content": "is as given in", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 477, + 505, + 491 + ], + "spans": [ + { + "bbox": [ + 105, + 477, + 207, + 491 + ], + "score": 1.0, + "content": "Definition 2.4, and each", + "type": "text" + }, + { + "bbox": [ + 207, + 479, + 219, + 489 + ], + "score": 0.87, + "content": "\\mathbf { E } _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 219, + 477, + 241, + 491 + ], + "score": 1.0, + "content": "is an", + "type": "text" + }, + { + "bbox": [ + 242, + 479, + 277, + 488 + ], + "score": 0.89, + "content": "e n \\times e n", + "type": "inline_equation" + }, + { + "bbox": [ + 277, + 477, + 317, + 491 + ], + "score": 1.0, + "content": "matrix in", + "type": "text" + }, + { + "bbox": [ + 318, + 478, + 337, + 488 + ], + "score": 0.87, + "content": "B B ^ { * }", + "type": "inline_equation" + }, + { + "bbox": [ + 337, + 477, + 505, + 491 + ], + "score": 1.0, + "content": ". Therefore, to compute Mv, we can use", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 489, + 483, + 500 + ], + "spans": [ + { + "bbox": [ + 105, + 489, + 483, + 500 + ], + "score": 1.0, + "content": "associativity of matrix multiplication to multiply the vector by one of these matrices at a time.", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 23, + "bbox_fs": [ + 105, + 466, + 505, + 500 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 505, + 505, + 583 + ], + "lines": [ + { + "bbox": [ + 106, + 504, + 505, + 518 + ], + "spans": [ + { + "bbox": [ + 106, + 504, + 505, + 518 + ], + "score": 1.0, + "content": "Since all of these factors are sparse, we use the naïve sparse matrix-vector multiplication algorithm", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 516, + 506, + 529 + ], + "spans": [ + { + "bbox": [ + 105, + 516, + 506, + 529 + ], + "score": 1.0, + "content": "(begin with a 0-vector and perform the corresponding multiplication and addition for each nonzero", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 526, + 504, + 540 + ], + "spans": [ + { + "bbox": [ + 105, + 526, + 217, + 540 + ], + "score": 1.0, + "content": "matrix entry). S (and thus", + "type": "text" + }, + { + "bbox": [ + 217, + 527, + 231, + 538 + ], + "score": 0.85, + "content": "\\mathbf { \\vec { S } } ^ { T }", + "type": "inline_equation" + }, + { + "bbox": [ + 231, + 526, + 257, + 540 + ], + "score": 1.0, + "content": ") have", + "type": "text" + }, + { + "bbox": [ + 258, + 529, + 265, + 537 + ], + "score": 0.61, + "content": "n", + "type": "inline_equation" + }, + { + "bbox": [ + 266, + 526, + 489, + 540 + ], + "score": 1.0, + "content": "NNZ. Therefore, matrix-vector multiplication by S or", + "type": "text" + }, + { + "bbox": [ + 490, + 527, + 504, + 538 + ], + "score": 0.87, + "content": "\\mathbf { S } ^ { T }", + "type": "inline_equation" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 538, + 505, + 551 + ], + "spans": [ + { + "bbox": [ + 105, + 538, + 142, + 551 + ], + "score": 1.0, + "content": "requires", + "type": "text" + }, + { + "bbox": [ + 142, + 539, + 164, + 550 + ], + "score": 0.92, + "content": "O ( n )", + "type": "inline_equation" + }, + { + "bbox": [ + 165, + 538, + 505, + 551 + ], + "score": 1.0, + "content": "operations, which is dominated by the butterfly matrix-vector multiplication. Each", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 548, + 505, + 563 + ], + "spans": [ + { + "bbox": [ + 106, + 550, + 118, + 560 + ], + "score": 0.86, + "content": "\\mathbf { E } _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 119, + 548, + 250, + 563 + ], + "score": 1.0, + "content": "can be further decomposed into", + "type": "text" + }, + { + "bbox": [ + 251, + 550, + 290, + 561 + ], + "score": 0.89, + "content": "2 \\log ( n e )", + "type": "inline_equation" + }, + { + "bbox": [ + 290, + 548, + 381, + 563 + ], + "score": 1.0, + "content": "matrices with at most", + "type": "text" + }, + { + "bbox": [ + 381, + 550, + 398, + 560 + ], + "score": 0.49, + "content": "2 n e", + "type": "inline_equation" + }, + { + "bbox": [ + 399, + 548, + 505, + 563 + ], + "score": 1.0, + "content": "non-zero entries each (by", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 560, + 506, + 574 + ], + "spans": [ + { + "bbox": [ + 105, + 560, + 368, + 574 + ], + "score": 1.0, + "content": "Observation E.1). Therefore, matrix vector multiplication by each", + "type": "text" + }, + { + "bbox": [ + 369, + 561, + 380, + 572 + ], + "score": 0.89, + "content": "\\mathbf { E } _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 381, + 560, + 416, + 574 + ], + "score": 1.0, + "content": "requires", + "type": "text" + }, + { + "bbox": [ + 416, + 560, + 476, + 573 + ], + "score": 0.94, + "content": "O ( n e \\log ( n e ) )", + "type": "inline_equation" + }, + { + "bbox": [ + 476, + 560, + 506, + 574 + ], + "score": 1.0, + "content": ". Since", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 571, + 505, + 584 + ], + "spans": [ + { + "bbox": [ + 106, + 571, + 143, + 584 + ], + "score": 1.0, + "content": "there are", + "type": "text" + }, + { + "bbox": [ + 144, + 573, + 152, + 581 + ], + "score": 0.78, + "content": "w", + "type": "inline_equation" + }, + { + "bbox": [ + 153, + 571, + 174, + 584 + ], + "score": 1.0, + "content": "such", + "type": "text" + }, + { + "bbox": [ + 174, + 572, + 186, + 582 + ], + "score": 0.88, + "content": "\\mathbf { E } _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 186, + 571, + 273, + 584 + ], + "score": 1.0, + "content": ", we require a total of", + "type": "text" + }, + { + "bbox": [ + 273, + 571, + 340, + 583 + ], + "score": 0.91, + "content": "O ( w n \\bar { e } \\log ( n e ) )", + "type": "inline_equation" + }, + { + "bbox": [ + 340, + 571, + 388, + 584 + ], + "score": 1.0, + "content": "operations.", + "type": "text" + }, + { + "bbox": [ + 494, + 572, + 505, + 582 + ], + "score": 1.0, + "content": "□", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 28, + "bbox_fs": [ + 105, + 504, + 506, + 584 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 595, + 470, + 607 + ], + "lines": [ + { + "bbox": [ + 105, + 595, + 471, + 609 + ], + "spans": [ + { + "bbox": [ + 105, + 595, + 332, + 609 + ], + "score": 1.0, + "content": "Now, we are ready to show that our definition of classes", + "type": "text" + }, + { + "bbox": [ + 333, + 595, + 365, + 608 + ], + "score": 0.92, + "content": "( B B ^ { * } ) _ { e } ^ { w }", + "type": "inline_equation" + }, + { + "bbox": [ + 365, + 595, + 471, + 609 + ], + "score": 1.0, + "content": "forms a natural hierarchy.", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 32, + "bbox_fs": [ + 105, + 595, + 471, + 609 + ] + }, + { + "type": "text", + "bbox": [ + 108, + 612, + 396, + 623 + ], + "lines": [ + { + "bbox": [ + 105, + 610, + 397, + 627 + ], + "spans": [ + { + "bbox": [ + 105, + 610, + 397, + 627 + ], + "score": 1.0, + "content": "First, we must argue that all matrices are contained within the hierarchy.", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 33, + "bbox_fs": [ + 105, + 610, + 397, + 627 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 628, + 414, + 640 + ], + "lines": [ + { + "bbox": [ + 104, + 626, + 411, + 642 + ], + "spans": [ + { + "bbox": [ + 104, + 626, + 255, + 642 + ], + "score": 1.0, + "content": "Lemma E.3. Let M be an arbitrary", + "type": "text" + }, + { + "bbox": [ + 255, + 630, + 281, + 639 + ], + "score": 0.88, + "content": "n \\times n", + "type": "inline_equation" + }, + { + "bbox": [ + 281, + 626, + 335, + 642 + ], + "score": 1.0, + "content": "matrix. Then", + "type": "text" + }, + { + "bbox": [ + 336, + 627, + 411, + 641 + ], + "score": 0.91, + "content": "\\mathbf { M } \\in ( B B ^ { * } ) ^ { ( 2 n - 2 ) }", + "type": "inline_equation" + } + ], + "index": 34 + } + ], + "index": 34, + "bbox_fs": [ + 104, + 626, + 411, + 642 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 653, + 505, + 708 + ], + "lines": [ + { + "bbox": [ + 106, + 653, + 505, + 664 + ], + "spans": [ + { + "bbox": [ + 106, + 653, + 252, + 664 + ], + "score": 1.0, + "content": "Proof. Corollary E.3 in Appendix", + "type": "text" + }, + { + "bbox": [ + 253, + 653, + 262, + 663 + ], + "score": 0.38, + "content": "\\mathbf { K }", + "type": "inline_equation" + }, + { + "bbox": [ + 262, + 653, + 330, + 664 + ], + "score": 1.0, + "content": "shows that any", + "type": "text" + }, + { + "bbox": [ + 330, + 654, + 358, + 663 + ], + "score": 0.89, + "content": "n \\times n", + "type": "inline_equation" + }, + { + "bbox": [ + 358, + 653, + 505, + 664 + ], + "score": 1.0, + "content": "matrix can be written in the form", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 107, + 661, + 505, + 677 + ], + "spans": [ + { + "bbox": [ + 107, + 663, + 340, + 676 + ], + "score": 0.76, + "content": "\\mathbf { M _ { 1 } } \\mathbf { \\bar { M } ^ { \\prime } } _ { 1 } ^ { * } \\dots \\mathbf { M } _ { n - 1 } \\mathbf { M ^ { \\prime } } _ { n - 1 } ^ { * } \\mathbf { \\bar { M } M _ { n } } \\mathbf { M ^ { \\prime } } _ { n } ^ { * } \\dots \\mathbf { M } _ { 2 n - 2 } \\mathbf { M ^ { \\prime } } _ { n - 2 } ^ { * } \\mathbf { \\bar { M } M ^ { \\prime } } _ { n } ^ { * } \\dots \\mathbf { M } _ { n } ^ { * } \\dots \\mathbf { M ^ { \\prime } } _ { n } ^ { * }", + "type": "inline_equation" + }, + { + "bbox": [ + 340, + 661, + 371, + 677 + ], + "score": 1.0, + "content": ", where", + "type": "text" + }, + { + "bbox": [ + 371, + 664, + 406, + 675 + ], + "score": 0.64, + "content": "\\mathbf { M } _ { i } , \\mathbf { M } _ { i } ^ { \\prime }", + "type": "inline_equation" + }, + { + "bbox": [ + 406, + 661, + 505, + 677 + ], + "score": 1.0, + "content": "are orthogonal butterfly", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 673, + 505, + 688 + ], + "spans": [ + { + "bbox": [ + 105, + 673, + 344, + 688 + ], + "score": 1.0, + "content": "matrices and M is a diagonal matrix. We can combine", + "type": "text" + }, + { + "bbox": [ + 344, + 675, + 354, + 685 + ], + "score": 0.82, + "content": "D", + "type": "inline_equation" + }, + { + "bbox": [ + 354, + 673, + 378, + 688 + ], + "score": 1.0, + "content": "with", + "type": "text" + }, + { + "bbox": [ + 378, + 676, + 394, + 686 + ], + "score": 0.9, + "content": "M _ { n }", + "type": "inline_equation" + }, + { + "bbox": [ + 394, + 673, + 505, + 688 + ], + "score": 1.0, + "content": "to form another (possibly", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 106, + 686, + 505, + 698 + ], + "spans": [ + { + "bbox": [ + 106, + 686, + 374, + 698 + ], + "score": 1.0, + "content": "not orthogonal) butterfly matrix. This yields a decomposition of", + "type": "text" + }, + { + "bbox": [ + 374, + 686, + 387, + 696 + ], + "score": 0.35, + "content": "\\mathbf { M }", + "type": "inline_equation" + }, + { + "bbox": [ + 387, + 686, + 505, + 698 + ], + "score": 1.0, + "content": "as products of (possibly not", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 696, + 504, + 709 + ], + "spans": [ + { + "bbox": [ + 105, + 696, + 448, + 709 + ], + "score": 1.0, + "content": "orthogonal) butterfly matrices and their (conjugate) transposes, completing the proof.", + "type": "text" + }, + { + "bbox": [ + 496, + 699, + 504, + 706 + ], + "score": 0.998, + "content": "□", + "type": "text" + } + ], + "index": 39 + } + ], + "index": 37, + "bbox_fs": [ + 105, + 653, + 505, + 709 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 720, + 364, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 719, + 365, + 734 + ], + "spans": [ + { + "bbox": [ + 105, + 719, + 365, + 734 + ], + "score": 1.0, + "content": "Next, we argue that, up to a certain point, this hierarchy is strict.", + "type": "text" + } + ], + "index": 40 + } + ], + "index": 40, + "bbox_fs": [ + 105, + 719, + 365, + 734 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 105, + 82, + 504, + 106 + ], + "lines": [ + { + "bbox": [ + 106, + 82, + 505, + 95 + ], + "spans": [ + { + "bbox": [ + 106, + 82, + 223, + 95 + ], + "score": 1.0, + "content": "Lemma E.4. For every fixed", + "type": "text" + }, + { + "bbox": [ + 223, + 83, + 246, + 93 + ], + "score": 0.88, + "content": "c \\geq 1", + "type": "inline_equation" + }, + { + "bbox": [ + 247, + 82, + 294, + 95 + ], + "score": 1.0, + "content": ", there is an", + "type": "text" + }, + { + "bbox": [ + 294, + 84, + 318, + 93 + ], + "score": 0.88, + "content": "n \\times n", + "type": "inline_equation" + }, + { + "bbox": [ + 319, + 82, + 347, + 95 + ], + "score": 1.0, + "content": "matrix", + "type": "text" + }, + { + "bbox": [ + 347, + 83, + 365, + 94 + ], + "score": 0.89, + "content": "{ { \\bf { M } } _ { n } }", + "type": "inline_equation" + }, + { + "bbox": [ + 365, + 82, + 505, + 95 + ], + "score": 1.0, + "content": "(with n sufficiently large) such that", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 91, + 256, + 107 + ], + "spans": [ + { + "bbox": [ + 106, + 93, + 176, + 106 + ], + "score": 0.94, + "content": "\\mathbf { M } _ { n } \\in ( B B ^ { * } ) ^ { c + 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 177, + 91, + 192, + 107 + ], + "score": 1.0, + "content": "but", + "type": "text" + }, + { + "bbox": [ + 193, + 93, + 252, + 106 + ], + "score": 0.92, + "content": "\\mathbf { M } _ { n } \\notin ( B B ^ { * } ) ^ { c }", + "type": "inline_equation" + }, + { + "bbox": [ + 252, + 91, + 256, + 107 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 1 + } + ], + "index": 0.5 + }, + { + "type": "text", + "bbox": [ + 106, + 116, + 505, + 211 + ], + "lines": [ + { + "bbox": [ + 105, + 114, + 504, + 133 + ], + "spans": [ + { + "bbox": [ + 105, + 114, + 163, + 133 + ], + "score": 1.0, + "content": "Proof. Given", + "type": "text" + }, + { + "bbox": [ + 164, + 120, + 169, + 127 + ], + "score": 0.7, + "content": "c", + "type": "inline_equation" + }, + { + "bbox": [ + 169, + 114, + 186, + 133 + ], + "score": 1.0, + "content": ", fix", + "type": "text" + }, + { + "bbox": [ + 187, + 120, + 194, + 127 + ], + "score": 0.74, + "content": "n", + "type": "inline_equation" + }, + { + "bbox": [ + 194, + 114, + 312, + 133 + ], + "score": 1.0, + "content": "to be a power of 2 such that", + "type": "text" + }, + { + "bbox": [ + 312, + 118, + 360, + 132 + ], + "score": 0.93, + "content": "\\begin{array} { r } { c < \\frac { n } { 4 \\log _ { 2 } n } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 360, + 114, + 362, + 133 + ], + "score": 1.0, + "content": ".", + "type": "text" + }, + { + "bbox": [ + 362, + 118, + 504, + 129 + ], + "score": 1.0, + "content": "For sake of contradiction, assume", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 130, + 506, + 145 + ], + "spans": [ + { + "bbox": [ + 105, + 130, + 149, + 145 + ], + "score": 1.0, + "content": "that every", + "type": "text" + }, + { + "bbox": [ + 149, + 133, + 174, + 142 + ], + "score": 0.92, + "content": "n \\times n", + "type": "inline_equation" + }, + { + "bbox": [ + 175, + 130, + 216, + 145 + ], + "score": 1.0, + "content": "matrix in", + "type": "text" + }, + { + "bbox": [ + 216, + 131, + 256, + 144 + ], + "score": 0.92, + "content": "( B B ^ { * } ) ^ { c + 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 257, + 130, + 297, + 145 + ], + "score": 1.0, + "content": "is also in", + "type": "text" + }, + { + "bbox": [ + 297, + 132, + 327, + 144 + ], + "score": 0.91, + "content": "( B B ^ { * } ) ^ { c }", + "type": "inline_equation" + }, + { + "bbox": [ + 328, + 130, + 421, + 145 + ], + "score": 1.0, + "content": ". Let A be an arbitrary", + "type": "text" + }, + { + "bbox": [ + 421, + 133, + 447, + 142 + ], + "score": 0.89, + "content": "n \\times n", + "type": "inline_equation" + }, + { + "bbox": [ + 447, + 130, + 506, + 145 + ], + "score": 1.0, + "content": "matrix. From", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 104, + 141, + 506, + 158 + ], + "spans": [ + { + "bbox": [ + 104, + 141, + 159, + 158 + ], + "score": 1.0, + "content": "Lemma E.3,", + "type": "text" + }, + { + "bbox": [ + 159, + 143, + 234, + 156 + ], + "score": 0.91, + "content": "\\mathbf { A } \\in ( B B ^ { \\ast } ) ^ { ( 2 n - 2 ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 234, + 141, + 429, + 158 + ], + "score": 1.0, + "content": ". From our assumption, we can replace the first", + "type": "text" + }, + { + "bbox": [ + 429, + 144, + 474, + 155 + ], + "score": 0.58, + "content": "c + 1 B B ^ { * }", + "type": "inline_equation" + }, + { + "bbox": [ + 474, + 141, + 506, + 158 + ], + "score": 1.0, + "content": "factors", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 154, + 505, + 168 + ], + "spans": [ + { + "bbox": [ + 105, + 154, + 118, + 168 + ], + "score": 1.0, + "content": "of", + "type": "text" + }, + { + "bbox": [ + 118, + 155, + 128, + 165 + ], + "score": 0.27, + "content": "\\mathbf { A }", + "type": "inline_equation" + }, + { + "bbox": [ + 129, + 154, + 151, + 168 + ], + "score": 1.0, + "content": "with", + "type": "text" + }, + { + "bbox": [ + 152, + 157, + 158, + 165 + ], + "score": 0.7, + "content": "c", + "type": "inline_equation" + }, + { + "bbox": [ + 158, + 154, + 251, + 168 + ], + "score": 1.0, + "content": "(potentially different)", + "type": "text" + }, + { + "bbox": [ + 251, + 155, + 271, + 165 + ], + "score": 0.88, + "content": "B B ^ { * }", + "type": "inline_equation" + }, + { + "bbox": [ + 271, + 154, + 505, + 168 + ], + "score": 1.0, + "content": "factors and still recover A. We can repeat this process", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 166, + 505, + 178 + ], + "spans": [ + { + "bbox": [ + 106, + 166, + 192, + 178 + ], + "score": 1.0, + "content": "until we are left with", + "type": "text" + }, + { + "bbox": [ + 192, + 166, + 219, + 176 + ], + "score": 0.65, + "content": "c B B ^ { * }", + "type": "inline_equation" + }, + { + "bbox": [ + 219, + 166, + 309, + 178 + ], + "score": 1.0, + "content": "factors, implying that", + "type": "text" + }, + { + "bbox": [ + 309, + 166, + 361, + 178 + ], + "score": 0.93, + "content": "\\mathbf { A } \\in ( B B ^ { * } ) ^ { c }", + "type": "inline_equation" + }, + { + "bbox": [ + 361, + 166, + 505, + 178 + ], + "score": 1.0, + "content": ". From Observation E.1, we require", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 177, + 505, + 189 + ], + "spans": [ + { + "bbox": [ + 106, + 177, + 169, + 188 + ], + "score": 0.9, + "content": "4 c n \\log n < n ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 169, + 177, + 243, + 189 + ], + "score": 1.0, + "content": "(by our choice of", + "type": "text" + }, + { + "bbox": [ + 243, + 180, + 250, + 187 + ], + "score": 0.72, + "content": "n", + "type": "inline_equation" + }, + { + "bbox": [ + 250, + 177, + 505, + 189 + ], + "score": 1.0, + "content": ") parameters to completely describe A. This is a contradiction", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 187, + 506, + 201 + ], + "spans": [ + { + "bbox": [ + 106, + 188, + 129, + 201 + ], + "score": 1.0, + "content": "since", + "type": "text" + }, + { + "bbox": [ + 129, + 189, + 139, + 198 + ], + "score": 0.38, + "content": "\\mathbf { A }", + "type": "inline_equation" + }, + { + "bbox": [ + 139, + 188, + 196, + 201 + ], + "score": 1.0, + "content": "is an arbitrary", + "type": "text" + }, + { + "bbox": [ + 197, + 189, + 222, + 198 + ], + "score": 0.88, + "content": "n \\times n", + "type": "inline_equation" + }, + { + "bbox": [ + 222, + 188, + 322, + 201 + ], + "score": 1.0, + "content": "matrix, and therefore has", + "type": "text" + }, + { + "bbox": [ + 323, + 187, + 334, + 198 + ], + "score": 0.88, + "content": "n ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 335, + 188, + 506, + 201 + ], + "score": 1.0, + "content": "arbitrary parameters. Hence, there must be", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 104, + 197, + 506, + 213 + ], + "spans": [ + { + "bbox": [ + 104, + 197, + 130, + 213 + ], + "score": 1.0, + "content": "some", + "type": "text" + }, + { + "bbox": [ + 130, + 200, + 155, + 209 + ], + "score": 0.89, + "content": "n \\times n", + "type": "inline_equation" + }, + { + "bbox": [ + 156, + 197, + 196, + 213 + ], + "score": 1.0, + "content": "matrix in", + "type": "text" + }, + { + "bbox": [ + 196, + 198, + 236, + 211 + ], + "score": 0.92, + "content": "( B B ^ { * } ) ^ { c + 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 237, + 197, + 290, + 213 + ], + "score": 1.0, + "content": "that is not in", + "type": "text" + }, + { + "bbox": [ + 291, + 199, + 321, + 211 + ], + "score": 0.92, + "content": "( B B ^ { * } ) ^ { c }", + "type": "inline_equation" + }, + { + "bbox": [ + 321, + 197, + 325, + 213 + ], + "score": 1.0, + "content": ".", + "type": "text" + }, + { + "bbox": [ + 493, + 199, + 506, + 210 + ], + "score": 0.998, + "content": "□", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 5.5 + }, + { + "type": "title", + "bbox": [ + 107, + 225, + 351, + 239 + ], + "lines": [ + { + "bbox": [ + 105, + 225, + 352, + 241 + ], + "spans": [ + { + "bbox": [ + 105, + 225, + 262, + 241 + ], + "score": 1.0, + "content": "F ARITHMETIC CIRCUITS IN", + "type": "text" + }, + { + "bbox": [ + 262, + 226, + 285, + 238 + ], + "score": 0.84, + "content": "B B ^ { * }", + "type": "inline_equation" + }, + { + "bbox": [ + 285, + 225, + 352, + 241 + ], + "score": 1.0, + "content": "HIERARCHY", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 10 + }, + { + "type": "text", + "bbox": [ + 108, + 250, + 505, + 284 + ], + "lines": [ + { + "bbox": [ + 105, + 250, + 507, + 264 + ], + "spans": [ + { + "bbox": [ + 105, + 250, + 507, + 264 + ], + "score": 1.0, + "content": "In this appendix, we prove our main theoretical result, namely, our ability to capture general trans-", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 262, + 505, + 274 + ], + "spans": [ + { + "bbox": [ + 106, + 262, + 384, + 274 + ], + "score": 1.0, + "content": "formations, expressed as low-depth linear arithmetic circuits, in the", + "type": "text" + }, + { + "bbox": [ + 385, + 262, + 403, + 272 + ], + "score": 0.88, + "content": "B B ^ { * }", + "type": "inline_equation" + }, + { + "bbox": [ + 404, + 262, + 505, + 274 + ], + "score": 1.0, + "content": "hierarchy. This result is", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 273, + 203, + 284 + ], + "spans": [ + { + "bbox": [ + 106, + 273, + 203, + 284 + ], + "score": 1.0, + "content": "recorded in Theorem 1.", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 12 + }, + { + "type": "text", + "bbox": [ + 106, + 287, + 505, + 327 + ], + "lines": [ + { + "bbox": [ + 106, + 286, + 505, + 301 + ], + "spans": [ + { + "bbox": [ + 106, + 286, + 209, + 301 + ], + "score": 1.0, + "content": "Theorem 1. Let M be an", + "type": "text" + }, + { + "bbox": [ + 209, + 289, + 232, + 298 + ], + "score": 0.88, + "content": "n \\times n", + "type": "inline_equation" + }, + { + "bbox": [ + 232, + 286, + 505, + 301 + ], + "score": 1.0, + "content": "matrix such that matrix-vector multiplication of M times an arbitrary", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 298, + 505, + 312 + ], + "spans": [ + { + "bbox": [ + 105, + 298, + 363, + 312 + ], + "score": 1.0, + "content": "vector v can be represented as a be a linear arithmetic circuit", + "type": "text" + }, + { + "bbox": [ + 363, + 299, + 372, + 308 + ], + "score": 0.8, + "content": "C", + "type": "inline_equation" + }, + { + "bbox": [ + 372, + 298, + 429, + 312 + ], + "score": 1.0, + "content": "comprised of", + "type": "text" + }, + { + "bbox": [ + 429, + 300, + 435, + 308 + ], + "score": 0.31, + "content": "s", + "type": "inline_equation" + }, + { + "bbox": [ + 436, + 298, + 505, + 312 + ], + "score": 1.0, + "content": "gates (including", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 104, + 308, + 318, + 330 + ], + "spans": [ + { + "bbox": [ + 104, + 308, + 208, + 327 + ], + "score": 1.0, + "content": "inputs) and having depth", + "type": "text" + }, + { + "bbox": [ + 208, + 312, + 215, + 322 + ], + "score": 0.25, + "content": "d .", + "type": "inline_equation" + }, + { + "bbox": [ + 216, + 308, + 244, + 327 + ], + "score": 1.0, + "content": ". Then,", + "type": "text" + }, + { + "bbox": [ + 244, + 309, + 313, + 327 + ], + "score": 0.93, + "content": "\\mathbf { M } \\in \\left( B B ^ { * } \\right) _ { O ( \\frac { s } { n } ) } ^ { O ( d ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 314, + 314, + 318, + 330 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 15 + }, + { + "type": "text", + "bbox": [ + 106, + 335, + 373, + 347 + ], + "lines": [ + { + "bbox": [ + 105, + 333, + 372, + 349 + ], + "spans": [ + { + "bbox": [ + 105, + 333, + 372, + 349 + ], + "score": 1.0, + "content": "To prove Theorem 1, we make use of the following two theorems.", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 17 + }, + { + "type": "text", + "bbox": [ + 105, + 349, + 472, + 362 + ], + "lines": [ + { + "bbox": [ + 105, + 348, + 472, + 362 + ], + "spans": [ + { + "bbox": [ + 105, + 348, + 209, + 362 + ], + "score": 1.0, + "content": "Theorem 2. Let P be an", + "type": "text" + }, + { + "bbox": [ + 210, + 352, + 235, + 361 + ], + "score": 0.88, + "content": "n \\times n", + "type": "inline_equation" + }, + { + "bbox": [ + 235, + 348, + 429, + 362 + ], + "score": 1.0, + "content": "permutation matrix (with n a power of 2). Then", + "type": "text" + }, + { + "bbox": [ + 429, + 350, + 468, + 360 + ], + "score": 0.9, + "content": "\\mathbf { P } \\in B B ^ { * }", + "type": "inline_equation" + }, + { + "bbox": [ + 468, + 348, + 472, + 362 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 18 + }, + { + "type": "text", + "bbox": [ + 106, + 364, + 399, + 382 + ], + "lines": [ + { + "bbox": [ + 102, + 360, + 397, + 388 + ], + "spans": [ + { + "bbox": [ + 102, + 360, + 208, + 388 + ], + "score": 1.0, + "content": "Theorem 3. Let S be an", + "type": "text" + }, + { + "bbox": [ + 208, + 371, + 234, + 380 + ], + "score": 0.88, + "content": "n \\times n", + "type": "inline_equation" + }, + { + "bbox": [ + 234, + 360, + 327, + 388 + ], + "score": 1.0, + "content": "matrix of s NNZ. Then", + "type": "text" + }, + { + "bbox": [ + 327, + 364, + 392, + 382 + ], + "score": 0.92, + "content": "\\mathbf { S } \\in \\left( B B ^ { * } \\right) _ { 4 } ^ { 4 \\left\\lceil \\frac { s } { n } \\right\\rceil }", + "type": "inline_equation" + }, + { + "bbox": [ + 392, + 360, + 397, + 388 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 19 + }, + { + "type": "text", + "bbox": [ + 107, + 390, + 416, + 403 + ], + "lines": [ + { + "bbox": [ + 106, + 389, + 417, + 404 + ], + "spans": [ + { + "bbox": [ + 106, + 389, + 243, + 404 + ], + "score": 1.0, + "content": "Theorem 2 is proven in Appendix", + "type": "text" + }, + { + "bbox": [ + 243, + 391, + 252, + 401 + ], + "score": 0.31, + "content": "\\mathbf { G }", + "type": "inline_equation" + }, + { + "bbox": [ + 252, + 389, + 417, + 404 + ], + "score": 1.0, + "content": ", and Theorem 3 is proven in Appendix I.", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 20 + }, + { + "type": "text", + "bbox": [ + 107, + 414, + 505, + 437 + ], + "lines": [ + { + "bbox": [ + 105, + 414, + 505, + 426 + ], + "spans": [ + { + "bbox": [ + 105, + 414, + 262, + 426 + ], + "score": 1.0, + "content": "Proof of Theorem 1. We will represent", + "type": "text" + }, + { + "bbox": [ + 263, + 415, + 272, + 424 + ], + "score": 0.81, + "content": "C", + "type": "inline_equation" + }, + { + "bbox": [ + 272, + 414, + 333, + 426 + ], + "score": 1.0, + "content": "as a product of", + "type": "text" + }, + { + "bbox": [ + 334, + 415, + 340, + 424 + ], + "score": 0.78, + "content": "d", + "type": "inline_equation" + }, + { + "bbox": [ + 340, + 414, + 428, + 426 + ], + "score": 1.0, + "content": "matrices, each of size", + "type": "text" + }, + { + "bbox": [ + 428, + 414, + 456, + 425 + ], + "score": 0.93, + "content": "s ^ { \\prime } \\times s ^ { \\prime }", + "type": "inline_equation" + }, + { + "bbox": [ + 456, + 414, + 486, + 426 + ], + "score": 1.0, + "content": ", where", + "type": "text" + }, + { + "bbox": [ + 486, + 415, + 495, + 424 + ], + "score": 0.85, + "content": "s ^ { \\prime }", + "type": "inline_equation" + }, + { + "bbox": [ + 495, + 414, + 505, + 426 + ], + "score": 1.0, + "content": "is", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 425, + 333, + 438 + ], + "spans": [ + { + "bbox": [ + 106, + 425, + 323, + 438 + ], + "score": 1.0, + "content": "the smallest power of 2 that is greater than or equal to", + "type": "text" + }, + { + "bbox": [ + 324, + 428, + 329, + 435 + ], + "score": 0.71, + "content": "s", + "type": "inline_equation" + }, + { + "bbox": [ + 329, + 425, + 333, + 438 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 21.5 + }, + { + "type": "text", + "bbox": [ + 107, + 442, + 505, + 488 + ], + "lines": [ + { + "bbox": [ + 105, + 442, + 505, + 455 + ], + "spans": [ + { + "bbox": [ + 105, + 442, + 252, + 455 + ], + "score": 1.0, + "content": "To introduce some notation, define", + "type": "text" + }, + { + "bbox": [ + 252, + 444, + 294, + 453 + ], + "score": 0.87, + "content": "w _ { 1 } , \\dots , w _ { d }", + "type": "inline_equation" + }, + { + "bbox": [ + 295, + 442, + 335, + 455 + ], + "score": 1.0, + "content": "such that", + "type": "text" + }, + { + "bbox": [ + 336, + 444, + 349, + 453 + ], + "score": 0.85, + "content": "w _ { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 349, + 442, + 505, + 455 + ], + "score": 1.0, + "content": "represents the number of gates in the", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 453, + 507, + 472 + ], + "spans": [ + { + "bbox": [ + 106, + 456, + 114, + 465 + ], + "score": 0.74, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 114, + 453, + 163, + 472 + ], + "score": 1.0, + "content": "’th layer of", + "type": "text" + }, + { + "bbox": [ + 164, + 456, + 173, + 466 + ], + "score": 0.81, + "content": "C", + "type": "inline_equation" + }, + { + "bbox": [ + 173, + 453, + 219, + 472 + ], + "score": 1.0, + "content": "(note that", + "type": "text" + }, + { + "bbox": [ + 219, + 454, + 306, + 468 + ], + "score": 0.91, + "content": "\\begin{array} { r } { s = n + \\sum _ { k = 1 } ^ { d } w _ { k } ) } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 306, + 453, + 369, + 472 + ], + "score": 1.0, + "content": ". Also, define", + "type": "text" + }, + { + "bbox": [ + 369, + 457, + 407, + 467 + ], + "score": 0.88, + "content": "z _ { 1 } , \\dotsc z _ { d }", + "type": "inline_equation" + }, + { + "bbox": [ + 407, + 453, + 450, + 472 + ], + "score": 1.0, + "content": "such that", + "type": "text" + }, + { + "bbox": [ + 451, + 457, + 486, + 466 + ], + "score": 0.88, + "content": "z _ { 1 } ~ = ~ n", + "type": "inline_equation" + }, + { + "bbox": [ + 486, + 453, + 507, + 472 + ], + "score": 1.0, + "content": "and", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 466, + 506, + 479 + ], + "spans": [ + { + "bbox": [ + 106, + 468, + 183, + 478 + ], + "score": 0.89, + "content": "z _ { k } = w _ { k - 1 } + z _ { k - 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 184, + 466, + 189, + 479 + ], + "score": 1.0, + "content": "(", + "type": "text" + }, + { + "bbox": [ + 189, + 468, + 199, + 477 + ], + "score": 0.83, + "content": "z _ { k }", + "type": "inline_equation" + }, + 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This is a contradiction", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 187, + 506, + 201 + ], + "spans": [ + { + "bbox": [ + 106, + 188, + 129, + 201 + ], + "score": 1.0, + "content": "since", + "type": "text" + }, + { + "bbox": [ + 129, + 189, + 139, + 198 + ], + "score": 0.38, + "content": "\\mathbf { A }", + "type": "inline_equation" + }, + { + "bbox": [ + 139, + 188, + 196, + 201 + ], + "score": 1.0, + "content": "is an arbitrary", + "type": "text" + }, + { + "bbox": [ + 197, + 189, + 222, + 198 + ], + "score": 0.88, + "content": "n \\times n", + "type": "inline_equation" + }, + { + "bbox": [ + 222, + 188, + 322, + 201 + ], + "score": 1.0, + "content": "matrix, and therefore has", + "type": "text" + }, + { + "bbox": [ + 323, + 187, + 334, + 198 + ], + "score": 0.88, + "content": "n ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 335, + 188, + 506, + 201 + ], + "score": 1.0, + "content": "arbitrary parameters. Hence, there must be", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 104, + 197, + 506, + 213 + ], + "spans": [ + { + "bbox": [ + 104, + 197, + 130, + 213 + ], + "score": 1.0, + "content": "some", + "type": "text" + }, + { + "bbox": [ + 130, + 200, + 155, + 209 + ], + "score": 0.89, + "content": "n \\times n", + "type": "inline_equation" + }, + { + "bbox": [ + 156, + 197, + 196, + 213 + ], + "score": 1.0, + "content": "matrix in", + "type": "text" + }, + { + "bbox": [ + 196, + 198, + 236, + 211 + ], + "score": 0.92, + "content": "( B B ^ { * } ) ^ { c + 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 237, + 197, + 290, + 213 + ], + "score": 1.0, + "content": "that is not in", + "type": "text" + }, + { + "bbox": [ + 291, + 199, + 321, + 211 + ], + "score": 0.92, + "content": "( B B ^ { * } ) ^ { c }", + "type": "inline_equation" + }, + { + "bbox": [ + 321, + 197, + 325, + 213 + ], + "score": 1.0, + "content": ".", + "type": "text" + }, + { + "bbox": [ + 493, + 199, + 506, + 210 + ], + "score": 0.998, + "content": "□", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 5.5, + "bbox_fs": [ + 104, + 114, + 506, + 213 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 225, + 351, + 239 + ], + "lines": [ + { + "bbox": [ + 105, + 225, + 352, + 241 + ], + "spans": [ + { + "bbox": [ + 105, + 225, + 262, + 241 + ], + "score": 1.0, + "content": "F ARITHMETIC CIRCUITS IN", + "type": "text" + }, + { + "bbox": [ + 262, + 226, + 285, + 238 + ], + "score": 0.84, + "content": "B B ^ { * }", + "type": "inline_equation" + }, + { + "bbox": [ + 285, + 225, + 352, + 241 + ], + "score": 1.0, + "content": "HIERARCHY", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 10 + }, + { + "type": "text", + "bbox": [ + 108, + 250, + 505, + 284 + ], + "lines": [ + { + "bbox": [ + 105, + 250, + 507, + 264 + ], + "spans": [ + { + "bbox": [ + 105, + 250, + 507, + 264 + ], + "score": 1.0, + "content": "In this appendix, we prove our main theoretical result, namely, our ability to capture general trans-", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 262, + 505, + 274 + ], + "spans": [ + { + "bbox": [ + 106, + 262, + 384, + 274 + ], + "score": 1.0, + "content": "formations, expressed as low-depth linear arithmetic circuits, in the", + "type": "text" + }, + { + "bbox": [ + 385, + 262, + 403, + 272 + ], + "score": 0.88, + "content": "B B ^ { * }", + "type": "inline_equation" + }, + { + "bbox": [ + 404, + 262, + 505, + 274 + ], + "score": 1.0, + "content": "hierarchy. This result is", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 273, + 203, + 284 + ], + "spans": [ + { + "bbox": [ + 106, + 273, + 203, + 284 + ], + "score": 1.0, + "content": "recorded in Theorem 1.", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 12, + "bbox_fs": [ + 105, + 250, + 507, + 284 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 287, + 505, + 327 + ], + "lines": [ + { + "bbox": [ + 106, + 286, + 505, + 301 + ], + "spans": [ + { + "bbox": [ + 106, + 286, + 209, + 301 + ], + "score": 1.0, + "content": "Theorem 1. Let M be an", + "type": "text" + }, + { + "bbox": [ + 209, + 289, + 232, + 298 + ], + "score": 0.88, + "content": "n \\times n", + "type": "inline_equation" + }, + { + "bbox": [ + 232, + 286, + 505, + 301 + ], + "score": 1.0, + "content": "matrix such that matrix-vector multiplication of M times an arbitrary", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 298, + 505, + 312 + ], + "spans": [ + { + "bbox": [ + 105, + 298, + 363, + 312 + ], + "score": 1.0, + "content": "vector v can be represented as a be a linear arithmetic circuit", + "type": "text" + }, + { + "bbox": [ + 363, + 299, + 372, + 308 + ], + "score": 0.8, + "content": "C", + "type": "inline_equation" + }, + { + "bbox": [ + 372, + 298, + 429, + 312 + ], + "score": 1.0, + "content": "comprised of", + "type": "text" + }, + { + "bbox": [ + 429, + 300, + 435, + 308 + ], + "score": 0.31, + "content": "s", + "type": "inline_equation" + }, + { + "bbox": [ + 436, + 298, + 505, + 312 + ], + "score": 1.0, + "content": "gates (including", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 104, + 308, + 318, + 330 + ], + "spans": [ + { + "bbox": [ + 104, + 308, + 208, + 327 + ], + "score": 1.0, + "content": "inputs) and having depth", + "type": "text" + }, + { + "bbox": [ + 208, + 312, + 215, + 322 + ], + "score": 0.25, + "content": "d .", + "type": "inline_equation" + }, + { + "bbox": [ + 216, + 308, + 244, + 327 + ], + "score": 1.0, + "content": ". Then,", + "type": "text" + }, + { + "bbox": [ + 244, + 309, + 313, + 327 + ], + "score": 0.93, + "content": "\\mathbf { M } \\in \\left( B B ^ { * } \\right) _ { O ( \\frac { s } { n } ) } ^ { O ( d ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 314, + 314, + 318, + 330 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 15, + "bbox_fs": [ + 104, + 286, + 505, + 330 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 335, + 373, + 347 + ], + "lines": [ + { + "bbox": [ + 105, + 333, + 372, + 349 + ], + "spans": [ + { + "bbox": [ + 105, + 333, + 372, + 349 + ], + "score": 1.0, + "content": "To prove Theorem 1, we make use of the following two theorems.", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 17, + "bbox_fs": [ + 105, + 333, + 372, + 349 + ] + }, + { + "type": "text", + "bbox": [ + 105, + 349, + 472, + 362 + ], + "lines": [ + { + "bbox": [ + 105, + 348, + 472, + 362 + ], + "spans": [ + { + "bbox": [ + 105, + 348, + 209, + 362 + ], + "score": 1.0, + "content": "Theorem 2. Let P be an", + "type": "text" + }, + { + "bbox": [ + 210, + 352, + 235, + 361 + ], + "score": 0.88, + "content": "n \\times n", + "type": "inline_equation" + }, + { + "bbox": [ + 235, + 348, + 429, + 362 + ], + "score": 1.0, + "content": "permutation matrix (with n a power of 2). Then", + "type": "text" + }, + { + "bbox": [ + 429, + 350, + 468, + 360 + ], + "score": 0.9, + "content": "\\mathbf { P } \\in B B ^ { * }", + "type": "inline_equation" + }, + { + "bbox": [ + 468, + 348, + 472, + 362 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 18, + "bbox_fs": [ + 105, + 348, + 472, + 362 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 364, + 399, + 382 + ], + "lines": [ + { + "bbox": [ + 102, + 360, + 397, + 388 + ], + "spans": [ + { + "bbox": [ + 102, + 360, + 208, + 388 + ], + "score": 1.0, + "content": "Theorem 3. Let S be an", + "type": "text" + }, + { + "bbox": [ + 208, + 371, + 234, + 380 + ], + "score": 0.88, + "content": "n \\times n", + "type": "inline_equation" + }, + { + "bbox": [ + 234, + 360, + 327, + 388 + ], + "score": 1.0, + "content": "matrix of s NNZ. Then", + "type": "text" + }, + { + "bbox": [ + 327, + 364, + 392, + 382 + ], + "score": 0.92, + "content": "\\mathbf { S } \\in \\left( B B ^ { * } \\right) _ { 4 } ^ { 4 \\left\\lceil \\frac { s } { n } \\right\\rceil }", + "type": "inline_equation" + }, + { + "bbox": [ + 392, + 360, + 397, + 388 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 19, + "bbox_fs": [ + 102, + 360, + 397, + 388 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 390, + 416, + 403 + ], + "lines": [ + { + "bbox": [ + 106, + 389, + 417, + 404 + ], + "spans": [ + { + "bbox": [ + 106, + 389, + 243, + 404 + ], + "score": 1.0, + "content": "Theorem 2 is proven in Appendix", + "type": "text" + }, + { + "bbox": [ + 243, + 391, + 252, + 401 + ], + "score": 0.31, + "content": "\\mathbf { G }", + "type": "inline_equation" + }, + { + "bbox": [ + 252, + 389, + 417, + 404 + ], + "score": 1.0, + "content": ", and Theorem 3 is proven in Appendix I.", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 20, + "bbox_fs": [ + 106, + 389, + 417, + 404 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 414, + 505, + 437 + ], + "lines": [ + { + "bbox": [ + 105, + 414, + 505, + 426 + ], + "spans": [ + { + "bbox": [ + 105, + 414, + 262, + 426 + ], + "score": 1.0, + "content": "Proof of Theorem 1. We will represent", + "type": "text" + }, + { + "bbox": [ + 263, + 415, + 272, + 424 + ], + "score": 0.81, + "content": "C", + "type": "inline_equation" + }, + { + "bbox": [ + 272, + 414, + 333, + 426 + ], + "score": 1.0, + "content": "as a product of", + "type": "text" + }, + { + "bbox": [ + 334, + 415, + 340, + 424 + ], + "score": 0.78, + "content": "d", + "type": "inline_equation" + }, + { + "bbox": [ + 340, + 414, + 428, + 426 + ], + "score": 1.0, + "content": "matrices, each of size", + "type": "text" + }, + { + "bbox": [ + 428, + 414, + 456, + 425 + ], + "score": 0.93, + "content": "s ^ { \\prime } \\times s ^ { \\prime }", + "type": "inline_equation" + }, + { + "bbox": [ + 456, + 414, + 486, + 426 + ], + "score": 1.0, + "content": ", where", + "type": "text" + }, + { + "bbox": [ + 486, + 415, + 495, + 424 + ], + "score": 0.85, + "content": "s ^ { \\prime }", + "type": "inline_equation" + }, + { + "bbox": [ + 495, + 414, + 505, + 426 + ], + "score": 1.0, + "content": "is", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 425, + 333, + 438 + ], + "spans": [ + { + "bbox": [ + 106, + 425, + 323, + 438 + ], + "score": 1.0, + "content": "the smallest power of 2 that is greater than or equal to", + "type": "text" + }, + { + "bbox": [ + 324, + 428, + 329, + 435 + ], + "score": 0.71, + "content": "s", + "type": "inline_equation" + }, + { + "bbox": [ + 329, + 425, + 333, + 438 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 21.5, + "bbox_fs": [ + 105, + 414, + 505, + 438 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 442, + 505, + 488 + ], + "lines": [ + { + "bbox": [ + 105, + 442, + 505, + 455 + ], + "spans": [ + { + "bbox": [ + 105, + 442, + 252, + 455 + ], + "score": 1.0, + "content": "To introduce some notation, define", + "type": "text" + }, + { + "bbox": [ + 252, + 444, + 294, + 453 + ], + "score": 0.87, + "content": "w _ { 1 } , \\dots , w _ { d }", + "type": "inline_equation" + }, + { + "bbox": [ + 295, + 442, + 335, + 455 + ], + "score": 1.0, + "content": "such that", + "type": "text" + }, + { + "bbox": [ + 336, + 444, + 349, + 453 + ], + "score": 0.85, + "content": "w _ { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 349, + 442, + 505, + 455 + ], + "score": 1.0, + "content": "represents the number of gates in the", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 453, + 507, + 472 + ], + "spans": [ + { + "bbox": [ + 106, + 456, + 114, + 465 + ], + "score": 0.74, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 114, + 453, + 163, + 472 + ], + "score": 1.0, + "content": "’th layer of", + "type": "text" + }, + { + "bbox": [ + 164, + 456, + 173, + 466 + ], + "score": 0.81, + "content": "C", + "type": "inline_equation" + }, + { + "bbox": [ + 173, + 453, + 219, + 472 + ], + "score": 1.0, + "content": "(note that", + "type": "text" + }, + { + "bbox": [ + 219, + 454, + 306, + 468 + ], + "score": 0.91, + "content": "\\begin{array} { r } { s = n + \\sum _ { k = 1 } ^ { d } w _ { k } ) } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 306, + 453, + 369, + 472 + ], + "score": 1.0, + "content": ". Also, define", + "type": "text" + }, + { + "bbox": [ + 369, + 457, + 407, + 467 + ], + "score": 0.88, + "content": "z _ { 1 } , \\dotsc z _ { d }", + "type": "inline_equation" + }, + { + "bbox": [ + 407, + 453, + 450, + 472 + ], + "score": 1.0, + "content": "such that", + "type": "text" + }, + { + "bbox": [ + 451, + 457, + 486, + 466 + ], + "score": 0.88, + "content": "z _ { 1 } ~ = ~ n", + "type": "inline_equation" + }, + { + "bbox": [ + 486, + 453, + 507, + 472 + ], + "score": 1.0, + "content": "and", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 466, + 506, + 479 + ], + "spans": [ + { + "bbox": [ + 106, + 468, + 183, + 478 + ], + "score": 0.89, + "content": "z _ { k } = w _ { k - 1 } + z _ { k - 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 184, + 466, + 189, + 479 + ], + "score": 1.0, + "content": "(", + "type": "text" + }, + { + "bbox": [ + 189, + 468, + 199, + 477 + ], + "score": 0.83, + "content": "z _ { k }", + "type": "inline_equation" + }, + 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"score": 1.0, + "content": "denote the", + "type": "text" + }, + { + "bbox": [ + 177, + 495, + 182, + 504 + ], + "score": 0.74, + "content": "i ^ { \\because }", + "type": "inline_equation" + }, + { + "bbox": [ + 183, + 493, + 286, + 507 + ], + "score": 1.0, + "content": "’th gate (and its output) of", + "type": "text" + }, + { + "bbox": [ + 287, + 495, + 295, + 504 + ], + "score": 0.77, + "content": "C", + "type": "inline_equation" + }, + { + "bbox": [ + 296, + 493, + 300, + 507 + ], + "score": 0.0, + "content": "", + "type": "text" + }, + { + "bbox": [ + 300, + 494, + 342, + 505 + ], + "score": 0.85, + "content": "0 \\leq i < s", + "type": "inline_equation" + }, + { + "bbox": [ + 342, + 493, + 421, + 507 + ], + "score": 1.0, + "content": "), defined such that:", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 27, + "bbox_fs": [ + 105, + 493, + 421, + 507 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 236, + 511, + 375, + 539 + ], + "lines": [ + { + "bbox": [ + 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Each", + "type": "text" + }, + { + "bbox": [ + 443, + 133, + 460, + 144 + ], + "score": 0.9, + "content": "{ { \\bf { M } } _ { k } }", + "type": "inline_equation" + }, + { + "bbox": [ + 460, + 132, + 506, + 145 + ], + "score": 1.0, + "content": "is a matrix", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 143, + 506, + 156 + ], + "spans": [ + { + "bbox": [ + 105, + 143, + 147, + 156 + ], + "score": 1.0, + "content": "with total", + "type": "text" + }, + { + "bbox": [ + 148, + 144, + 215, + 154 + ], + "score": 0.9, + "content": "2 w _ { k } + z _ { k } < 2 s ^ { \\prime }", + "type": "inline_equation" + }, + { + "bbox": [ + 215, + 143, + 434, + 156 + ], + "score": 1.0, + "content": "NNZ. From Theorem 3, we can, therefore, represent", + "type": "text" + }, + { + "bbox": [ + 434, + 144, + 451, + 154 + ], + "score": 0.88, + "content": "{ { \\bf { M } } _ { k } }", + "type": "inline_equation" + }, + { + "bbox": [ + 452, + 143, + 506, + 156 + ], + "score": 1.0, + "content": "as a product", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 104, + 152, + 506, + 167 + ], + "spans": [ + { + "bbox": [ + 104, + 152, + 118, + 167 + ], + "score": 1.0, + "content": "of", + "type": "text" + }, + { + "bbox": [ + 118, + 154, + 140, + 165 + ], + "score": 0.86, + "content": "O ( 1 )", + "type": "inline_equation" + }, + { + "bbox": [ + 140, + 152, + 213, + 167 + ], + "score": 1.0, + "content": "matrices (of size", + "type": "text" + }, + { + "bbox": [ + 213, + 154, + 227, + 164 + ], + "score": 0.8, + "content": "2 s ^ { \\prime }", + "type": "inline_equation" + }, + { + "bbox": [ + 227, + 152, + 243, + 167 + ], + "score": 1.0, + "content": ") in", + "type": "text" + }, + { + "bbox": [ + 243, + 154, + 262, + 164 + ], + "score": 0.88, + "content": "B B ^ { * }", + "type": "inline_equation" + }, + { + "bbox": [ + 263, + 152, + 344, + 167 + ], + "score": 1.0, + "content": ". From Theorem 2,", + "type": "text" + }, + { + "bbox": [ + 345, + 154, + 387, + 165 + ], + "score": 0.89, + "content": "\\mathbf { P } \\in B B ^ { * }", + "type": "inline_equation" + }, + { + "bbox": [ + 387, + 152, + 434, + 167 + ], + "score": 1.0, + "content": ". Note that", + "type": "text" + }, + { + "bbox": [ + 435, + 154, + 489, + 165 + ], + "score": 0.91, + "content": "s \\leq s ^ { \\prime } < 2 s", + "type": "inline_equation" + }, + { + "bbox": [ + 489, + 152, + 506, + 167 + ], + "score": 1.0, + "content": ", so", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 107, + 163, + 154, + 180 + ], + "spans": [ + { + "bbox": [ + 107, + 165, + 148, + 177 + ], + "score": 0.91, + "content": "s ^ { \\prime } = \\Theta ( s )", + "type": "inline_equation" + }, + { + "bbox": [ + 149, + 163, + 154, + 180 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 4.5 + }, + { + "type": "text", + "bbox": [ + 106, + 181, + 504, + 210 + ], + "lines": [ + { + "bbox": [ + 106, + 181, + 506, + 195 + ], + "spans": [ + { + "bbox": [ + 106, + 181, + 245, + 195 + ], + "score": 1.0, + "content": "Our final decomposition will have", + "type": "text" + }, + { + "bbox": [ + 246, + 182, + 288, + 194 + ], + "score": 0.86, + "content": "O ( d ) B B ^ { * }", + "type": "inline_equation" + }, + { + "bbox": [ + 288, + 181, + 468, + 195 + ], + "score": 1.0, + "content": "factors, and requires an expansion from size", + "type": "text" + }, + { + "bbox": [ + 468, + 185, + 475, + 192 + ], + "score": 0.75, + "content": "n", + "type": "inline_equation" + }, + { + "bbox": [ + 476, + 181, + 506, + 195 + ], + "score": 1.0, + "content": "to size", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 192, + 506, + 212 + ], + "spans": [ + { + "bbox": [ + 106, + 195, + 120, + 206 + ], + "score": 0.86, + "content": "2 s ^ { \\prime }", + "type": "inline_equation" + }, + { + "bbox": [ + 120, + 192, + 225, + 212 + ], + "score": 1.0, + "content": ", or an expansion factor of", + "type": "text" + }, + { + "bbox": [ + 226, + 196, + 249, + 209 + ], + "score": 0.92, + "content": "O ( { \\textstyle { \\frac { s } { n } } } )", + "type": "inline_equation" + }, + { + "bbox": [ + 250, + 192, + 298, + 212 + ], + "score": 1.0, + "content": ". Therefore,", + "type": "text" + }, + { + "bbox": [ + 299, + 193, + 368, + 211 + ], + "score": 0.93, + "content": "\\mathbf { M } \\in \\left( B B ^ { * } \\right) _ { O ( \\frac { s } { n } ) } ^ { O ( d ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 368, + 192, + 416, + 212 + ], + "score": 1.0, + "content": ", as desired.", + "type": "text" + }, + { + "bbox": [ + 494, + 195, + 506, + 208 + ], + "score": 0.998, + "content": "□", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 8.5 + }, + { + "type": "text", + "bbox": [ + 106, + 219, + 505, + 242 + ], + "lines": [ + { + "bbox": [ + 106, + 219, + 506, + 233 + ], + "spans": [ + { + "bbox": [ + 106, + 219, + 271, + 233 + ], + "score": 1.0, + "content": "Remark F.1. By applying Observation", + "type": "text" + }, + { + "bbox": [ + 271, + 221, + 286, + 230 + ], + "score": 0.61, + "content": "E . 2", + "type": "inline_equation" + }, + { + "bbox": [ + 286, + 219, + 380, + 233 + ], + "score": 1.0, + "content": ", we see that Theorem", + "type": "text" + }, + { + "bbox": [ + 380, + 221, + 387, + 230 + ], + "score": 0.43, + "content": "I", + "type": "inline_equation" + }, + { + "bbox": [ + 387, + 219, + 426, + 233 + ], + "score": 1.0, + "content": "gives an", + "type": "text" + }, + { + "bbox": [ + 426, + 220, + 474, + 232 + ], + "score": 0.91, + "content": "O ( s d \\log s )", + "type": "inline_equation" + }, + { + "bbox": [ + 474, + 219, + 506, + 233 + ], + "score": 1.0, + "content": "matrix", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 231, + 263, + 243 + ], + "spans": [ + { + "bbox": [ + 106, + 231, + 248, + 243 + ], + "score": 1.0, + "content": "vector multiplication algorithm for", + "type": "text" + }, + { + "bbox": [ + 248, + 231, + 261, + 241 + ], + "score": 0.25, + "content": "\\mathbf { M }", + "type": "inline_equation" + }, + { + "bbox": [ + 261, + 231, + 263, + 243 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 10.5 + }, + { + "type": "title", + "bbox": [ + 108, + 258, + 247, + 272 + ], + "lines": [ + { + "bbox": [ + 106, + 258, + 248, + 274 + ], + "spans": [ + { + "bbox": [ + 106, + 258, + 225, + 274 + ], + "score": 1.0, + "content": "G PERMUTATIONS IN", + "type": "text" + }, + { + "bbox": [ + 225, + 259, + 248, + 271 + ], + "score": 0.78, + "content": "B B ^ { * }", + "type": "inline_equation" + } + ], + "index": 12 + } + ], + "index": 12 + }, + { + "type": "text", + "bbox": [ + 107, + 284, + 506, + 318 + ], + "lines": [ + { + "bbox": [ + 105, + 284, + 506, + 297 + ], + "spans": [ + { + "bbox": [ + 105, + 284, + 483, + 297 + ], + "score": 1.0, + "content": "In this appendix, we prove Theorem 2. In addition, we will also show that permutations are in", + "type": "text" + }, + { + "bbox": [ + 483, + 285, + 502, + 295 + ], + "score": 0.89, + "content": "B ^ { * } B", + "type": "inline_equation" + }, + { + "bbox": [ + 503, + 284, + 506, + 297 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 295, + 505, + 308 + ], + "spans": [ + { + "bbox": [ + 105, + 295, + 162, + 308 + ], + "score": 1.0, + "content": "where the set", + "type": "text" + }, + { + "bbox": [ + 162, + 296, + 182, + 306 + ], + "score": 0.9, + "content": "B ^ { * } B", + "type": "inline_equation" + }, + { + "bbox": [ + 182, + 295, + 286, + 308 + ], + "score": 1.0, + "content": "is defined analogously to", + "type": "text" + }, + { + "bbox": [ + 286, + 296, + 305, + 306 + ], + "score": 0.89, + "content": "B B ^ { * }", + "type": "inline_equation" + }, + { + "bbox": [ + 306, + 295, + 409, + 308 + ], + "score": 1.0, + "content": "(i.e. matrices of the form", + "type": "text" + }, + { + "bbox": [ + 410, + 296, + 466, + 308 + ], + "score": 0.93, + "content": "\\mathbf { M } = \\mathbf { M } _ { 1 } ^ { * } \\mathbf { M } _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 466, + 295, + 505, + 308 + ], + "score": 1.0, + "content": "for some", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 107, + 306, + 169, + 320 + ], + "spans": [ + { + "bbox": [ + 107, + 307, + 164, + 318 + ], + "score": 0.86, + "content": "\\mathbf { M } _ { 1 } , \\mathbf { M } _ { 2 } \\in B )", + "type": "inline_equation" + }, + { + "bbox": [ + 164, + 306, + 169, + 320 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 14 + }, + { + "type": "text", + "bbox": [ + 107, + 323, + 503, + 346 + ], + "lines": [ + { + "bbox": [ + 105, + 323, + 502, + 336 + ], + "spans": [ + { + "bbox": [ + 105, + 323, + 333, + 336 + ], + "score": 1.0, + "content": "To prove Theorem 2, we decompose permutation matrix", + "type": "text" + }, + { + "bbox": [ + 334, + 324, + 343, + 334 + ], + "score": 0.7, + "content": "\\mathbf { P }", + "type": "inline_equation" + }, + { + "bbox": [ + 343, + 323, + 362, + 336 + ], + "score": 1.0, + "content": "into", + "type": "text" + }, + { + "bbox": [ + 362, + 324, + 400, + 334 + ], + "score": 0.89, + "content": "\\mathbf { P } = \\mathbf { L R }", + "type": "inline_equation" + }, + { + "bbox": [ + 400, + 323, + 424, + 336 + ], + "score": 1.0, + "content": ", with", + "type": "text" + }, + { + "bbox": [ + 424, + 324, + 452, + 334 + ], + "score": 0.92, + "content": "\\mathbf { L } \\in B", + "type": "inline_equation" + }, + { + "bbox": [ + 452, + 323, + 469, + 336 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 470, + 324, + 502, + 334 + ], + "score": 0.9, + "content": "\\mathbf { R } \\in B ^ { * }", + "type": "inline_equation" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 334, + 360, + 347 + ], + "spans": [ + { + "bbox": [ + 106, + 334, + 360, + 347 + ], + "score": 1.0, + "content": "Throughout the proof, we make use of the following definition.", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 16.5 + }, + { + "type": "text", + "bbox": [ + 106, + 349, + 505, + 408 + ], + "lines": [ + { + "bbox": [ + 105, + 349, + 505, + 363 + ], + "spans": [ + { + "bbox": [ + 105, + 349, + 224, + 363 + ], + "score": 1.0, + "content": "Definition G.1. Let L be an", + "type": "text" + }, + { + "bbox": [ + 224, + 352, + 250, + 361 + ], + "score": 0.88, + "content": "n \\times n", + "type": "inline_equation" + }, + { + "bbox": [ + 250, + 349, + 505, + 363 + ], + "score": 1.0, + "content": "permutation matrix (n a power of 2). We say that L meets the", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 359, + 506, + 374 + ], + "spans": [ + { + "bbox": [ + 106, + 361, + 117, + 371 + ], + "score": 0.8, + "content": "2 ^ { j }", + "type": "inline_equation" + }, + { + "bbox": [ + 117, + 359, + 338, + 374 + ], + "score": 1.0, + "content": "balance condition if L can be divided into chunks of", + "type": "text" + }, + { + "bbox": [ + 339, + 361, + 349, + 371 + ], + "score": 0.85, + "content": "2 ^ { j }", + "type": "inline_equation" + }, + { + "bbox": [ + 349, + 359, + 506, + 374 + ], + "score": 1.0, + "content": "(with each chunk having all columns", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 369, + 506, + 388 + ], + "spans": [ + { + "bbox": [ + 106, + 373, + 111, + 382 + ], + "score": 0.32, + "content": "i", + "type": "inline_equation" + }, + { + "bbox": [ + 112, + 369, + 155, + 388 + ], + "score": 1.0, + "content": "such that", + "type": "text" + }, + { + "bbox": [ + 155, + 371, + 174, + 385 + ], + "score": 0.91, + "content": "\\left\\lfloor { \\frac { i } { 2 ^ { j } } } \\right\\rfloor", + "type": "inline_equation" + }, + { + "bbox": [ + 175, + 369, + 344, + 388 + ], + "score": 1.0, + "content": "has the same value) such that for every", + "type": "text" + }, + { + "bbox": [ + 345, + 372, + 401, + 383 + ], + "score": 0.91, + "content": "0 \\leq m < 2 ^ { j }", + "type": "inline_equation" + }, + { + "bbox": [ + 402, + 369, + 506, + 388 + ], + "score": 1.0, + "content": ", each chunk has exactly", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 384, + 504, + 397 + ], + "spans": [ + { + "bbox": [ + 105, + 384, + 124, + 397 + ], + "score": 1.0, + "content": "one", + "type": "text" + }, + { + "bbox": [ + 124, + 385, + 179, + 397 + ], + "score": 0.9, + "content": "\\mathbf { L } [ : , k ] = \\mathbf { e } _ { \\pi _ { k } }", + "type": "inline_equation" + }, + { + "bbox": [ + 179, + 384, + 200, + 397 + ], + "score": 1.0, + "content": "with", + "type": "text" + }, + { + "bbox": [ + 201, + 385, + 236, + 396 + ], + "score": 0.85, + "content": "\\pi _ { k } \\equiv m", + "type": "inline_equation" + }, + { + "bbox": [ + 236, + 384, + 269, + 397 + ], + "score": 1.0, + "content": "( mod", + "type": "text" + }, + { + "bbox": [ + 270, + 384, + 280, + 395 + ], + "score": 0.37, + "content": "2 ^ { j }", + "type": "inline_equation" + }, + { + "bbox": [ + 281, + 384, + 493, + 397 + ], + "score": 1.0, + "content": "). We say that L is modular-balanced if it meets the", + "type": "text" + }, + { + "bbox": [ + 493, + 384, + 504, + 394 + ], + "score": 0.77, + "content": "2 ^ { j }", + "type": "inline_equation" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 394, + 268, + 409 + ], + "spans": [ + { + "bbox": [ + 105, + 394, + 216, + 409 + ], + "score": 1.0, + "content": "balance condition for each", + "type": "text" + }, + { + "bbox": [ + 216, + 396, + 263, + 407 + ], + "score": 0.91, + "content": "2 \\leq 2 ^ { \\overset { . } { j } } \\leq n", + "type": "inline_equation" + }, + { + "bbox": [ + 264, + 394, + 268, + 409 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 20 + }, + { + "type": "image", + "bbox": [ + 138, + 424, + 473, + 552 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 138, + 424, + 473, + 552 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 138, + 424, + 473, + 552 + ], + "spans": [ + { + "bbox": [ + 138, + 424, + 473, + 552 + ], + "score": 0.97, + "type": "image", + "image_path": "e1a749d098c79becb5562ba1896bfb4d508efdd4f05072b76cda0be851a989ca.jpg" + } + ] + } + ], + "index": 24, + "virtual_lines": [ + { + "bbox": [ + 138, + 424, + 473, + 466.6666666666667 + ], + "spans": [], + "index": 23 + }, + { + "bbox": [ + 138, + 466.6666666666667, + 473, + 509.33333333333337 + ], + "spans": [], + "index": 24 + }, + { + "bbox": [ + 138, + 509.33333333333337, + 473, + 552.0 + ], + "spans": [], + "index": 25 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 106, + 565, + 505, + 588 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 106, + 565, + 505, + 578 + ], + "spans": [ + { + "bbox": [ + 106, + 565, + 505, + 578 + ], + "score": 1.0, + "content": "Figure 7: First step of decomposition of modular-balanced matrix L. Here, the red entries must be", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 577, + 269, + 589 + ], + "spans": [ + { + "bbox": [ + 105, + 577, + 269, + 589 + ], + "score": 1.0, + "content": "permuted into the main diagonal blocks.", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 26.5 + } + ], + "index": 25.25 + }, + { + "type": "text", + "bbox": [ + 106, + 603, + 399, + 615 + ], + "lines": [ + { + "bbox": [ + 106, + 603, + 400, + 617 + ], + "spans": [ + { + "bbox": [ + 106, + 603, + 213, + 617 + ], + "score": 1.0, + "content": "Lemma G.1. Let L be an", + "type": "text" + }, + { + "bbox": [ + 213, + 605, + 239, + 614 + ], + "score": 0.88, + "content": "n \\times n", + "type": "inline_equation" + }, + { + "bbox": [ + 239, + 603, + 369, + 617 + ], + "score": 1.0, + "content": "modular-balanced matrix. Then", + "type": "text" + }, + { + "bbox": [ + 370, + 604, + 397, + 614 + ], + "score": 0.86, + "content": "\\mathbf { L } \\in B", + "type": "inline_equation" + }, + { + "bbox": [ + 397, + 603, + 400, + 617 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 28 + }, + { + "type": "text", + "bbox": [ + 107, + 630, + 506, + 664 + ], + "lines": [ + { + "bbox": [ + 105, + 629, + 507, + 643 + ], + "spans": [ + { + "bbox": [ + 105, + 629, + 252, + 643 + ], + "score": 1.0, + "content": "Proof. We proceed by induction on", + "type": "text" + }, + { + "bbox": [ + 253, + 633, + 259, + 640 + ], + "score": 0.69, + "content": "n", + "type": "inline_equation" + }, + { + "bbox": [ + 260, + 629, + 322, + 643 + ], + "score": 1.0, + "content": ". The base case", + "type": "text" + }, + { + "bbox": [ + 323, + 631, + 348, + 640 + ], + "score": 0.88, + "content": "n = 2", + "type": "inline_equation" + }, + { + "bbox": [ + 349, + 629, + 507, + 643 + ], + "score": 1.0, + "content": "is trivial. As our inductive hypothesis,", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 102, + 637, + 505, + 659 + ], + "spans": [ + { + "bbox": [ + 102, + 637, + 323, + 659 + ], + "score": 1.0, + "content": "we assume that all modular-balanced matrices of size", + "type": "text" + }, + { + "bbox": [ + 323, + 642, + 350, + 654 + ], + "score": 0.92, + "content": "{ \\frac { n } { 2 } } \\times { \\frac { n } { 2 } }", + "type": "inline_equation" + }, + { + "bbox": [ + 351, + 637, + 468, + 659 + ], + "score": 1.0, + "content": "are butterfly matrices of size", + "type": "text" + }, + { + "bbox": [ + 469, + 642, + 476, + 654 + ], + "score": 0.85, + "content": "\\frac { n } { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 477, + 637, + 505, + 659 + ], + "score": 1.0, + "content": ". 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Each", + "type": "text" + }, + { + "bbox": [ + 443, + 133, + 460, + 144 + ], + "score": 0.9, + "content": "{ { \\bf { M } } _ { k } }", + "type": "inline_equation" + }, + { + "bbox": [ + 460, + 132, + 506, + 145 + ], + "score": 1.0, + "content": "is a matrix", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 143, + 506, + 156 + ], + "spans": [ + { + "bbox": [ + 105, + 143, + 147, + 156 + ], + "score": 1.0, + "content": "with total", + "type": "text" + }, + { + "bbox": [ + 148, + 144, + 215, + 154 + ], + "score": 0.9, + "content": "2 w _ { k } + z _ { k } < 2 s ^ { \\prime }", + "type": "inline_equation" + }, + { + "bbox": [ + 215, + 143, + 434, + 156 + ], + "score": 1.0, + "content": "NNZ. 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Note that", + "type": "text" + }, + { + "bbox": [ + 435, + 154, + 489, + 165 + ], + "score": 0.91, + "content": "s \\leq s ^ { \\prime } < 2 s", + "type": "inline_equation" + }, + { + "bbox": [ + 489, + 152, + 506, + 167 + ], + "score": 1.0, + "content": ", so", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 107, + 163, + 154, + 180 + ], + "spans": [ + { + "bbox": [ + 107, + 165, + 148, + 177 + ], + "score": 0.91, + "content": "s ^ { \\prime } = \\Theta ( s )", + "type": "inline_equation" + }, + { + "bbox": [ + 149, + 163, + 154, + 180 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 4.5, + "bbox_fs": [ + 104, + 109, + 507, + 180 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 181, + 504, + 210 + ], + "lines": [ + { + "bbox": [ + 106, + 181, + 506, + 195 + ], + "spans": [ + { + "bbox": [ + 106, + 181, + 245, + 195 + ], + "score": 1.0, + "content": "Our final decomposition will have", + "type": "text" + }, + { + "bbox": [ + 246, + 182, + 288, + 194 + ], + "score": 0.86, + "content": "O ( d ) B B ^ { * }", + "type": "inline_equation" + }, + { + "bbox": [ + 288, + 181, + 468, + 195 + ], + "score": 1.0, + "content": "factors, and requires an expansion from size", + "type": "text" + }, + { + "bbox": [ + 468, + 185, + 475, + 192 + ], + "score": 0.75, + "content": "n", + "type": "inline_equation" + }, + { + "bbox": [ + 476, + 181, + 506, + 195 + ], + "score": 1.0, + "content": "to size", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 192, + 506, + 212 + ], + "spans": [ + { + "bbox": [ + 106, + 195, + 120, + 206 + ], + "score": 0.86, + "content": "2 s ^ { \\prime }", + "type": "inline_equation" + }, + { + "bbox": [ + 120, + 192, + 225, + 212 + ], + "score": 1.0, + "content": ", or an expansion factor of", + "type": "text" + }, + { + "bbox": [ + 226, + 196, + 249, + 209 + ], + "score": 0.92, + "content": "O ( { \\textstyle { \\frac { s } { n } } } )", + "type": "inline_equation" + }, + { + "bbox": [ + 250, + 192, + 298, + 212 + ], + "score": 1.0, + "content": ". 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By applying Observation", + "type": "text" + }, + { + "bbox": [ + 271, + 221, + 286, + 230 + ], + "score": 0.61, + "content": "E . 2", + "type": "inline_equation" + }, + { + "bbox": [ + 286, + 219, + 380, + 233 + ], + "score": 1.0, + "content": ", we see that Theorem", + "type": "text" + }, + { + "bbox": [ + 380, + 221, + 387, + 230 + ], + "score": 0.43, + "content": "I", + "type": "inline_equation" + }, + { + "bbox": [ + 387, + 219, + 426, + 233 + ], + "score": 1.0, + "content": "gives an", + "type": "text" + }, + { + "bbox": [ + 426, + 220, + 474, + 232 + ], + "score": 0.91, + "content": "O ( s d \\log s )", + "type": "inline_equation" + }, + { + "bbox": [ + 474, + 219, + 506, + 233 + ], + "score": 1.0, + "content": "matrix", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 231, + 263, + 243 + ], + "spans": [ + { + "bbox": [ + 106, + 231, + 248, + 243 + ], + "score": 1.0, + "content": "vector multiplication algorithm for", + "type": "text" + }, + { + "bbox": [ + 248, + 231, + 261, + 241 + ], + "score": 0.25, + "content": "\\mathbf { M }", + "type": "inline_equation" + }, + { + "bbox": [ + 261, + 231, + 263, + 243 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 10.5, + "bbox_fs": [ + 106, + 219, + 506, + 243 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 258, + 247, + 272 + ], + "lines": [ + { + "bbox": [ + 106, + 258, + 248, + 274 + ], + "spans": [ + { + "bbox": [ + 106, + 258, + 225, + 274 + ], + "score": 1.0, + "content": "G PERMUTATIONS IN", + "type": "text" + }, + { + "bbox": [ + 225, + 259, + 248, + 271 + ], + "score": 0.78, + "content": "B B ^ { * }", + "type": "inline_equation" + } + ], + "index": 12 + } + ], + "index": 12 + }, + { + "type": "text", + "bbox": [ + 107, + 284, + 506, + 318 + ], + "lines": [ + { + "bbox": [ + 105, + 284, + 506, + 297 + ], + "spans": [ + { + "bbox": [ + 105, + 284, + 483, + 297 + ], + "score": 1.0, + "content": "In this appendix, we prove Theorem 2. In addition, we will also show that permutations are in", + "type": "text" + }, + { + "bbox": [ + 483, + 285, + 502, + 295 + ], + "score": 0.89, + "content": "B ^ { * } B", + "type": "inline_equation" + }, + { + "bbox": [ + 503, + 284, + 506, + 297 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 295, + 505, + 308 + ], + "spans": [ + { + "bbox": [ + 105, + 295, + 162, + 308 + ], + "score": 1.0, + "content": "where the set", + "type": "text" + }, + { + "bbox": [ + 162, + 296, + 182, + 306 + ], + "score": 0.9, + "content": "B ^ { * } B", + "type": "inline_equation" + }, + { + "bbox": [ + 182, + 295, + 286, + 308 + ], + "score": 1.0, + "content": "is defined analogously to", + "type": "text" + }, + { + "bbox": [ + 286, + 296, + 305, + 306 + ], + "score": 0.89, + "content": "B B ^ { * }", + "type": "inline_equation" + }, + { + "bbox": [ + 306, + 295, + 409, + 308 + ], + "score": 1.0, + "content": "(i.e. matrices of the form", + "type": "text" + }, + { + "bbox": [ + 410, + 296, + 466, + 308 + ], + "score": 0.93, + "content": "\\mathbf { M } = \\mathbf { M } _ { 1 } ^ { * } \\mathbf { M } _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 466, + 295, + 505, + 308 + ], + "score": 1.0, + "content": "for some", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 107, + 306, + 169, + 320 + ], + "spans": [ + { + "bbox": [ + 107, + 307, + 164, + 318 + ], + "score": 0.86, + "content": "\\mathbf { M } _ { 1 } , \\mathbf { M } _ { 2 } \\in B )", + "type": "inline_equation" + }, + { + "bbox": [ + 164, + 306, + 169, + 320 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 14, + "bbox_fs": [ + 105, + 284, + 506, + 320 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 323, + 503, + 346 + ], + "lines": [ + { + "bbox": [ + 105, + 323, + 502, + 336 + ], + "spans": [ + { + "bbox": [ + 105, + 323, + 333, + 336 + ], + "score": 1.0, + "content": "To prove Theorem 2, we decompose permutation matrix", + "type": "text" + }, + { + "bbox": [ + 334, + 324, + 343, + 334 + ], + "score": 0.7, + "content": "\\mathbf { P }", + "type": "inline_equation" + }, + { + "bbox": [ + 343, + 323, + 362, + 336 + ], + "score": 1.0, + "content": "into", + "type": "text" + }, + { + "bbox": [ + 362, + 324, + 400, + 334 + ], + "score": 0.89, + "content": "\\mathbf { P } = \\mathbf { L R }", + "type": "inline_equation" + }, + { + "bbox": [ + 400, + 323, + 424, + 336 + ], + "score": 1.0, + "content": ", with", + "type": "text" + }, + { + "bbox": [ + 424, + 324, + 452, + 334 + ], + "score": 0.92, + "content": "\\mathbf { L } \\in B", + "type": "inline_equation" + }, + { + "bbox": [ + 452, + 323, + 469, + 336 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 470, + 324, + 502, + 334 + ], + "score": 0.9, + "content": "\\mathbf { R } \\in B ^ { * }", + "type": "inline_equation" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 334, + 360, + 347 + ], + "spans": [ + { + "bbox": [ + 106, + 334, + 360, + 347 + ], + "score": 1.0, + "content": "Throughout the proof, we make use of the following definition.", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 16.5, + "bbox_fs": [ + 105, + 323, + 502, + 347 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 349, + 505, + 408 + ], + "lines": [ + { + "bbox": [ + 105, + 349, + 505, + 363 + ], + "spans": [ + { + "bbox": [ + 105, + 349, + 224, + 363 + ], + "score": 1.0, + "content": "Definition G.1. Let L be an", + "type": "text" + }, + { + "bbox": [ + 224, + 352, + 250, + 361 + ], + "score": 0.88, + "content": "n \\times n", + "type": "inline_equation" + }, + { + "bbox": [ + 250, + 349, + 505, + 363 + ], + "score": 1.0, + "content": "permutation matrix (n a power of 2). We say that L meets the", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 359, + 506, + 374 + ], + "spans": [ + { + "bbox": [ + 106, + 361, + 117, + 371 + ], + "score": 0.8, + "content": "2 ^ { j }", + "type": "inline_equation" + }, + { + "bbox": [ + 117, + 359, + 338, + 374 + ], + "score": 1.0, + "content": "balance condition if L can be divided into chunks of", + "type": "text" + }, + { + "bbox": [ + 339, + 361, + 349, + 371 + ], + "score": 0.85, + "content": "2 ^ { j }", + "type": "inline_equation" + }, + { + "bbox": [ + 349, + 359, + 506, + 374 + ], + "score": 1.0, + "content": "(with each chunk having all columns", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 369, + 506, + 388 + ], + "spans": [ + { + "bbox": [ + 106, + 373, + 111, + 382 + ], + "score": 0.32, + "content": "i", + "type": "inline_equation" + }, + { + "bbox": [ + 112, + 369, + 155, + 388 + ], + "score": 1.0, + "content": "such that", + "type": "text" + }, + { + "bbox": [ + 155, + 371, + 174, + 385 + ], + "score": 0.91, + "content": "\\left\\lfloor { \\frac { i } { 2 ^ { j } } } \\right\\rfloor", + "type": "inline_equation" + }, + { + "bbox": [ + 175, + 369, + 344, + 388 + ], + "score": 1.0, + "content": "has the same value) such that for every", + "type": "text" + }, + { + "bbox": [ + 345, + 372, + 401, + 383 + ], + "score": 0.91, + "content": "0 \\leq m < 2 ^ { j }", + "type": "inline_equation" + }, + { + "bbox": [ + 402, + 369, + 506, + 388 + ], + "score": 1.0, + "content": ", each chunk has exactly", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 384, + 504, + 397 + ], + "spans": [ + { + "bbox": [ + 105, + 384, + 124, + 397 + ], + "score": 1.0, + "content": "one", + "type": "text" + }, + { + "bbox": [ + 124, + 385, + 179, + 397 + ], + "score": 0.9, + "content": "\\mathbf { L } [ : , k ] = \\mathbf { e } _ { \\pi _ { k } }", + "type": "inline_equation" + }, + { + "bbox": [ + 179, + 384, + 200, + 397 + ], + "score": 1.0, + "content": "with", + "type": "text" + }, + { + "bbox": [ + 201, + 385, + 236, + 396 + ], + "score": 0.85, + "content": "\\pi _ { k } \\equiv m", + "type": "inline_equation" + }, + { + "bbox": [ + 236, + 384, + 269, + 397 + ], + "score": 1.0, + "content": "( mod", + "type": "text" + }, + { + "bbox": [ + 270, + 384, + 280, + 395 + ], + "score": 0.37, + "content": "2 ^ { j }", + "type": "inline_equation" + }, + { + "bbox": [ + 281, + 384, + 493, + 397 + ], + "score": 1.0, + "content": "). We say that L is modular-balanced if it meets the", + "type": "text" + }, + { + "bbox": [ + 493, + 384, + 504, + 394 + ], + "score": 0.77, + "content": "2 ^ { j }", + "type": "inline_equation" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 394, + 268, + 409 + ], + "spans": [ + { + "bbox": [ + 105, + 394, + 216, + 409 + ], + "score": 1.0, + "content": "balance condition for each", + "type": "text" + }, + { + "bbox": [ + 216, + 396, + 263, + 407 + ], + "score": 0.91, + "content": "2 \\leq 2 ^ { \\overset { . } { j } } \\leq n", + "type": "inline_equation" + }, + { + "bbox": [ + 264, + 394, + 268, + 409 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 20, + "bbox_fs": [ + 105, + 349, + 506, + 409 + ] + }, + { + "type": "image", + "bbox": [ + 138, + 424, + 473, + 552 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 138, + 424, + 473, + 552 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 138, + 424, + 473, + 552 + ], + "spans": [ + { + "bbox": [ + 138, + 424, + 473, + 552 + ], + "score": 0.97, + "type": "image", + "image_path": "e1a749d098c79becb5562ba1896bfb4d508efdd4f05072b76cda0be851a989ca.jpg" + } + ] + } + ], + "index": 24, + "virtual_lines": [ + { + "bbox": [ + 138, + 424, + 473, + 466.6666666666667 + ], + "spans": [], + "index": 23 + }, + { + "bbox": [ + 138, + 466.6666666666667, + 473, + 509.33333333333337 + ], + "spans": [], + "index": 24 + }, + { + "bbox": [ + 138, + 509.33333333333337, + 473, + 552.0 + ], + "spans": [], + "index": 25 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 106, + 565, + 505, + 588 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 106, + 565, + 505, + 578 + ], + "spans": [ + { + "bbox": [ + 106, + 565, + 505, + 578 + ], + "score": 1.0, + "content": "Figure 7: First step of decomposition of modular-balanced matrix L. Here, the red entries must be", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 577, + 269, + 589 + ], + "spans": [ + { + "bbox": [ + 105, + 577, + 269, + 589 + ], + "score": 1.0, + "content": "permuted into the main diagonal blocks.", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 26.5 + } + ], + "index": 25.25 + }, + { + "type": "text", + "bbox": [ + 106, + 603, + 399, + 615 + ], + "lines": [ + { + "bbox": [ + 106, + 603, + 400, + 617 + ], + "spans": [ + { + "bbox": [ + 106, + 603, + 213, + 617 + ], + "score": 1.0, + "content": "Lemma G.1. Let L be an", + "type": "text" + }, + { + "bbox": [ + 213, + 605, + 239, + 614 + ], + "score": 0.88, + "content": "n \\times n", + "type": "inline_equation" + }, + { + "bbox": [ + 239, + 603, + 369, + 617 + ], + "score": 1.0, + "content": "modular-balanced matrix. Then", + "type": "text" + }, + { + "bbox": [ + 370, + 604, + 397, + 614 + ], + "score": 0.86, + "content": "\\mathbf { L } \\in B", + "type": "inline_equation" + }, + { + "bbox": [ + 397, + 603, + 400, + 617 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 28, + "bbox_fs": [ + 106, + 603, + 400, + 617 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 630, + 506, + 664 + ], + "lines": [ + { + "bbox": [ + 105, + 629, + 507, + 643 + ], + "spans": [ + { + "bbox": [ + 105, + 629, + 252, + 643 + ], + "score": 1.0, + "content": "Proof. We proceed by induction on", + "type": "text" + }, + { + "bbox": [ + 253, + 633, + 259, + 640 + ], + "score": 0.69, + "content": "n", + "type": "inline_equation" + }, + { + "bbox": [ + 260, + 629, + 322, + 643 + ], + "score": 1.0, + "content": ". The base case", + "type": "text" + }, + { + "bbox": [ + 323, + 631, + 348, + 640 + ], + "score": 0.88, + "content": "n = 2", + "type": "inline_equation" + }, + { + "bbox": [ + 349, + 629, + 507, + 643 + ], + "score": 1.0, + "content": "is trivial. As our inductive hypothesis,", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 102, + 637, + 505, + 659 + ], + "spans": [ + { + "bbox": [ + 102, + 637, + 323, + 659 + ], + "score": 1.0, + "content": "we assume that all modular-balanced matrices of size", + "type": "text" + }, + { + "bbox": [ + 323, + 642, + 350, + 654 + ], + "score": 0.92, + "content": "{ \\frac { n } { 2 } } \\times { \\frac { n } { 2 } }", + "type": "inline_equation" + }, + { + "bbox": [ + 351, + 637, + 468, + 659 + ], + "score": 1.0, + "content": "are butterfly matrices of size", + "type": "text" + }, + { + "bbox": [ + 469, + 642, + 476, + 654 + ], + "score": 0.85, + "content": "\\frac { n } { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 477, + 637, + 505, + 659 + ], + "score": 1.0, + "content": ". From", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 651, + 378, + 665 + ], + "spans": [ + { + "bbox": [ + 105, + 651, + 274, + 665 + ], + "score": 1.0, + "content": "Definition 2.3, it is sufficient to show that", + "type": "text" + }, + { + "bbox": [ + 283, + 651, + 378, + 665 + ], + "score": 1.0, + "content": "can be decomposed as:", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 30, + "bbox_fs": [ + 102, + 629, + 507, + 665 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 263, + 669, + 347, + 715 + ], + "lines": [ + { + "bbox": [ + 263, + 669, + 347, + 715 + ], + "spans": [ + { + "bbox": [ + 263, + 669, + 347, + 715 + ], + "score": 0.93, + "content": "\\begin{array} { r } { \\mathbf { L } = \\mathbf { B } _ { n } \\underbrace { \\left[ \\begin{array} { c c } { \\mathbf { L } _ { 1 } } & { 0 } \\\\ { 0 } & { \\mathbf { L } _ { 2 } } \\end{array} \\right] } _ { \\mathbf { L } ^ { \\prime } } , } \\end{array}", + "type": "interline_equation", + "image_path": "df1b27189f25d83f3c2706f4f55e1d3299d3ac12ed64ecc586c74c3ec0a96383.jpg" + } + ] + } + ], + "index": 32.5, + "virtual_lines": [ + { + "bbox": [ + 263, + 669, + 347, + 692.0 + ], + "spans": [], + "index": 32 + }, + { + "bbox": [ + 263, + 692.0, + 347, + 715.0 + ], + "spans": [], + "index": 33 + } + ] + }, + { + "type": "text", + "bbox": [ + 102, + 720, + 470, + 734 + ], + "lines": [ + { + "bbox": [ + 104, + 717, + 473, + 737 + ], + "spans": [ + { + "bbox": [ + 104, + 717, + 133, + 737 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 133, + 721, + 147, + 732 + ], + "score": 0.89, + "content": "\\mathbf { B } _ { n }", + "type": "inline_equation" + }, + { + "bbox": [ + 148, + 717, + 255, + 737 + ], + "score": 1.0, + "content": "is a butterfly factor of size", + "type": "text" + }, + { + "bbox": [ + 256, + 723, + 263, + 730 + ], + "score": 0.77, + "content": "n", + "type": "inline_equation" + }, + { + "bbox": [ + 263, + 717, + 301, + 737 + ], + "score": 1.0, + "content": "and each", + "type": "text" + }, + { + "bbox": [ + 302, + 721, + 313, + 733 + ], + "score": 0.89, + "content": "\\mathbf { L } _ { j }", + "type": "inline_equation" + }, + { + "bbox": [ + 314, + 717, + 336, + 737 + ], + "score": 1.0, + "content": "is an", + "type": "text" + }, + { + "bbox": [ + 337, + 721, + 364, + 734 + ], + "score": 0.92, + "content": "{ \\frac { n } { 2 } } \\times { \\frac { n } { 2 } }", + "type": "inline_equation" + }, + { + "bbox": [ + 364, + 717, + 473, + 737 + ], + "score": 1.0, + "content": "modular-balanced matrix.", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 34, + "bbox_fs": [ + 104, + 717, + 473, + 737 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 82, + 220, + 93 + ], + "lines": [ + { + "bbox": [ + 106, + 81, + 221, + 96 + ], + "spans": [ + { + "bbox": [ + 106, + 81, + 135, + 96 + ], + "score": 1.0, + "content": "Define", + "type": "text" + }, + { 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"type": "text" + } + ], + "index": 3 + } + ], + "index": 2.5 + }, + { + "type": "text", + "bbox": [ + 106, + 142, + 505, + 214 + ], + "lines": [ + { + "bbox": [ + 105, + 141, + 506, + 157 + ], + "spans": [ + { + "bbox": [ + 105, + 141, + 240, + 157 + ], + "score": 1.0, + "content": "For sake of contradiction, assume", + "type": "text" + }, + { + "bbox": [ + 241, + 144, + 253, + 154 + ], + "score": 0.87, + "content": "\\mathbf { L } _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 253, + 141, + 416, + 157 + ], + "score": 1.0, + "content": "is not modular-balanced. Then, for some", + "type": "text" + }, + { + "bbox": [ + 417, + 142, + 447, + 156 + ], + "score": 0.93, + "content": "2 ^ { j } \\leq { \\frac { n } { 2 } }", + "type": "inline_equation" + }, + { + "bbox": [ + 448, + 141, + 506, + 157 + ], + "score": 1.0, + "content": ", there are two", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 106, + 155, + 505, + 171 + ], + "spans": [ + { + "bbox": [ + 106, + 155, + 142, + 171 + ], + "score": 1.0, + "content": "columns", + "type": "text" + }, + { + "bbox": [ + 142, + 158, + 165, + 168 + ], + "score": 0.89, + "content": "c _ { 1 } , c _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 166, + 155, + 205, + 171 + ], + "score": 1.0, + "content": "such that", + "type": "text" + }, + { + "bbox": [ + 205, + 155, + 257, + 169 + ], + "score": 0.93, + "content": "\\textstyle \\left\\lfloor { \\frac { c _ { 1 } } { 2 ^ { j } } } \\right\\rfloor = \\left\\lfloor { \\frac { c _ { 2 } } { 2 ^ { j } } } \\right\\rfloor", + "type": "inline_equation" + }, + { + "bbox": [ + 258, + 155, + 445, + 171 + ], + "score": 1.0, + "content": "and such that indices of the non-zero entries of", + "type": "text" + }, + { + "bbox": [ + 446, + 157, + 458, + 167 + ], + "score": 0.88, + "content": "\\mathbf { L } _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 458, + 155, + 505, + 171 + ], + "score": 1.0, + "content": "in columns", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 168, + 506, + 181 + ], + "spans": [ + { + "bbox": [ + 106, + 171, + 116, + 180 + ], + "score": 0.83, + "content": "c _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 117, + 168, + 134, + 181 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 134, + 171, + 144, + 180 + ], + "score": 0.84, + "content": "c _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 145, + 168, + 228, + 181 + ], + "score": 1.0, + "content": "are the same modulo", + "type": "text" + }, + { + "bbox": [ + 229, + 169, + 239, + 179 + ], + "score": 0.83, + "content": "2 ^ { j }", + "type": "inline_equation" + }, + { + "bbox": [ + 239, + 168, + 369, + 181 + ], + "score": 1.0, + "content": ". However, from the definition of", + "type": "text" + }, + { + "bbox": [ + 369, + 169, + 381, + 180 + ], + "score": 0.88, + "content": "\\mathbf { L } _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 381, + 168, + 506, + 181 + ], + "score": 1.0, + "content": ", this implies that the indices of", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 178, + 505, + 194 + ], + "spans": [ + { + "bbox": [ + 105, + 178, + 200, + 194 + ], + "score": 1.0, + "content": "the non-zero entries of", + "type": "text" + }, + { + "bbox": [ + 200, + 181, + 208, + 190 + ], + "score": 0.56, + "content": "\\mathbf { L }", + "type": "inline_equation" + }, + { + "bbox": [ + 209, + 178, + 256, + 194 + ], + "score": 1.0, + "content": "in columns", + "type": "text" + }, + { + "bbox": [ + 256, + 182, + 266, + 191 + ], + "score": 0.86, + "content": "c _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 267, + 178, + 284, + 194 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 285, + 182, + 295, + 191 + ], + "score": 0.85, + "content": "c _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 295, + 178, + 401, + 194 + ], + "score": 1.0, + "content": "are also the same modulo", + "type": "text" + }, + { + "bbox": [ + 401, + 180, + 411, + 190 + ], + "score": 0.85, + "content": "2 ^ { j }", + "type": "inline_equation" + }, + { + "bbox": [ + 411, + 178, + 470, + 194 + ], + "score": 1.0, + "content": ", contradicting", + "type": "text" + }, + { + "bbox": [ + 470, + 180, + 479, + 190 + ], + "score": 0.52, + "content": "\\mathbf { L }", + "type": "inline_equation" + }, + { + "bbox": [ + 479, + 178, + 505, + 194 + ], + "score": 1.0, + "content": "being", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 190, + 505, + 204 + ], + "spans": [ + { + "bbox": [ + 105, + 190, + 214, + 204 + ], + "score": 1.0, + "content": "modular-balanced. Hence,", + "type": "text" + }, + { + "bbox": [ + 214, + 191, + 227, + 202 + ], + "score": 0.89, + "content": "\\mathbf { L } _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 227, + 190, + 505, + 204 + ], + "score": 1.0, + "content": "is modular-balanced. An analogous argument (that instead considers", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 200, + 393, + 217 + ], + "spans": [ + { + "bbox": [ + 105, + 200, + 143, + 217 + ], + "score": 1.0, + "content": "columns", + "type": "text" + }, + { + "bbox": [ + 143, + 202, + 205, + 215 + ], + "score": 0.92, + "content": "\\begin{array} { r } { c _ { 1 } + \\frac { n } { 2 } , c _ { 2 } + \\frac { n } { 2 } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 205, + 200, + 217, + 217 + ], + "score": 1.0, + "content": "of", + "type": "text" + }, + { + "bbox": [ + 217, + 203, + 225, + 212 + ], + "score": 0.55, + "content": "\\mathbf { L }", + "type": "inline_equation" + }, + { + "bbox": [ + 226, + 200, + 274, + 217 + ], + "score": 1.0, + "content": ") shows that", + "type": "text" + }, + { + "bbox": [ + 274, + 202, + 287, + 213 + ], + "score": 0.87, + "content": "\\mathbf { L } _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 287, + 200, + 393, + 217 + ], + "score": 1.0, + "content": "is also modular-balanced.", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 6.5 + }, + { + "type": "text", + "bbox": [ + 106, + 218, + 505, + 288 + ], + "lines": [ + { + "bbox": [ + 106, + 218, + 505, + 231 + ], + "spans": [ + { + "bbox": [ + 106, + 218, + 288, + 231 + ], + "score": 1.0, + "content": "To complete the proof, we must argue that", + "type": "text" + }, + { + "bbox": [ + 288, + 219, + 303, + 230 + ], + "score": 0.89, + "content": "\\mathbf { B } _ { n }", + "type": "inline_equation" + }, + { + "bbox": [ + 303, + 218, + 419, + 231 + ], + "score": 1.0, + "content": "is a butterfly factor of size", + "type": "text" + }, + { + "bbox": [ + 419, + 221, + 426, + 229 + ], + "score": 0.65, + "content": "n", + "type": "inline_equation" + }, + { + "bbox": [ + 427, + 218, + 481, + 231 + ], + "score": 1.0, + "content": ". 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We make use of", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 497, + 194, + 510 + ], + "spans": [ + { + "bbox": [ + 106, + 497, + 194, + 510 + ], + "score": 1.0, + "content": "the following lemma.", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 29.5 + }, + { + "type": "image", + "bbox": [ + 148, + 520, + 462, + 646 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 148, + 520, + 462, + 646 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 148, + 520, + 462, + 646 + ], + "spans": [ + { + "bbox": [ + 148, + 520, + 462, + 646 + ], + "score": 0.972, + "type": "image", + "image_path": "a11fbc418e35ae750d3922a0ae3f160b759972fa724560634f39157fd9066c93.jpg" + } + ] + } + ], + "index": 32, + "virtual_lines": [ + { + "bbox": [ + 148, + 520, + 462, + 562.0 + ], + "spans": [], + "index": 31 + }, + { + "bbox": [ + 148, + 562.0, + 462, + 604.0 + ], + "spans": [], + "index": 32 + }, + { + "bbox": [ + 148, + 604.0, + 462, + 646.0 + ], + "spans": [], + "index": 33 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 105, + 660, + 506, + 684 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 105, + 660, + 506, + 674 + ], + "spans": [ + { + "bbox": [ + 105, + 660, + 234, + 674 + ], + "score": 1.0, + "content": "Figure 8: First step of balancing", + "type": "text" + }, + { + "bbox": [ + 235, + 662, + 258, + 672 + ], + "score": 0.88, + "content": "8 \\times 8", + "type": "inline_equation" + }, + { + "bbox": [ + 259, + 660, + 435, + 674 + ], + "score": 1.0, + "content": "bit reversal permutation (a component of the", + "type": "text" + }, + { + "bbox": [ + 435, + 661, + 459, + 672 + ], + "score": 0.87, + "content": "8 \\times 8", + "type": "inline_equation" + }, + { + "bbox": [ + 460, + 660, + 506, + 674 + ], + "score": 1.0, + "content": "DFT). Red", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 106, + 671, + 251, + 685 + ], + "spans": [ + { + "bbox": [ + 106, + 671, + 251, + 685 + ], + "score": 1.0, + "content": "signifies edges that must be flipped.", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 34.5 + } + ], + "index": 33.25 + }, + { + "type": "text", + "bbox": [ + 106, + 695, + 506, + 733 + ], + "lines": [ + { + "bbox": [ + 105, + 694, + 506, + 709 + ], + "spans": [ + { + "bbox": [ + 105, + 694, + 219, + 709 + ], + "score": 1.0, + "content": "Lemma G.2. Let M be a", + "type": "text" + }, + { + "bbox": [ + 219, + 696, + 246, + 707 + ], + "score": 0.9, + "content": "k \\times k", + "type": "inline_equation" + }, + { + "bbox": [ + 246, + 694, + 299, + 709 + ], + "score": 1.0, + "content": "matrix with", + "type": "text" + }, + { + "bbox": [ + 300, + 697, + 306, + 706 + ], + "score": 0.29, + "content": "I", + "type": "inline_equation" + }, + { + "bbox": [ + 307, + 694, + 506, + 709 + ], + "score": 1.0, + "content": "non-zero entry per column, such that for each", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 106, + 705, + 506, + 722 + ], + "spans": [ + { + "bbox": [ + 106, + 707, + 157, + 720 + ], + "score": 0.92, + "content": "\\textstyle 0 \\leq m < { \\frac { k } { 2 } }", + "type": "inline_equation" + }, + { + "bbox": [ + 157, + 705, + 439, + 722 + ], + "score": 1.0, + "content": ", there are exactly 2 columns with non-zero entry in a row with index", + "type": "text" + }, + { + "bbox": [ + 439, + 708, + 462, + 718 + 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Then, for some", + "type": "text" + }, + { + "bbox": [ + 417, + 142, + 447, + 156 + ], + "score": 0.93, + "content": "2 ^ { j } \\leq { \\frac { n } { 2 } }", + "type": "inline_equation" + }, + { + "bbox": [ + 448, + 141, + 506, + 157 + ], + "score": 1.0, + "content": ", there are two", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 106, + 155, + 505, + 171 + ], + "spans": [ + { + "bbox": [ + 106, + 155, + 142, + 171 + ], + "score": 1.0, + "content": "columns", + "type": "text" + }, + { + "bbox": [ + 142, + 158, + 165, + 168 + ], + "score": 0.89, + "content": "c _ { 1 } , c _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 166, + 155, + 205, + 171 + ], + "score": 1.0, + "content": "such that", + "type": "text" + }, + { + "bbox": [ + 205, + 155, + 257, + 169 + ], + "score": 0.93, + "content": "\\textstyle \\left\\lfloor { \\frac { c _ { 1 } } { 2 ^ { j } } } \\right\\rfloor = \\left\\lfloor { \\frac { c _ { 2 } } { 2 ^ { j } } } \\right\\rfloor", + "type": "inline_equation" + }, + { + "bbox": [ + 258, + 155, + 445, + 171 + ], + "score": 1.0, + "content": "and such that indices of the non-zero entries of", + "type": "text" + }, + { + "bbox": [ + 446, + 157, + 458, + 167 + ], + "score": 0.88, + "content": "\\mathbf { L } _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 458, + 155, + 505, + 171 + ], + "score": 1.0, + "content": "in columns", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 168, + 506, + 181 + ], + "spans": [ + { + "bbox": [ + 106, + 171, + 116, + 180 + ], + "score": 0.83, + "content": "c _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 117, + 168, + 134, + 181 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 134, + 171, + 144, + 180 + ], + "score": 0.84, + "content": "c _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 145, + 168, + 228, + 181 + ], + "score": 1.0, + "content": "are the same modulo", + "type": "text" + }, + { + "bbox": [ + 229, + 169, + 239, + 179 + ], + "score": 0.83, + "content": "2 ^ { j }", + "type": "inline_equation" + }, + { + "bbox": [ + 239, + 168, + 369, + 181 + ], + "score": 1.0, + "content": ". 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Therefore,", + "type": "text" + }, + { + "bbox": [ + 374, + 421, + 462, + 436 + ], + "score": 0.93, + "content": "\\begin{array} { r } { \\mathbf { L } ^ { \\prime } \\left[ i + \\frac { n } { 2 } , : \\right] = \\mathbf { L } \\left[ i , : \\right] } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 463, + 421, + 506, + 437 + ], + "score": 1.0, + "content": ", so we set", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 434, + 238, + 449 + ], + "spans": [ + { + "bbox": [ + 106, + 434, + 238, + 449 + ], + "score": 0.88, + "content": "\\begin{array} { r } { \\mathbf { B } \\left[ i , i + \\frac { n } { 2 } \\right] = \\mathbf { B } \\left[ i + \\frac { n } { 2 } , i \\right] = 1 . } \\end{array}", + "type": "inline_equation" + } + ], + "index": 26 + } + ], + "index": 24.5, + "bbox_fs": [ + 104, + 396, + 507, + 449 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 452, + 504, + 475 + ], + "lines": [ + { + "bbox": [ + 105, + 452, + 505, + 465 + ], + "spans": [ + { + "bbox": [ + 105, + 452, + 256, + 465 + ], + "score": 1.0, + "content": "In both cases, the non-zero entries of", + "type": "text" + }, + { + "bbox": [ + 256, + 453, + 266, + 462 + ], + "score": 0.44, + "content": "\\mathbf { B }", + "type": "inline_equation" + }, + { + "bbox": [ + 266, + 452, + 505, + 465 + ], + "score": 1.0, + "content": "fall into the correct diagonal bands (the main diagonal, and", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 463, + 505, + 477 + ], + "spans": [ + { + "bbox": [ + 105, + 463, + 147, + 476 + ], + "score": 1.0, + "content": "the bands", + "type": "text" + }, + { + "bbox": [ + 147, + 464, + 155, + 477 + ], + "score": 0.87, + "content": "\\frac { n } { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 156, + 463, + 217, + 476 + ], + "score": 1.0, + "content": "away). Hence,", + "type": "text" + }, + { + "bbox": [ + 217, + 464, + 226, + 473 + ], + "score": 0.48, + "content": "\\mathbf { B }", + "type": "inline_equation" + }, + { + "bbox": [ + 227, + 463, + 334, + 476 + ], + "score": 1.0, + "content": "is a butterfly factor of size", + "type": "text" + }, + { + "bbox": [ + 334, + 466, + 341, + 473 + ], + "score": 0.69, + "content": "n", + "type": "inline_equation" + }, + { + "bbox": [ + 341, + 463, + 345, + 476 + ], + "score": 1.0, + "content": ".", + "type": "text" + }, + { + "bbox": [ + 495, + 465, + 505, + 474 + ], + "score": 0.999, + "content": "□", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 27.5, + "bbox_fs": [ + 105, + 452, + 505, + 477 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 486, + 505, + 509 + ], + "lines": [ + { + "bbox": [ + 105, + 486, + 506, + 499 + ], + "spans": [ + { + "bbox": [ + 105, + 486, + 294, + 499 + ], + "score": 1.0, + "content": "Now, we consider the process of transforming", + "type": "text" + }, + { + "bbox": [ + 295, + 487, + 304, + 497 + ], + "score": 0.62, + "content": "\\mathbf { P }", + "type": "inline_equation" + }, + { + "bbox": [ + 304, + 486, + 506, + 499 + ], + "score": 1.0, + "content": "into a modular-balanced matrix. We make use of", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 497, + 194, + 510 + ], + "spans": [ + { + "bbox": [ + 106, + 497, + 194, + 510 + ], + "score": 1.0, + "content": "the following lemma.", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 29.5, + "bbox_fs": [ + 105, + 486, + 506, + 510 + ] + }, + { + "type": "image", + "bbox": [ + 148, + 520, + 462, + 646 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 148, + 520, + 462, + 646 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 148, + 520, + 462, + 646 + ], + "spans": [ + { + "bbox": [ + 148, + 520, + 462, + 646 + ], + "score": 0.972, + "type": "image", + "image_path": "a11fbc418e35ae750d3922a0ae3f160b759972fa724560634f39157fd9066c93.jpg" + } + ] + } + ], + "index": 32, + "virtual_lines": [ + { + "bbox": [ + 148, + 520, + 462, + 562.0 + ], + "spans": [], + "index": 31 + }, + { + "bbox": [ + 148, + 562.0, + 462, + 604.0 + ], + "spans": [], + "index": 32 + }, + { + "bbox": [ + 148, + 604.0, + 462, + 646.0 + ], + "spans": [], + "index": 33 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 105, + 660, + 506, + 684 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 105, + 660, + 506, + 674 + ], + "spans": [ + { + "bbox": [ + 105, + 660, + 234, + 674 + ], + "score": 1.0, + "content": "Figure 8: First step of balancing", + "type": "text" + }, + { + "bbox": [ + 235, + 662, + 258, + 672 + ], + "score": 0.88, + "content": "8 \\times 8", + "type": "inline_equation" + }, + { + "bbox": [ + 259, + 660, + 435, + 674 + ], + "score": 1.0, + "content": "bit reversal permutation (a component of the", + "type": "text" + }, + { + "bbox": [ + 435, + 661, + 459, + 672 + ], + "score": 0.87, + "content": "8 \\times 8", + "type": "inline_equation" + }, + { + "bbox": [ + 460, + 660, + 506, + 674 + ], + "score": 1.0, + "content": "DFT). Red", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 106, + 671, + 251, + 685 + ], + "spans": [ + { + "bbox": [ + 106, + 671, + 251, + 685 + ], + "score": 1.0, + "content": "signifies edges that must be flipped.", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 34.5 + } + ], + "index": 33.25 + }, + { + "type": "text", + "bbox": [ + 106, + 695, + 506, + 733 + ], + "lines": [ + { + "bbox": [ + 105, + 694, + 506, + 709 + ], + "spans": [ + { + "bbox": [ + 105, + 694, + 219, + 709 + ], + "score": 1.0, + "content": "Lemma G.2. Let M be a", + "type": "text" + }, + { + "bbox": [ + 219, + 696, + 246, + 707 + ], + "score": 0.9, + "content": "k \\times k", + "type": "inline_equation" + }, + { + "bbox": [ + 246, + 694, + 299, + 709 + ], + "score": 1.0, + "content": "matrix with", + "type": "text" + }, + { + "bbox": [ + 300, + 697, + 306, + 706 + ], + "score": 0.29, + "content": "I", + "type": "inline_equation" + }, + { + "bbox": [ + 307, + 694, + 506, + 709 + ], + "score": 1.0, + "content": "non-zero entry per column, such that for each", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 106, + 705, + 506, + 722 + ], + "spans": [ + { + "bbox": [ + 106, + 707, + 157, + 720 + ], + "score": 0.92, + "content": "\\textstyle 0 \\leq m < { \\frac { k } { 2 } }", + "type": "inline_equation" + }, + { + "bbox": [ + 157, + 705, + 439, + 722 + ], + "score": 1.0, + "content": ", there are exactly 2 columns with non-zero entry in a row with index", + "type": "text" + }, + { + "bbox": [ + 439, + 708, + 462, + 718 + ], + "score": 0.76, + "content": "\\equiv m", + "type": "inline_equation" + }, + { + "bbox": [ + 463, + 705, + 491, + 722 + ], + "score": 1.0, + "content": "\u0000mod", + "type": "text" + }, + { + "bbox": [ + 491, + 707, + 498, + 720 + ], + "score": 0.75, + "content": "\\frac { k } { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 499, + 705, + 506, + 722 + ], + "score": 1.0, + "content": "\u0001 .", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 719, + 506, + 734 + ], + "spans": [ + { + "bbox": [ + 105, + 719, + 230, + 734 + ], + "score": 1.0, + "content": "Then, there is a butterfly factor", + "type": "text" + }, + { + "bbox": [ + 230, + 720, + 244, + 732 + ], + "score": 0.89, + "content": "\\mathbf { B } _ { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 245, + 719, + 284, + 734 + ], + "score": 1.0, + "content": "such that", + "type": "text" + }, + { + "bbox": [ + 284, + 720, + 336, + 732 + ], + "score": 0.92, + "content": "\\mathbf { M } \\mathbf { B } _ { k } = \\mathbf { M } ^ { \\prime }", + "type": "inline_equation" + }, + { + "bbox": [ + 336, + 719, + 366, + 734 + ], + "score": 1.0, + "content": ", where", + "type": "text" + }, + { + "bbox": [ + 366, + 720, + 381, + 731 + ], + "score": 0.79, + "content": "\\mathbf { M } ^ { \\prime }", + "type": "inline_equation" + }, + { + "bbox": [ + 381, + 719, + 422, + 734 + ], + "score": 1.0, + "content": "meets the", + "type": "text" + }, + { + "bbox": [ + 422, + 719, + 430, + 734 + ], + "score": 0.85, + "content": "\\frac { k } { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 430, + 719, + 506, + 734 + ], + "score": 1.0, + "content": "balance condition.", + "type": "text" + } + ], + "index": 38 + } + ], + "index": 37, + "bbox_fs": [ + 105, + 694, + 506, + 734 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 81, + 505, + 129 + ], + "lines": [ + { + "bbox": [ + 104, + 81, + 506, + 96 + ], + "spans": [ + { + "bbox": [ + 104, + 81, + 259, + 95 + ], + "score": 1.0, + "content": "Proof. We construct a directed graph", + "type": "text" + }, + { + "bbox": [ + 260, + 83, + 269, + 93 + ], + "score": 0.8, + "content": "G", + "type": "inline_equation" + }, + { + "bbox": [ + 270, + 81, + 329, + 95 + ], + "score": 1.0, + "content": "with nodes in", + "type": "text" + }, + { + "bbox": [ + 329, + 81, + 343, + 96 + ], + "score": 0.9, + "content": "\\left[ \\frac { k } { 2 } \\right]", + "type": "inline_equation" + }, + { + "bbox": [ + 344, + 81, + 386, + 95 + ], + "score": 1.0, + "content": ". For each", + "type": "text" + }, + { + "bbox": [ + 386, + 81, + 430, + 95 + ], + "score": 0.91, + "content": "\\begin{array} { r } { 0 \\leq i < \\frac { k } { 2 } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 430, + 81, + 506, + 95 + ], + "score": 1.0, + "content": "we add a directed", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 104, + 91, + 505, + 110 + ], + "spans": [ + { + "bbox": [ + 104, + 91, + 172, + 110 + ], + "score": 1.0, + "content": "edge from node", + "type": "text" + }, + { + "bbox": [ + 172, + 94, + 219, + 109 + ], + "score": 0.71, + "content": "\\left( s \\mod \\frac { k } { 2 } \\right)", + "type": "inline_equation" + }, + { + "bbox": [ + 219, + 91, + 255, + 110 + ], + "score": 1.0, + "content": "\u0001 to node", + "type": "text" + }, + { + "bbox": [ + 256, + 95, + 304, + 109 + ], + "score": 0.71, + "content": "\\left( t \\mod \\frac { k } { 2 } \\right)", + "type": "inline_equation" + }, + { + "bbox": [ + 304, + 91, + 314, + 110 + ], + "score": 1.0, + "content": "if", + "type": "text" + }, + { + "bbox": [ + 315, + 96, + 365, + 108 + ], + "score": 0.88, + "content": "\\mathbf { M } [ : , i ] = \\mathbf { e } _ { s }", + "type": "inline_equation" + }, + { + "bbox": [ + 366, + 91, + 383, + 110 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 384, + 95, + 456, + 109 + ], + "score": 0.88, + "content": "\\begin{array} { r } { \\mathbf { M } \\left[ : , i + \\frac { k } { 2 } \\right] = \\mathbf { e } _ { t } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 457, + 91, + 505, + 110 + ], + "score": 1.0, + "content": ". Each node", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 105, + 506, + 119 + ], + "spans": [ + { + "bbox": [ + 105, + 105, + 361, + 119 + ], + "score": 1.0, + "content": "has (undirected) degree exactly 2 by the structure of M. Hence,", + "type": "text" + }, + { + "bbox": [ + 361, + 107, + 370, + 117 + ], + "score": 0.81, + "content": "G", + "type": "inline_equation" + }, + { + "bbox": [ + 370, + 105, + 506, + 119 + ], + "score": 1.0, + "content": "is a union of disjoint (undirected)", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 117, + 138, + 132 + ], + "spans": [ + { + "bbox": [ + 105, + 117, + 138, + 132 + ], + "score": 1.0, + "content": "cycles.", + "type": "text" + } + ], + "index": 3 + } + ], + "index": 1.5 + }, + { + "type": "text", + "bbox": [ + 106, + 134, + 505, + 203 + ], + "lines": [ + { + "bbox": [ + 105, + 133, + 506, + 148 + ], + "spans": [ + { + "bbox": [ + 105, + 133, + 162, + 148 + ], + "score": 1.0, + "content": "If M met the", + "type": "text" + }, + { + "bbox": [ + 162, + 133, + 169, + 147 + ], + "score": 0.86, + "content": "\\frac { k } { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 170, + 133, + 506, + 148 + ], + "score": 1.0, + "content": "balance condition, then each node would additionally have in-degree exactly 1 and", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 106, + 146, + 505, + 158 + ], + "spans": [ + { + "bbox": [ + 106, + 146, + 282, + 158 + ], + "score": 1.0, + "content": "out-degree exactly 1. By reversing edges of", + "type": "text" + }, + { + "bbox": [ + 282, + 146, + 291, + 156 + ], + "score": 0.81, + "content": "G", + "type": "inline_equation" + }, + { + "bbox": [ + 292, + 146, + 505, + 158 + ], + "score": 1.0, + "content": "such that each (undirected) cycle becomes a directed", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 157, + 506, + 169 + ], + "spans": [ + { + "bbox": [ + 105, + 157, + 474, + 169 + ], + "score": 1.0, + "content": "cycle, we can achieve this. However, reversing edges corresponds to swapping columns of", + "type": "text" + }, + { + "bbox": [ + 474, + 157, + 487, + 167 + ], + "score": 0.29, + "content": "\\mathbf { M }", + "type": "inline_equation" + }, + { + "bbox": [ + 487, + 157, + 506, + 169 + ], + "score": 1.0, + "content": "that", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 107, + 166, + 505, + 182 + ], + "spans": [ + { + "bbox": [ + 107, + 166, + 121, + 182 + ], + "score": 1.0, + "content": "are", + "type": "text" + }, + { + "bbox": [ + 121, + 167, + 129, + 181 + ], + "score": 0.87, + "content": "\\frac { k } { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 129, + 166, + 171, + 182 + ], + "score": 1.0, + "content": "apart. Let", + "type": "text" + }, + { + "bbox": [ + 171, + 168, + 185, + 179 + ], + "score": 0.88, + "content": "\\mathbf { B } _ { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 186, + 166, + 407, + 182 + ], + "score": 1.0, + "content": "be the permutation matrix that performs all such swaps.", + "type": "text" + }, + { + "bbox": [ + 408, + 168, + 422, + 179 + ], + "score": 0.88, + "content": "\\mathbf { B } _ { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 423, + 166, + 505, + 182 + ], + "score": 1.0, + "content": "has non-zero entries", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 179, + 505, + 194 + ], + "spans": [ + { + "bbox": [ + 105, + 179, + 324, + 194 + ], + "score": 1.0, + "content": "only along the main diagonal and the diagonal bands", + "type": "text" + }, + { + "bbox": [ + 325, + 180, + 332, + 194 + ], + "score": 0.86, + "content": "\\frac { k } { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 333, + 179, + 505, + 194 + ], + "score": 1.0, + "content": "away, and thus is a butterfly factor of size", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 192, + 505, + 204 + ], + "spans": [ + { + "bbox": [ + 106, + 192, + 113, + 201 + ], + "score": 0.78, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 114, + 192, + 118, + 204 + ], + "score": 1.0, + "content": ".", + "type": "text" + }, + { + "bbox": [ + 494, + 192, + 505, + 203 + ], + "score": 0.999, + "content": "□", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 6.5 + }, + { + "type": "text", + "bbox": [ + 107, + 217, + 300, + 228 + ], + "lines": [ + { + "bbox": [ + 106, + 216, + 298, + 230 + ], + "spans": [ + { + "bbox": [ + 106, + 216, + 289, + 230 + ], + "score": 1.0, + "content": "We are ready to present the decomposition of", + "type": "text" + }, + { + "bbox": [ + 289, + 218, + 298, + 227 + ], + "score": 0.77, + "content": "\\mathbf { P }", + "type": "inline_equation" + } + ], + "index": 10 + } + ], + "index": 10 + }, + { + "type": "text", + "bbox": [ + 106, + 232, + 505, + 254 + ], + "lines": [ + { + "bbox": [ + 105, + 232, + 506, + 245 + ], + "spans": [ + { + "bbox": [ + 105, + 232, + 181, + 245 + ], + "score": 1.0, + "content": "Lemma G.3. Let", + "type": "text" + }, + { + "bbox": [ + 181, + 233, + 190, + 243 + ], + "score": 0.52, + "content": "\\mathbf { P }", + "type": "inline_equation" + }, + { + "bbox": [ + 190, + 232, + 217, + 245 + ], + "score": 1.0, + "content": "be an", + "type": "text" + }, + { + "bbox": [ + 217, + 234, + 243, + 243 + ], + "score": 0.9, + "content": "n \\times n", + "type": "inline_equation" + }, + { + "bbox": [ + 244, + 232, + 434, + 245 + ], + "score": 1.0, + "content": "permutation matrix. Then we can decompose", + "type": "text" + }, + { + "bbox": [ + 434, + 233, + 443, + 243 + ], + "score": 0.58, + "content": "\\mathbf { P }", + "type": "inline_equation" + }, + { + "bbox": [ + 444, + 232, + 463, + 245 + ], + "score": 1.0, + "content": "into", + "type": "text" + }, + { + "bbox": [ + 464, + 232, + 502, + 243 + ], + "score": 0.87, + "content": "\\mathbf { P } = \\mathbf { L R }", + "type": "inline_equation" + }, + { + "bbox": [ + 503, + 232, + 506, + 245 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 243, + 281, + 255 + ], + "spans": [ + { + "bbox": [ + 106, + 243, + 245, + 255 + ], + "score": 1.0, + "content": "where L is modular-balanced and", + "type": "text" + }, + { + "bbox": [ + 245, + 244, + 277, + 254 + ], + "score": 0.91, + "content": "\\mathbf { R } \\in B ^ { * }", + "type": "inline_equation" + }, + { + "bbox": [ + 278, + 243, + 281, + 255 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 11.5 + }, + { + "type": "text", + "bbox": [ + 106, + 268, + 505, + 290 + ], + "lines": [ + { + "bbox": [ + 105, + 267, + 505, + 282 + ], + "spans": [ + { + "bbox": [ + 105, + 267, + 468, + 282 + ], + "score": 1.0, + "content": "Proof. We repeatedly apply Lemma G.2. First, we conclude that there is a butterfly factor", + "type": "text" + }, + { + "bbox": [ + 468, + 269, + 483, + 280 + ], + "score": 0.89, + "content": "\\mathbf { B } _ { n }", + "type": "inline_equation" + }, + { + "bbox": [ + 483, + 267, + 505, + 282 + ], + "score": 1.0, + "content": "such", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 278, + 126, + 293 + ], + "spans": [ + { + "bbox": [ + 105, + 278, + 126, + 293 + ], + "score": 1.0, + "content": "that", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 13.5 + }, + { + "type": "interline_equation", + "bbox": [ + 280, + 290, + 330, + 304 + ], + "lines": [ + { + "bbox": [ + 280, + 290, + 330, + 304 + ], + "spans": [ + { + "bbox": [ + 280, + 290, + 330, + 304 + ], + "score": 0.9, + "content": "\\mathbf { P B } _ { n } = \\mathbf { P } ^ { \\prime } ,", + "type": "interline_equation", + "image_path": "a295912c9fdc3d8079021a4e4a58ef21f0cab90f50c2e152683860e89417f17c.jpg" + } + ] + } + ], + "index": 15, + "virtual_lines": [ + { + "bbox": [ + 280, + 290, + 330, + 304 + ], + "spans": [], + "index": 15 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 307, + 505, + 343 + ], + "lines": [ + { + "bbox": [ + 105, + 307, + 504, + 321 + ], + "spans": [ + { + "bbox": [ + 105, + 307, + 134, + 319 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 134, + 308, + 146, + 318 + ], + "score": 0.85, + "content": "\\mathbf { P ^ { \\prime } }", + "type": "inline_equation" + }, + { + "bbox": [ + 147, + 307, + 191, + 319 + ], + "score": 1.0, + "content": "meets the", + "type": "text" + }, + { + "bbox": [ + 192, + 308, + 200, + 321 + ], + "score": 0.86, + "content": "\\frac { n } { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 200, + 307, + 432, + 319 + ], + "score": 1.0, + "content": "balance condition. Now, we consider the first and last", + "type": "text" + }, + { + "bbox": [ + 432, + 308, + 440, + 321 + ], + "score": 0.84, + "content": "\\frac { n } { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 441, + 307, + 492, + 319 + ], + "score": 1.0, + "content": "columns of", + "type": "text" + }, + { + "bbox": [ + 492, + 307, + 504, + 317 + ], + "score": 0.85, + "content": "\\mathbf { P ^ { \\prime } }", + "type": "inline_equation" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 318, + 505, + 331 + ], + "spans": [ + { + "bbox": [ + 106, + 318, + 505, + 331 + ], + "score": 1.0, + "content": "independently. We can again apply Lemma G.2 (twice) to conclude that there are butterfly factors", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 108, + 328, + 212, + 345 + ], + "spans": [ + { + "bbox": [ + 108, + 329, + 170, + 344 + ], + "score": 0.91, + "content": "\\left[ \\mathbf { B } _ { \\frac { n } { 2 } } \\right] _ { 1 } , \\left[ \\mathbf { B } _ { \\frac { n } { 2 } } \\right] _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 171, + 328, + 212, + 345 + ], + "score": 1.0, + "content": "such that", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 17 + }, + { + "type": "interline_equation", + "bbox": [ + 208, + 348, + 401, + 378 + ], + "lines": [ + { + "bbox": [ + 208, + 348, + 401, + 378 + ], + "spans": [ + { + "bbox": [ + 208, + 348, + 401, + 378 + ], + "score": 0.93, + "content": "\\mathbf { P B } _ { n } \\left[ \\begin{array} { c c } { \\left[ \\mathbf { B } _ { \\frac { n } { 2 } } \\right] _ { 1 } } & { 0 } \\\\ { 0 } & { \\left[ \\mathbf { B } _ { \\frac { n } { 2 } } \\right] _ { 2 } } \\end{array} \\right] = \\mathbf { P B } _ { n } ^ { ( n ) } \\mathbf { B } _ { \\frac { n } { 2 } } ^ { ( n ) } = \\mathbf { P } ^ { \\prime \\prime } ,", + "type": "interline_equation", + "image_path": "f466ce619a581a644535e0693f9477d60a9d39de0b2dd7d8d8bfd4b730939922.jpg" + } + ] + } + ], + "index": 19.5, + "virtual_lines": [ + { + "bbox": [ + 208, + 348, + 401, + 363.0 + ], + "spans": [], + "index": 19 + }, + { + "bbox": [ + 208, + 363.0, + 401, + 378.0 + ], + "spans": [], + "index": 20 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 383, + 304, + 397 + ], + "lines": [ + { + "bbox": [ + 105, + 382, + 306, + 399 + ], + "spans": [ + { + "bbox": [ + 105, + 382, + 133, + 399 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 133, + 384, + 147, + 394 + ], + "score": 0.87, + "content": "\\mathbf { P } ^ { \\prime \\prime }", + "type": "inline_equation" + }, + { + "bbox": [ + 147, + 382, + 189, + 399 + ], + "score": 1.0, + "content": "meets the", + "type": "text" + }, + { + "bbox": [ + 189, + 385, + 197, + 397 + ], + "score": 0.86, + "content": "\\frac { n } { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 198, + 382, + 216, + 399 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 216, + 385, + 224, + 397 + ], + "score": 0.87, + "content": "\\frac { n } { 4 }", + "type": "inline_equation" + }, + { + "bbox": [ + 224, + 382, + 306, + 399 + ], + "score": 1.0, + "content": "balance conditions.", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 21 + }, + { + "type": "text", + "bbox": [ + 108, + 401, + 504, + 423 + ], + "lines": [ + { + "bbox": [ + 106, + 400, + 505, + 413 + ], + "spans": [ + { + "bbox": [ + 106, + 400, + 505, + 413 + ], + "score": 1.0, + "content": "We continue this process until we obtain a matrix that meets all of the balance conditions. Our final", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 412, + 203, + 424 + ], + "spans": [ + { + "bbox": [ + 106, + 412, + 203, + 424 + ], + "score": 1.0, + "content": "equation is of the form:", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 22.5 + }, + { + "type": "interline_equation", + "bbox": [ + 233, + 422, + 377, + 442 + ], + "lines": [ + { + "bbox": [ + 233, + 422, + 377, + 442 + ], + "spans": [ + { + "bbox": [ + 233, + 422, + 377, + 442 + ], + "score": 0.93, + "content": "\\mathbf { P } \\cdot \\mathbf { B } _ { n } ^ { \\left( n \\right) } \\mathbf { B } _ { \\frac { n } { 2 } } ^ { \\left( n \\right) } \\cdot \\ldots \\mathbf { B } _ { 2 } ^ { \\left( n \\right) } = \\mathbf { P B } = \\mathbf { L } ,", + "type": "interline_equation", + "image_path": "0c8bf0bb2c78363fdef2b9b2a33689d32ea3a5cba19e7d214bbf410262be555e.jpg" + } + ] + } + ], + "index": 24, + "virtual_lines": [ + { + "bbox": [ + 233, + 422, + 377, + 442 + ], + "spans": [], + "index": 24 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 445, + 507, + 468 + ], + "lines": [ + { + "bbox": [ + 105, + 444, + 507, + 459 + ], + "spans": [ + { + "bbox": [ + 105, + 444, + 132, + 459 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 133, + 446, + 141, + 456 + ], + "score": 0.83, + "content": "\\boldsymbol { B }", + "type": "inline_equation" + }, + { + "bbox": [ + 141, + 444, + 238, + 459 + ], + "score": 1.0, + "content": "is a butterfly matrix and", + "type": "text" + }, + { + "bbox": [ + 238, + 446, + 246, + 456 + ], + "score": 0.73, + "content": "\\mathbf { L }", + "type": "inline_equation" + }, + { + "bbox": [ + 247, + 444, + 383, + 459 + ], + "score": 1.0, + "content": "is a modular-balanced matrix. Let", + "type": "text" + }, + { + "bbox": [ + 383, + 445, + 451, + 456 + ], + "score": 0.91, + "content": "\\mathbf { R } = \\mathbf { B } ^ { - 1 } = \\mathbf { B } ^ { * }", + "type": "inline_equation" + }, + { + "bbox": [ + 451, + 444, + 478, + 459 + ], + "score": 1.0, + "content": "(since", + "type": "text" + }, + { + "bbox": [ + 479, + 446, + 488, + 456 + ], + "score": 0.69, + "content": "\\mathbf { B }", + "type": "inline_equation" + }, + { + "bbox": [ + 488, + 444, + 507, + 459 + ], + "score": 1.0, + "content": "is a", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 457, + 505, + 468 + ], + "spans": [ + { + "bbox": [ + 105, + 457, + 324, + 468 + ], + "score": 1.0, + "content": "permutation matrix, and thus is orthogonal) and hence", + "type": "text" + }, + { + "bbox": [ + 325, + 457, + 358, + 467 + ], + "score": 0.91, + "content": "\\mathbf { R } \\in B ^ { * }", + "type": "inline_equation" + }, + { + "bbox": [ + 358, + 457, + 385, + 468 + ], + "score": 1.0, + "content": ". Then", + "type": "text" + }, + { + "bbox": [ + 385, + 457, + 423, + 467 + ], + "score": 0.89, + "content": "\\mathbf { P } = \\mathbf { L R }", + "type": "inline_equation" + }, + { + "bbox": [ + 424, + 457, + 472, + 468 + ], + "score": 1.0, + "content": ", as desired.", + "type": "text" + }, + { + "bbox": [ + 495, + 457, + 505, + 467 + ], + "score": 0.999, + "content": "□", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 25.5 + }, + { + "type": "text", + "bbox": [ + 106, + 481, + 348, + 493 + ], + "lines": [ + { + "bbox": [ + 106, + 481, + 349, + 495 + ], + "spans": [ + { + "bbox": [ + 106, + 481, + 349, + 495 + ], + "score": 1.0, + "content": "Theorem 2 follows immediately from Lemmas G.3 and G.1.", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 27 + }, + { + "type": "text", + "bbox": [ + 108, + 498, + 504, + 522 + ], + "lines": [ + { + "bbox": [ + 105, + 497, + 505, + 513 + ], + "spans": [ + { + "bbox": [ + 105, + 497, + 284, + 513 + ], + "score": 1.0, + "content": "We now show that permutations are also in", + "type": "text" + }, + { + "bbox": [ + 285, + 499, + 304, + 509 + ], + "score": 0.89, + "content": "B ^ { * } B", + "type": "inline_equation" + }, + { + "bbox": [ + 304, + 497, + 505, + 513 + ], + "score": 1.0, + "content": ". We start with the relationship between butterfly", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 510, + 274, + 522 + ], + "spans": [ + { + "bbox": [ + 106, + 510, + 274, + 522 + ], + "score": 1.0, + "content": "matrices and the bit-reversal permutation.", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 28.5 + }, + { + "type": "text", + "bbox": [ + 106, + 525, + 505, + 547 + ], + "lines": [ + { + "bbox": [ + 106, + 525, + 506, + 538 + ], + "spans": [ + { + "bbox": [ + 106, + 525, + 179, + 538 + ], + "score": 1.0, + "content": "Lemma G.4. Let", + "type": "text" + }, + { + "bbox": [ + 179, + 525, + 196, + 536 + ], + "score": 0.89, + "content": "\\mathbf { P } _ { \\mathrm { b r } }", + "type": "inline_equation" + }, + { + "bbox": [ + 196, + 525, + 223, + 538 + ], + "score": 1.0, + "content": "be the", + "type": "text" + }, + { + "bbox": [ + 224, + 526, + 249, + 536 + ], + "score": 0.89, + "content": "n \\times n", + "type": "inline_equation" + }, + { + "bbox": [ + 249, + 525, + 506, + 538 + ], + "score": 1.0, + "content": "bit-reversal permutation matrix where n is some power of 2, and", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 536, + 491, + 549 + ], + "spans": [ + { + "bbox": [ + 106, + 536, + 119, + 549 + ], + "score": 1.0, + "content": "let", + "type": "text" + }, + { + "bbox": [ + 119, + 536, + 155, + 547 + ], + "score": 0.92, + "content": "{ \\bf M } _ { 1 } \\in { B }", + "type": "inline_equation" + }, + { + "bbox": [ + 155, + 536, + 180, + 549 + ], + "score": 1.0, + "content": "be an", + "type": "text" + }, + { + "bbox": [ + 180, + 538, + 205, + 547 + ], + "score": 0.89, + "content": "n \\times n", + "type": "inline_equation" + }, + { + "bbox": [ + 206, + 536, + 415, + 549 + ], + "score": 1.0, + "content": "butterfly matrix. Then there is some butterfly matrix", + "type": "text" + }, + { + "bbox": [ + 415, + 537, + 451, + 547 + ], + "score": 0.92, + "content": "{ \\bf M } _ { 2 } \\in { B }", + "type": "inline_equation" + }, + { + "bbox": [ + 451, + 536, + 491, + 549 + ], + "score": 1.0, + "content": "such that", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 30.5 + }, + { + "type": "interline_equation", + "bbox": [ + 264, + 554, + 346, + 568 + ], + "lines": [ + { + "bbox": [ + 264, + 554, + 346, + 568 + ], + "spans": [ + { + "bbox": [ + 264, + 554, + 346, + 568 + ], + "score": 0.91, + "content": "\\mathbf { M } _ { 1 } ^ { * } = \\mathbf { P } _ { \\mathrm { b r } } \\mathbf { M } _ { 2 } \\mathbf { P } _ { \\mathrm { b r } } .", + "type": "interline_equation", + "image_path": "38ebc511817bb20e47cd2363018c0b54183cacf0471e56cd3378fc2bf70e9b43.jpg" + } + ] + } + ], + "index": 32, + "virtual_lines": [ + { + "bbox": [ + 264, + 554, + 346, + 568 + ], + "spans": [], + "index": 32 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 579, + 506, + 702 + ], + "lines": [ + { + "bbox": [ + 105, + 579, + 506, + 594 + ], + "spans": [ + { + "bbox": [ + 105, + 579, + 249, + 594 + ], + "score": 1.0, + "content": "Proof sketch. For any input vector", + "type": "text" + }, + { + "bbox": [ + 249, + 582, + 257, + 591 + ], + "score": 0.54, + "content": "\\mathbf { x }", + "type": "inline_equation" + }, + { + "bbox": [ + 258, + 579, + 298, + 594 + ], + "score": 1.0, + "content": "of length", + "type": "text" + }, + { + "bbox": [ + 298, + 583, + 305, + 591 + ], + "score": 0.79, + "content": "n", + "type": "inline_equation" + }, + { + "bbox": [ + 305, + 579, + 355, + 594 + ], + "score": 1.0, + "content": ", to perform", + "type": "text" + }, + { + "bbox": [ + 356, + 581, + 378, + 592 + ], + "score": 0.91, + "content": "\\mathbf { M } _ { 1 } ^ { * } \\mathbf { x }", + "type": "inline_equation" + }, + { + "bbox": [ + 379, + 579, + 454, + 594 + ], + "score": 1.0, + "content": ", we trace through", + "type": "text" + }, + { + "bbox": [ + 454, + 581, + 481, + 592 + ], + "score": 0.91, + "content": "\\log _ { 2 } n", + "type": "inline_equation" + }, + { + "bbox": [ + 481, + 579, + 506, + 594 + ], + "score": 1.0, + "content": "steps", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 591, + 506, + 604 + ], + "spans": [ + { + "bbox": [ + 105, + 591, + 341, + 604 + ], + "score": 1.0, + "content": "of the multiplication algorithm. At each step, we perform", + "type": "text" + }, + { + "bbox": [ + 341, + 592, + 365, + 601 + ], + "score": 0.89, + "content": "2 \\times 2", + "type": "inline_equation" + }, + { + "bbox": [ + 365, + 591, + 506, + 604 + ], + "score": 1.0, + "content": "matrix multiplication on elements", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 601, + 506, + 615 + ], + "spans": [ + { + "bbox": [ + 105, + 601, + 117, + 615 + ], + "score": 1.0, + "content": "of", + "type": "text" + }, + { + "bbox": [ + 117, + 604, + 125, + 612 + ], + "score": 0.59, + "content": "\\mathbf { x }", + "type": "inline_equation" + }, + { + "bbox": [ + 126, + 601, + 200, + 615 + ], + "score": 1.0, + "content": "whose indices are", + "type": "text" + }, + { + "bbox": [ + 200, + 603, + 218, + 614 + ], + "score": 0.89, + "content": "n / 2", + "type": "inline_equation" + }, + { + "bbox": [ + 218, + 601, + 317, + 615 + ], + "score": 1.0, + "content": "apart (e.g. indices 0 and", + "type": "text" + }, + { + "bbox": [ + 318, + 603, + 345, + 614 + ], + "score": 0.65, + "content": "n / 2 , 1", + "type": "inline_equation" + }, + { + "bbox": [ + 345, + 601, + 363, + 615 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 363, + 602, + 398, + 614 + ], + "score": 0.92, + "content": "n / 2 + 1", + "type": "inline_equation" + }, + { + "bbox": [ + 398, + 601, + 444, + 615 + ], + "score": 1.0, + "content": ", etc.), then", + "type": "text" + }, + { + "bbox": [ + 444, + 602, + 462, + 614 + ], + "score": 0.89, + "content": "n / 4", + "type": "inline_equation" + }, + { + "bbox": [ + 462, + 601, + 506, + 615 + ], + "score": 1.0, + "content": "apart, and", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 613, + 506, + 625 + ], + "spans": [ + { + "bbox": [ + 105, + 613, + 424, + 625 + ], + "score": 1.0, + "content": "so on, till indices are that 1 apart. If we apply the bit-reversal permutation on", + "type": "text" + }, + { + "bbox": [ + 424, + 615, + 432, + 623 + ], + "score": 0.29, + "content": "\\mathbf { x }", + "type": "inline_equation" + }, + { + "bbox": [ + 432, + 613, + 506, + 625 + ], + "score": 1.0, + "content": ", then indices that", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 624, + 506, + 637 + ], + "spans": [ + { + "bbox": [ + 105, + 624, + 121, + 637 + ], + "score": 1.0, + "content": "are", + "type": "text" + }, + { + "bbox": [ + 121, + 624, + 138, + 636 + ], + "score": 0.88, + "content": "n / 2", + "type": "inline_equation" + }, + { + "bbox": [ + 139, + 624, + 308, + 637 + ], + "score": 1.0, + "content": "apart will become 1 apart, indices that are", + "type": "text" + }, + { + "bbox": [ + 309, + 624, + 326, + 636 + ], + "score": 0.89, + "content": "n / 4", + "type": "inline_equation" + }, + { + "bbox": [ + 326, + 624, + 506, + 637 + ], + "score": 1.0, + "content": "apart will become 2 apart, and so on. So the", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 635, + 506, + 649 + ], + "spans": [ + { + "bbox": [ + 105, + 635, + 206, + 649 + ], + "score": 1.0, + "content": "multiplication algorithm", + "type": "text" + }, + { + "bbox": [ + 207, + 635, + 230, + 647 + ], + "score": 0.92, + "content": "\\mathbf { M } _ { 1 } ^ { * } \\mathbf { x }", + "type": "inline_equation" + }, + { + "bbox": [ + 230, + 635, + 506, + 649 + ], + "score": 1.0, + "content": "is equivalent to applying bit-reversal, then multiplying the permuted", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 646, + 506, + 659 + ], + "spans": [ + { + "bbox": [ + 105, + 646, + 276, + 659 + ], + "score": 1.0, + "content": "vector with another butterfly matrix (i.e.", + "type": "text" + }, + { + "bbox": [ + 276, + 647, + 301, + 657 + ], + "score": 0.89, + "content": "2 \\times 2", + "type": "inline_equation" + }, + { + "bbox": [ + 301, + 646, + 506, + 659 + ], + "score": 1.0, + "content": "matrix multiplication on indices that are 1 apart,", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 106, + 657, + 506, + 669 + ], + "spans": [ + { + "bbox": [ + 106, + 658, + 283, + 669 + ], + "score": 1.0, + "content": "then 2 apart, and so on, till indices that are", + "type": "text" + }, + { + "bbox": [ + 283, + 657, + 300, + 669 + ], + "score": 0.87, + "content": "n / 2", + "type": "inline_equation" + }, + { + "bbox": [ + 301, + 658, + 506, + 669 + ], + "score": 1.0, + "content": "apart). Finally we need to do another bit-reversal", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 104, + 667, + 507, + 681 + ], + "spans": [ + { + "bbox": [ + 104, + 667, + 486, + 681 + ], + "score": 1.0, + "content": "permutation to put all the indices back to the original order. If we call this other butterfly matrix", + "type": "text" + }, + { + "bbox": [ + 486, + 668, + 502, + 679 + ], + "score": 0.89, + "content": "\\mathbf { M } _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 502, + 667, + 507, + 681 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 678, + 506, + 691 + ], + "spans": [ + { + "bbox": [ + 105, + 678, + 209, + 691 + ], + "score": 1.0, + "content": "then we have shown that", + "type": "text" + }, + { + "bbox": [ + 209, + 679, + 299, + 691 + ], + "score": 0.91, + "content": "\\mathbf { M } _ { 1 } ^ { * } \\mathbf { x } = \\mathbf { P } _ { \\mathrm { b r } } \\mathbf { M } _ { 2 } \\mathbf { P } _ { \\mathrm { b r } } \\mathbf { x }", + "type": "inline_equation" + }, + { + "bbox": [ + 299, + 678, + 375, + 691 + ], + "score": 1.0, + "content": ". This holds for all", + "type": "text" + }, + { + "bbox": [ + 375, + 680, + 383, + 689 + ], + "score": 0.65, + "content": "\\mathbf { x }", + "type": "inline_equation" + }, + { + "bbox": [ + 384, + 678, + 469, + 691 + ], + "score": 1.0, + "content": "(for the same matrix", + "type": "text" + }, + { + "bbox": [ + 470, + 680, + 486, + 690 + ], + "score": 0.86, + "content": "\\mathbf { M } _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 487, + 678, + 506, + 691 + ], + "score": 1.0, + "content": "), so", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 689, + 505, + 703 + ], + "spans": [ + { + "bbox": [ + 105, + 689, + 141, + 703 + ], + "score": 1.0, + "content": "we have", + "type": "text" + }, + { + "bbox": [ + 141, + 690, + 218, + 702 + ], + "score": 0.92, + "content": "\\mathbf { M } _ { 1 } ^ { * } = \\mathbf { P } _ { \\mathrm { b r } } \\mathbf { M } _ { 2 } \\mathbf { P } _ { \\mathrm { b r } }", + "type": "inline_equation" + }, + { + "bbox": [ + 218, + 689, + 222, + 703 + ], + "score": 1.0, + "content": ".", + "type": "text" + }, + { + "bbox": [ + 494, + 691, + 505, + 700 + ], + "score": 0.999, + "content": "□", + "type": "text" + } + ], + "index": 43 + } + ], + "index": 38 + }, + { + "type": "text", + "bbox": [ + 106, + 709, + 504, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 709, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 106, + 709, + 505, + 722 + ], + "score": 1.0, + "content": "Remark G.5. Lemma G.4 explains the connection between the two most common fast Fourier", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 106, + 721, + 505, + 734 + ], + "spans": [ + { + "bbox": [ + 106, + 721, + 505, + 734 + ], + "score": 1.0, + "content": "transform algorithm, decimation in time and decimation in frequency. Using the decimation-in-time", + "type": "text" + } + ], + "index": 45 + } + ], + "index": 44.5 + } + ], + "page_idx": 26, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 107, + 27, + 293, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 26, + 294, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 26, + 294, + 38 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2020", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 300, + 751, + 311, + 760 + ], + "lines": [ + { + "bbox": [ + 298, + 750, + 313, + 765 + ], + "spans": [ + { + "bbox": [ + 298, + 750, + 313, + 765 + ], + "score": 1.0, + "content": "", + "type": "text", + "height": 15, + "width": 15 + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 81, + 505, + 129 + ], + "lines": [ + { + "bbox": [ + 104, + 81, + 506, + 96 + ], + "spans": [ + { + "bbox": [ + 104, + 81, + 259, + 95 + ], + "score": 1.0, + "content": "Proof. We construct a directed graph", + "type": "text" + }, + { + "bbox": [ + 260, + 83, + 269, + 93 + ], + "score": 0.8, + "content": "G", + "type": "inline_equation" + }, + { + "bbox": [ + 270, + 81, + 329, + 95 + ], + "score": 1.0, + "content": "with nodes in", + "type": "text" + }, + { + "bbox": [ + 329, + 81, + 343, + 96 + ], + "score": 0.9, + "content": "\\left[ \\frac { k } { 2 } \\right]", + "type": "inline_equation" + }, + { + "bbox": [ + 344, + 81, + 386, + 95 + ], + "score": 1.0, + "content": ". For each", + "type": "text" + }, + { + "bbox": [ + 386, + 81, + 430, + 95 + ], + "score": 0.91, + "content": "\\begin{array} { r } { 0 \\leq i < \\frac { k } { 2 } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 430, + 81, + 506, + 95 + ], + "score": 1.0, + "content": "we add a directed", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 104, + 91, + 505, + 110 + ], + "spans": [ + { + "bbox": [ + 104, + 91, + 172, + 110 + ], + "score": 1.0, + "content": "edge from node", + "type": "text" + }, + { + "bbox": [ + 172, + 94, + 219, + 109 + ], + "score": 0.71, + "content": "\\left( s \\mod \\frac { k } { 2 } \\right)", + "type": "inline_equation" + }, + { + "bbox": [ + 219, + 91, + 255, + 110 + ], + "score": 1.0, + "content": "\u0001 to node", + "type": "text" + }, + { + "bbox": [ + 256, + 95, + 304, + 109 + ], + "score": 0.71, + "content": "\\left( t \\mod \\frac { k } { 2 } \\right)", + "type": "inline_equation" + }, + { + "bbox": [ + 304, + 91, + 314, + 110 + ], + "score": 1.0, + "content": "if", + "type": "text" + }, + { + "bbox": [ + 315, + 96, + 365, + 108 + ], + "score": 0.88, + "content": "\\mathbf { M } [ : , i ] = \\mathbf { e } _ { s }", + "type": "inline_equation" + }, + { + "bbox": [ + 366, + 91, + 383, + 110 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 384, + 95, + 456, + 109 + ], + "score": 0.88, + "content": "\\begin{array} { r } { \\mathbf { M } \\left[ : , i + \\frac { k } { 2 } \\right] = \\mathbf { e } _ { t } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 457, + 91, + 505, + 110 + ], + "score": 1.0, + "content": ". Each node", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 105, + 506, + 119 + ], + "spans": [ + { + "bbox": [ + 105, + 105, + 361, + 119 + ], + "score": 1.0, + "content": "has (undirected) degree exactly 2 by the structure of M. Hence,", + "type": "text" + }, + { + "bbox": [ + 361, + 107, + 370, + 117 + ], + "score": 0.81, + "content": "G", + "type": "inline_equation" + }, + { + "bbox": [ + 370, + 105, + 506, + 119 + ], + "score": 1.0, + "content": "is a union of disjoint (undirected)", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 117, + 138, + 132 + ], + "spans": [ + { + "bbox": [ + 105, + 117, + 138, + 132 + ], + "score": 1.0, + "content": "cycles.", + "type": "text" + } + ], + "index": 3 + } + ], + "index": 1.5, + "bbox_fs": [ + 104, + 81, + 506, + 132 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 134, + 505, + 203 + ], + "lines": [ + { + "bbox": [ + 105, + 133, + 506, + 148 + ], + "spans": [ + { + "bbox": [ + 105, + 133, + 162, + 148 + ], + "score": 1.0, + "content": "If M met the", + "type": "text" + }, + { + "bbox": [ + 162, + 133, + 169, + 147 + ], + "score": 0.86, + "content": "\\frac { k } { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 170, + 133, + 506, + 148 + ], + "score": 1.0, + "content": "balance condition, then each node would additionally have in-degree exactly 1 and", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 106, + 146, + 505, + 158 + ], + "spans": [ + { + "bbox": [ + 106, + 146, + 282, + 158 + ], + "score": 1.0, + "content": "out-degree exactly 1. By reversing edges of", + "type": "text" + }, + { + "bbox": [ + 282, + 146, + 291, + 156 + ], + "score": 0.81, + "content": "G", + "type": "inline_equation" + }, + { + "bbox": [ + 292, + 146, + 505, + 158 + ], + "score": 1.0, + "content": "such that each (undirected) cycle becomes a directed", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 157, + 506, + 169 + ], + "spans": [ + { + "bbox": [ + 105, + 157, + 474, + 169 + ], + "score": 1.0, + "content": "cycle, we can achieve this. However, reversing edges corresponds to swapping columns of", + "type": "text" + }, + { + "bbox": [ + 474, + 157, + 487, + 167 + ], + "score": 0.29, + "content": "\\mathbf { M }", + "type": "inline_equation" + }, + { + "bbox": [ + 487, + 157, + 506, + 169 + ], + "score": 1.0, + "content": "that", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 107, + 166, + 505, + 182 + ], + "spans": [ + { + "bbox": [ + 107, + 166, + 121, + 182 + ], + "score": 1.0, + "content": "are", + "type": "text" + }, + { + "bbox": [ + 121, + 167, + 129, + 181 + ], + "score": 0.87, + "content": "\\frac { k } { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 129, + 166, + 171, + 182 + ], + "score": 1.0, + "content": "apart. Let", + "type": "text" + }, + { + "bbox": [ + 171, + 168, + 185, + 179 + ], + "score": 0.88, + "content": "\\mathbf { B } _ { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 186, + 166, + 407, + 182 + ], + "score": 1.0, + "content": "be the permutation matrix that performs all such swaps.", + "type": "text" + }, + { + "bbox": [ + 408, + 168, + 422, + 179 + ], + "score": 0.88, + "content": "\\mathbf { B } _ { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 423, + 166, + 505, + 182 + ], + "score": 1.0, + "content": "has non-zero entries", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 179, + 505, + 194 + ], + "spans": [ + { + "bbox": [ + 105, + 179, + 324, + 194 + ], + "score": 1.0, + "content": "only along the main diagonal and the diagonal bands", + "type": "text" + }, + { + "bbox": [ + 325, + 180, + 332, + 194 + ], + "score": 0.86, + "content": "\\frac { k } { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 333, + 179, + 505, + 194 + ], + "score": 1.0, + "content": "away, and thus is a butterfly factor of size", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 192, + 505, + 204 + ], + "spans": [ + { + "bbox": [ + 106, + 192, + 113, + 201 + ], + "score": 0.78, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 114, + 192, + 118, + 204 + ], + "score": 1.0, + "content": ".", + "type": "text" + }, + { + "bbox": [ + 494, + 192, + 505, + 203 + ], + "score": 0.999, + "content": "□", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 6.5, + "bbox_fs": [ + 105, + 133, + 506, + 204 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 217, + 300, + 228 + ], + "lines": [ + { + "bbox": [ + 106, + 216, + 298, + 230 + ], + "spans": [ + { + "bbox": [ + 106, + 216, + 289, + 230 + ], + "score": 1.0, + "content": "We are ready to present the decomposition of", + "type": "text" + }, + { + "bbox": [ + 289, + 218, + 298, + 227 + ], + "score": 0.77, + "content": "\\mathbf { P }", + "type": "inline_equation" + } + ], + "index": 10 + } + ], + "index": 10, + "bbox_fs": [ + 106, + 216, + 298, + 230 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 232, + 505, + 254 + ], + "lines": [ + { + "bbox": [ + 105, + 232, + 506, + 245 + ], + "spans": [ + { + "bbox": [ + 105, + 232, + 181, + 245 + ], + "score": 1.0, + "content": "Lemma G.3. Let", + "type": "text" + }, + { + "bbox": [ + 181, + 233, + 190, + 243 + ], + "score": 0.52, + "content": "\\mathbf { P }", + "type": "inline_equation" + }, + { + "bbox": [ + 190, + 232, + 217, + 245 + ], + "score": 1.0, + "content": "be an", + "type": "text" + }, + { + "bbox": [ + 217, + 234, + 243, + 243 + ], + "score": 0.9, + "content": "n \\times n", + "type": "inline_equation" + }, + { + "bbox": [ + 244, + 232, + 434, + 245 + ], + "score": 1.0, + "content": "permutation matrix. Then we can decompose", + "type": "text" + }, + { + "bbox": [ + 434, + 233, + 443, + 243 + ], + "score": 0.58, + "content": "\\mathbf { P }", + "type": "inline_equation" + }, + { + "bbox": [ + 444, + 232, + 463, + 245 + ], + "score": 1.0, + "content": "into", + "type": "text" + }, + { + "bbox": [ + 464, + 232, + 502, + 243 + ], + "score": 0.87, + "content": "\\mathbf { P } = \\mathbf { L R }", + "type": "inline_equation" + }, + { + "bbox": [ + 503, + 232, + 506, + 245 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 243, + 281, + 255 + ], + "spans": [ + { + "bbox": [ + 106, + 243, + 245, + 255 + ], + "score": 1.0, + "content": "where L is modular-balanced and", + "type": "text" + }, + { + "bbox": [ + 245, + 244, + 277, + 254 + ], + "score": 0.91, + "content": "\\mathbf { R } \\in B ^ { * }", + "type": "inline_equation" + }, + { + "bbox": [ + 278, + 243, + 281, + 255 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 11.5, + "bbox_fs": [ + 105, + 232, + 506, + 255 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 268, + 505, + 290 + ], + "lines": [ + { + "bbox": [ + 105, + 267, + 505, + 282 + ], + "spans": [ + { + "bbox": [ + 105, + 267, + 468, + 282 + ], + "score": 1.0, + "content": "Proof. We repeatedly apply Lemma G.2. First, we conclude that there is a butterfly factor", + "type": "text" + }, + { + "bbox": [ + 468, + 269, + 483, + 280 + ], + "score": 0.89, + "content": "\\mathbf { B } _ { n }", + "type": "inline_equation" + }, + { + "bbox": [ + 483, + 267, + 505, + 282 + ], + "score": 1.0, + "content": "such", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 278, + 126, + 293 + ], + "spans": [ + { + "bbox": [ + 105, + 278, + 126, + 293 + ], + "score": 1.0, + "content": "that", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 13.5, + "bbox_fs": [ + 105, + 267, + 505, + 293 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 280, + 290, + 330, + 304 + ], + "lines": [ + { + "bbox": [ + 280, + 290, + 330, + 304 + ], + "spans": [ + { + "bbox": [ + 280, + 290, + 330, + 304 + ], + "score": 0.9, + "content": "\\mathbf { P B } _ { n } = \\mathbf { P } ^ { \\prime } ,", + "type": "interline_equation", + "image_path": "a295912c9fdc3d8079021a4e4a58ef21f0cab90f50c2e152683860e89417f17c.jpg" + } + ] + } + ], + "index": 15, + "virtual_lines": [ + { + "bbox": [ + 280, + 290, + 330, + 304 + ], + "spans": [], + "index": 15 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 307, + 505, + 343 + ], + "lines": [ + { + "bbox": [ + 105, + 307, + 504, + 321 + ], + "spans": [ + { + "bbox": [ + 105, + 307, + 134, + 319 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 134, + 308, + 146, + 318 + ], + "score": 0.85, + "content": "\\mathbf { P ^ { \\prime } }", + "type": "inline_equation" + }, + { + "bbox": [ + 147, + 307, + 191, + 319 + ], + "score": 1.0, + "content": "meets the", + "type": "text" + }, + { + "bbox": [ + 192, + 308, + 200, + 321 + ], + "score": 0.86, + "content": "\\frac { n } { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 200, + 307, + 432, + 319 + ], + "score": 1.0, + "content": "balance condition. Now, we consider the first and last", + "type": "text" + }, + { + "bbox": [ + 432, + 308, + 440, + 321 + ], + "score": 0.84, + "content": "\\frac { n } { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 441, + 307, + 492, + 319 + ], + "score": 1.0, + "content": "columns of", + "type": "text" + }, + { + "bbox": [ + 492, + 307, + 504, + 317 + ], + "score": 0.85, + "content": "\\mathbf { P ^ { \\prime } }", + "type": "inline_equation" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 318, + 505, + 331 + ], + "spans": [ + { + "bbox": [ + 106, + 318, + 505, + 331 + ], + "score": 1.0, + "content": "independently. We can again apply Lemma G.2 (twice) to conclude that there are butterfly factors", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 108, + 328, + 212, + 345 + ], + "spans": [ + { + "bbox": [ + 108, + 329, + 170, + 344 + ], + "score": 0.91, + "content": "\\left[ \\mathbf { B } _ { \\frac { n } { 2 } } \\right] _ { 1 } , \\left[ \\mathbf { B } _ { \\frac { n } { 2 } } \\right] _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 171, + 328, + 212, + 345 + ], + "score": 1.0, + "content": "such that", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 17, + "bbox_fs": [ + 105, + 307, + 505, + 345 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 208, + 348, + 401, + 378 + ], + "lines": [ + { + "bbox": [ + 208, + 348, + 401, + 378 + ], + "spans": [ + { + "bbox": [ + 208, + 348, + 401, + 378 + ], + "score": 0.93, + "content": "\\mathbf { P B } _ { n } \\left[ \\begin{array} { c c } { \\left[ \\mathbf { B } _ { \\frac { n } { 2 } } \\right] _ { 1 } } & { 0 } \\\\ { 0 } & { \\left[ \\mathbf { B } _ { \\frac { n } { 2 } } \\right] _ { 2 } } \\end{array} \\right] = \\mathbf { P B } _ { n } ^ { ( n ) } \\mathbf { B } _ { \\frac { n } { 2 } } ^ { ( n ) } = \\mathbf { P } ^ { \\prime \\prime } ,", + "type": "interline_equation", + "image_path": "f466ce619a581a644535e0693f9477d60a9d39de0b2dd7d8d8bfd4b730939922.jpg" + } + ] + } + ], + "index": 19.5, + "virtual_lines": [ + { + "bbox": [ + 208, + 348, + 401, + 363.0 + ], + "spans": [], + "index": 19 + }, + { + "bbox": [ + 208, + 363.0, + 401, + 378.0 + ], + "spans": [], + "index": 20 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 383, + 304, + 397 + ], + "lines": [ + { + "bbox": [ + 105, + 382, + 306, + 399 + ], + "spans": [ + { + "bbox": [ + 105, + 382, + 133, + 399 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 133, + 384, + 147, + 394 + ], + "score": 0.87, + "content": "\\mathbf { P } ^ { \\prime \\prime }", + "type": "inline_equation" + }, + { + "bbox": [ + 147, + 382, + 189, + 399 + ], + "score": 1.0, + "content": "meets the", + "type": "text" + }, + { + "bbox": [ + 189, + 385, + 197, + 397 + ], + "score": 0.86, + "content": "\\frac { n } { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 198, + 382, + 216, + 399 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 216, + 385, + 224, + 397 + ], + "score": 0.87, + "content": "\\frac { n } { 4 }", + "type": "inline_equation" + }, + { + "bbox": [ + 224, + 382, + 306, + 399 + ], + "score": 1.0, + "content": "balance conditions.", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 21, + "bbox_fs": [ + 105, + 382, + 306, + 399 + ] + }, + { + "type": "text", + "bbox": [ + 108, + 401, + 504, + 423 + ], + "lines": [ + { + "bbox": [ + 106, + 400, + 505, + 413 + ], + "spans": [ + { + "bbox": [ + 106, + 400, + 505, + 413 + ], + "score": 1.0, + "content": "We continue this process until we obtain a matrix that meets all of the balance conditions. Our final", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 412, + 203, + 424 + ], + "spans": [ + { + "bbox": [ + 106, + 412, + 203, + 424 + ], + "score": 1.0, + "content": "equation is of the form:", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 22.5, + "bbox_fs": [ + 106, + 400, + 505, + 424 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 233, + 422, + 377, + 442 + ], + "lines": [ + { + "bbox": [ + 233, + 422, + 377, + 442 + ], + "spans": [ + { + "bbox": [ + 233, + 422, + 377, + 442 + ], + "score": 0.93, + "content": "\\mathbf { P } \\cdot \\mathbf { B } _ { n } ^ { \\left( n \\right) } \\mathbf { B } _ { \\frac { n } { 2 } } ^ { \\left( n \\right) } \\cdot \\ldots \\mathbf { B } _ { 2 } ^ { \\left( n \\right) } = \\mathbf { P B } = \\mathbf { L } ,", + "type": "interline_equation", + "image_path": "0c8bf0bb2c78363fdef2b9b2a33689d32ea3a5cba19e7d214bbf410262be555e.jpg" + } + ] + } + ], + "index": 24, + "virtual_lines": [ + { + "bbox": [ + 233, + 422, + 377, + 442 + ], + "spans": [], + "index": 24 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 445, + 507, + 468 + ], + "lines": [ + { + "bbox": [ + 105, + 444, + 507, + 459 + ], + "spans": [ + { + "bbox": [ + 105, + 444, + 132, + 459 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 133, + 446, + 141, + 456 + ], + "score": 0.83, + "content": "\\boldsymbol { B }", + "type": "inline_equation" + }, + { + "bbox": [ + 141, + 444, + 238, + 459 + ], + "score": 1.0, + "content": "is a butterfly matrix and", + "type": "text" + }, + { + "bbox": [ + 238, + 446, + 246, + 456 + ], + "score": 0.73, + "content": "\\mathbf { L }", + "type": "inline_equation" + }, + { + "bbox": [ + 247, + 444, + 383, + 459 + ], + "score": 1.0, + "content": "is a modular-balanced matrix. Let", + "type": "text" + }, + { + "bbox": [ + 383, + 445, + 451, + 456 + ], + "score": 0.91, + "content": "\\mathbf { R } = \\mathbf { B } ^ { - 1 } = \\mathbf { B } ^ { * }", + "type": "inline_equation" + }, + { + "bbox": [ + 451, + 444, + 478, + 459 + ], + "score": 1.0, + "content": "(since", + "type": "text" + }, + { + "bbox": [ + 479, + 446, + 488, + 456 + ], + "score": 0.69, + "content": "\\mathbf { B }", + "type": "inline_equation" + }, + { + "bbox": [ + 488, + 444, + 507, + 459 + ], + "score": 1.0, + "content": "is a", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 457, + 505, + 468 + ], + "spans": [ + { + "bbox": [ + 105, + 457, + 324, + 468 + ], + "score": 1.0, + "content": "permutation matrix, and thus is orthogonal) and hence", + "type": "text" + }, + { + "bbox": [ + 325, + 457, + 358, + 467 + ], + "score": 0.91, + "content": "\\mathbf { R } \\in B ^ { * }", + "type": "inline_equation" + }, + { + "bbox": [ + 358, + 457, + 385, + 468 + ], + "score": 1.0, + "content": ". Then", + "type": "text" + }, + { + "bbox": [ + 385, + 457, + 423, + 467 + ], + "score": 0.89, + "content": "\\mathbf { P } = \\mathbf { L R }", + "type": "inline_equation" + }, + { + "bbox": [ + 424, + 457, + 472, + 468 + ], + "score": 1.0, + "content": ", as desired.", + "type": "text" + }, + { + "bbox": [ + 495, + 457, + 505, + 467 + ], + "score": 0.999, + "content": "□", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 25.5, + "bbox_fs": [ + 105, + 444, + 507, + 468 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 481, + 348, + 493 + ], + "lines": [ + { + "bbox": [ + 106, + 481, + 349, + 495 + ], + "spans": [ + { + "bbox": [ + 106, + 481, + 349, + 495 + ], + "score": 1.0, + "content": "Theorem 2 follows immediately from Lemmas G.3 and G.1.", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 27, + "bbox_fs": [ + 106, + 481, + 349, + 495 + ] + }, + { + "type": "text", + "bbox": [ + 108, + 498, + 504, + 522 + ], + "lines": [ + { + "bbox": [ + 105, + 497, + 505, + 513 + ], + "spans": [ + { + "bbox": [ + 105, + 497, + 284, + 513 + ], + "score": 1.0, + "content": "We now show that permutations are also in", + "type": "text" + }, + { + "bbox": [ + 285, + 499, + 304, + 509 + ], + "score": 0.89, + "content": "B ^ { * } B", + "type": "inline_equation" + }, + { + "bbox": [ + 304, + 497, + 505, + 513 + ], + "score": 1.0, + "content": ". We start with the relationship between butterfly", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 510, + 274, + 522 + ], + "spans": [ + { + "bbox": [ + 106, + 510, + 274, + 522 + ], + "score": 1.0, + "content": "matrices and the bit-reversal permutation.", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 28.5, + "bbox_fs": [ + 105, + 497, + 505, + 522 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 525, + 505, + 547 + ], + "lines": [ + { + "bbox": [ + 106, + 525, + 506, + 538 + ], + "spans": [ + { + "bbox": [ + 106, + 525, + 179, + 538 + ], + "score": 1.0, + "content": "Lemma G.4. Let", + "type": "text" + }, + { + "bbox": [ + 179, + 525, + 196, + 536 + ], + "score": 0.89, + "content": "\\mathbf { P } _ { \\mathrm { b r } }", + "type": "inline_equation" + }, + { + "bbox": [ + 196, + 525, + 223, + 538 + ], + "score": 1.0, + "content": "be the", + "type": "text" + }, + { + "bbox": [ + 224, + 526, + 249, + 536 + ], + "score": 0.89, + "content": "n \\times n", + "type": "inline_equation" + }, + { + "bbox": [ + 249, + 525, + 506, + 538 + ], + "score": 1.0, + "content": "bit-reversal permutation matrix where n is some power of 2, and", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 536, + 491, + 549 + ], + "spans": [ + { + "bbox": [ + 106, + 536, + 119, + 549 + ], + "score": 1.0, + "content": "let", + "type": "text" + }, + { + "bbox": [ + 119, + 536, + 155, + 547 + ], + "score": 0.92, + "content": "{ \\bf M } _ { 1 } \\in { B }", + "type": "inline_equation" + }, + { + "bbox": [ + 155, + 536, + 180, + 549 + ], + "score": 1.0, + "content": "be an", + "type": "text" + }, + { + "bbox": [ + 180, + 538, + 205, + 547 + ], + "score": 0.89, + "content": "n \\times n", + "type": "inline_equation" + }, + { + "bbox": [ + 206, + 536, + 415, + 549 + ], + "score": 1.0, + "content": "butterfly matrix. Then there is some butterfly matrix", + "type": "text" + }, + { + "bbox": [ + 415, + 537, + 451, + 547 + ], + "score": 0.92, + "content": "{ \\bf M } _ { 2 } \\in { B }", + "type": "inline_equation" + }, + { + "bbox": [ + 451, + 536, + 491, + 549 + ], + "score": 1.0, + "content": "such that", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 30.5, + "bbox_fs": [ + 106, + 525, + 506, + 549 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 264, + 554, + 346, + 568 + ], + "lines": [ + { + "bbox": [ + 264, + 554, + 346, + 568 + ], + "spans": [ + { + "bbox": [ + 264, + 554, + 346, + 568 + ], + "score": 0.91, + "content": "\\mathbf { M } _ { 1 } ^ { * } = \\mathbf { P } _ { \\mathrm { b r } } \\mathbf { M } _ { 2 } \\mathbf { P } _ { \\mathrm { b r } } .", + "type": "interline_equation", + "image_path": "38ebc511817bb20e47cd2363018c0b54183cacf0471e56cd3378fc2bf70e9b43.jpg" + } + ] + } + ], + "index": 32, + "virtual_lines": [ + { + "bbox": [ + 264, + 554, + 346, + 568 + ], + "spans": [], + "index": 32 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 579, + 506, + 702 + ], + "lines": [ + { + "bbox": [ + 105, + 579, + 506, + 594 + ], + "spans": [ + { + "bbox": [ + 105, + 579, + 249, + 594 + ], + "score": 1.0, + "content": "Proof sketch. For any input vector", + "type": "text" + }, + { + "bbox": [ + 249, + 582, + 257, + 591 + ], + "score": 0.54, + "content": "\\mathbf { x }", + "type": "inline_equation" + }, + { + "bbox": [ + 258, + 579, + 298, + 594 + ], + "score": 1.0, + "content": "of length", + "type": "text" + }, + { + "bbox": [ + 298, + 583, + 305, + 591 + ], + "score": 0.79, + "content": "n", + "type": "inline_equation" + }, + { + "bbox": [ + 305, + 579, + 355, + 594 + ], + "score": 1.0, + "content": ", to perform", + "type": "text" + }, + { + "bbox": [ + 356, + 581, + 378, + 592 + ], + "score": 0.91, + "content": "\\mathbf { M } _ { 1 } ^ { * } \\mathbf { x }", + "type": "inline_equation" + }, + { + "bbox": [ + 379, + 579, + 454, + 594 + ], + "score": 1.0, + "content": ", we trace through", + "type": "text" + }, + { + "bbox": [ + 454, + 581, + 481, + 592 + ], + "score": 0.91, + "content": "\\log _ { 2 } n", + "type": "inline_equation" + }, + { + "bbox": [ + 481, + 579, + 506, + 594 + ], + "score": 1.0, + "content": "steps", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 591, + 506, + 604 + ], + "spans": [ + { + "bbox": [ + 105, + 591, + 341, + 604 + ], + "score": 1.0, + "content": "of the multiplication algorithm. At each step, we perform", + "type": "text" + }, + { + "bbox": [ + 341, + 592, + 365, + 601 + ], + "score": 0.89, + "content": "2 \\times 2", + "type": "inline_equation" + }, + { + "bbox": [ + 365, + 591, + 506, + 604 + ], + "score": 1.0, + "content": "matrix multiplication on elements", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 601, + 506, + 615 + ], + "spans": [ + { + "bbox": [ + 105, + 601, + 117, + 615 + ], + "score": 1.0, + "content": "of", + "type": "text" + }, + { + "bbox": [ + 117, + 604, + 125, + 612 + ], + "score": 0.59, + "content": "\\mathbf { x }", + "type": "inline_equation" + }, + { + "bbox": [ + 126, + 601, + 200, + 615 + ], + "score": 1.0, + "content": "whose indices are", + "type": "text" + }, + { + "bbox": [ + 200, + 603, + 218, + 614 + ], + "score": 0.89, + "content": "n / 2", + "type": "inline_equation" + }, + { + "bbox": [ + 218, + 601, + 317, + 615 + ], + "score": 1.0, + "content": "apart (e.g. indices 0 and", + "type": "text" + }, + { + "bbox": [ + 318, + 603, + 345, + 614 + ], + "score": 0.65, + "content": "n / 2 , 1", + "type": "inline_equation" + }, + { + "bbox": [ + 345, + 601, + 363, + 615 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 363, + 602, + 398, + 614 + ], + "score": 0.92, + "content": "n / 2 + 1", + "type": "inline_equation" + }, + { + "bbox": [ + 398, + 601, + 444, + 615 + ], + "score": 1.0, + "content": ", etc.), then", + "type": "text" + }, + { + "bbox": [ + 444, + 602, + 462, + 614 + ], + "score": 0.89, + "content": "n / 4", + "type": "inline_equation" + }, + { + "bbox": [ + 462, + 601, + 506, + 615 + ], + "score": 1.0, + "content": "apart, and", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 613, + 506, + 625 + ], + "spans": [ + { + "bbox": [ + 105, + 613, + 424, + 625 + ], + "score": 1.0, + "content": "so on, till indices are that 1 apart. If we apply the bit-reversal permutation on", + "type": "text" + }, + { + "bbox": [ + 424, + 615, + 432, + 623 + ], + "score": 0.29, + "content": "\\mathbf { x }", + "type": "inline_equation" + }, + { + "bbox": [ + 432, + 613, + 506, + 625 + ], + "score": 1.0, + "content": ", then indices that", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 624, + 506, + 637 + ], + "spans": [ + { + "bbox": [ + 105, + 624, + 121, + 637 + ], + "score": 1.0, + "content": "are", + "type": "text" + }, + { + "bbox": [ + 121, + 624, + 138, + 636 + ], + "score": 0.88, + "content": "n / 2", + "type": "inline_equation" + }, + { + "bbox": [ + 139, + 624, + 308, + 637 + ], + "score": 1.0, + "content": "apart will become 1 apart, indices that are", + "type": "text" + }, + { + "bbox": [ + 309, + 624, + 326, + 636 + ], + "score": 0.89, + "content": "n / 4", + "type": "inline_equation" + }, + { + "bbox": [ + 326, + 624, + 506, + 637 + ], + "score": 1.0, + "content": "apart will become 2 apart, and so on. So the", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 635, + 506, + 649 + ], + "spans": [ + { + "bbox": [ + 105, + 635, + 206, + 649 + ], + "score": 1.0, + "content": "multiplication algorithm", + "type": "text" + }, + { + "bbox": [ + 207, + 635, + 230, + 647 + ], + "score": 0.92, + "content": "\\mathbf { M } _ { 1 } ^ { * } \\mathbf { x }", + "type": "inline_equation" + }, + { + "bbox": [ + 230, + 635, + 506, + 649 + ], + "score": 1.0, + "content": "is equivalent to applying bit-reversal, then multiplying the permuted", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 646, + 506, + 659 + ], + "spans": [ + { + "bbox": [ + 105, + 646, + 276, + 659 + ], + "score": 1.0, + "content": "vector with another butterfly matrix (i.e.", + "type": "text" + }, + { + "bbox": [ + 276, + 647, + 301, + 657 + ], + "score": 0.89, + "content": "2 \\times 2", + "type": "inline_equation" + }, + { + "bbox": [ + 301, + 646, + 506, + 659 + ], + "score": 1.0, + "content": "matrix multiplication on indices that are 1 apart,", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 106, + 657, + 506, + 669 + ], + "spans": [ + { + "bbox": [ + 106, + 658, + 283, + 669 + ], + "score": 1.0, + "content": "then 2 apart, and so on, till indices that are", + "type": "text" + }, + { + "bbox": [ + 283, + 657, + 300, + 669 + ], + "score": 0.87, + "content": "n / 2", + "type": "inline_equation" + }, + { + "bbox": [ + 301, + 658, + 506, + 669 + ], + "score": 1.0, + "content": "apart). Finally we need to do another bit-reversal", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 104, + 667, + 507, + 681 + ], + "spans": [ + { + "bbox": [ + 104, + 667, + 486, + 681 + ], + "score": 1.0, + "content": "permutation to put all the indices back to the original order. If we call this other butterfly matrix", + "type": "text" + }, + { + "bbox": [ + 486, + 668, + 502, + 679 + ], + "score": 0.89, + "content": "\\mathbf { M } _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 502, + 667, + 507, + 681 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 678, + 506, + 691 + ], + "spans": [ + { + "bbox": [ + 105, + 678, + 209, + 691 + ], + "score": 1.0, + "content": "then we have shown that", + "type": "text" + }, + { + "bbox": [ + 209, + 679, + 299, + 691 + ], + "score": 0.91, + "content": "\\mathbf { M } _ { 1 } ^ { * } \\mathbf { x } = \\mathbf { P } _ { \\mathrm { b r } } \\mathbf { M } _ { 2 } \\mathbf { P } _ { \\mathrm { b r } } \\mathbf { x }", + "type": "inline_equation" + }, + { + "bbox": [ + 299, + 678, + 375, + 691 + ], + "score": 1.0, + "content": ". This holds for all", + "type": "text" + }, + { + "bbox": [ + 375, + 680, + 383, + 689 + ], + "score": 0.65, + "content": "\\mathbf { x }", + "type": "inline_equation" + }, + { + "bbox": [ + 384, + 678, + 469, + 691 + ], + "score": 1.0, + "content": "(for the same matrix", + "type": "text" + }, + { + "bbox": [ + 470, + 680, + 486, + 690 + ], + "score": 0.86, + "content": "\\mathbf { M } _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 487, + 678, + 506, + 691 + ], + "score": 1.0, + "content": "), so", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 689, + 505, + 703 + ], + "spans": [ + { + "bbox": [ + 105, + 689, + 141, + 703 + ], + "score": 1.0, + "content": "we have", + "type": "text" + }, + { + "bbox": [ + 141, + 690, + 218, + 702 + ], + "score": 0.92, + "content": "\\mathbf { M } _ { 1 } ^ { * } = \\mathbf { P } _ { \\mathrm { b r } } \\mathbf { M } _ { 2 } \\mathbf { P } _ { \\mathrm { b r } }", + "type": "inline_equation" + }, + { + "bbox": [ + 218, + 689, + 222, + 703 + ], + "score": 1.0, + "content": ".", + "type": "text" + }, + { + "bbox": [ + 494, + 691, + 505, + 700 + ], + "score": 0.999, + "content": "□", + "type": "text" + } + ], + "index": 43 + } + ], + "index": 38, + "bbox_fs": [ + 104, + 579, + 507, + 703 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 709, + 504, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 709, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 106, + 709, + 505, + 722 + ], + "score": 1.0, + "content": "Remark G.5. Lemma G.4 explains the connection between the two most common fast Fourier", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 106, + 721, + 505, + 734 + ], + "spans": [ + { + "bbox": [ + 106, + 721, + 505, + 734 + ], + "score": 1.0, + "content": "transform algorithm, decimation in time and decimation in frequency. Using the decimation-in-time", + "type": "text" + } + ], + "index": 45 + } + ], + "index": 44.5, + "bbox_fs": [ + 106, + 709, + 505, + 734 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 104, + 81, + 505, + 105 + ], + "lines": [ + { + "bbox": [ + 105, + 81, + 506, + 96 + ], + "spans": [ + { + "bbox": [ + 105, + 81, + 255, + 96 + ], + "score": 1.0, + "content": "FFT, we can write the DFT matrix", + "type": "text" + }, + { + "bbox": [ + 256, + 83, + 264, + 92 + ], + "score": 0.72, + "content": "\\mathbf { F }", + "type": "inline_equation" + }, + { + "bbox": [ + 265, + 81, + 401, + 96 + ], + "score": 1.0, + "content": "as product of a butterfly matrix", + "type": "text" + }, + { + "bbox": [ + 402, + 83, + 418, + 93 + ], + "score": 0.88, + "content": "{ { \\bf { M } } _ { 1 } }", + "type": "inline_equation" + }, + { + "bbox": [ + 419, + 81, + 506, + 96 + ], + "score": 1.0, + "content": "and the bit-reversal", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 104, + 93, + 222, + 106 + ], + "spans": [ + { + "bbox": [ + 104, + 93, + 222, + 106 + ], + "score": 1.0, + "content": "permutation (see Section J):", + "type": "text" + } + ], + "index": 1 + } + ], + "index": 0.5 + }, + { + "type": "interline_equation", + "bbox": [ + 277, + 105, + 333, + 118 + ], + "lines": [ + { + "bbox": [ + 277, + 105, + 333, + 118 + ], + "spans": [ + { + "bbox": [ + 277, + 105, + 333, + 118 + ], + "score": 0.89, + "content": "\\mathbf { F } = \\mathbf { M } _ { 1 } \\mathbf { P } _ { \\mathrm { b r } } .", + "type": "interline_equation", + "image_path": "57737368eaabd7699e4a2c05595918e0b33408dadc98eb87c72f1ea99064baae.jpg" + } + ] + } + ], + "index": 2, + "virtual_lines": [ + { + "bbox": [ + 277, + 105, + 333, + 118 + ], + "spans": [], + "index": 2 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 121, + 505, + 189 + ], + "lines": [ + { + "bbox": [ + 104, + 120, + 506, + 136 + ], + "spans": [ + { + "bbox": [ + 104, + 120, + 263, + 136 + ], + "score": 1.0, + "content": "Taking conjugate transpose, we obtain", + "type": "text" + }, + { + "bbox": [ + 264, + 122, + 321, + 134 + ], + "score": 0.93, + "content": "\\mathbf { F } ^ { * } = \\mathbf { P } _ { \\mathrm { b r } } \\mathbf { M } _ { 1 } ^ { * }", + "type": "inline_equation" + }, + { + "bbox": [ + 321, + 120, + 369, + 136 + ], + "score": 1.0, + "content": "(recall that", + "type": "text" + }, + { + "bbox": [ + 369, + 123, + 386, + 133 + ], + "score": 0.89, + "content": "\\mathbf { P } _ { \\mathrm { b r } }", + "type": "inline_equation" + }, + { + "bbox": [ + 387, + 120, + 506, + 136 + ], + "score": 1.0, + "content": "is its own transpose/inverse).", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 132, + 505, + 147 + ], + "spans": [ + { + "bbox": [ + 105, + 132, + 182, + 147 + ], + "score": 1.0, + "content": "On the other hand,", + "type": "text" + }, + { + "bbox": [ + 182, + 133, + 195, + 144 + ], + "score": 0.85, + "content": "\\mathbf { F } ^ { * }", + "type": "inline_equation" + }, + { + "bbox": [ + 195, + 132, + 505, + 147 + ], + "score": 1.0, + "content": "is just a scaled version of the inverse DFT matrix, so apply decimation-in-time", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 104, + 143, + 506, + 157 + ], + "spans": [ + { + "bbox": [ + 104, + 143, + 264, + 157 + ], + "score": 1.0, + "content": "FFT to the inverse DFT, we can write", + "type": "text" + }, + { + "bbox": [ + 264, + 144, + 324, + 155 + ], + "score": 0.91, + "content": "\\mathbf { F } ^ { * } = \\mathbf { M } _ { 2 } \\mathbf { P } _ { \\mathrm { b r } }", + "type": "inline_equation" + }, + { + "bbox": [ + 324, + 143, + 454, + 157 + ], + "score": 1.0, + "content": "for some other butterfly matrix", + "type": "text" + }, + { + "bbox": [ + 454, + 144, + 470, + 155 + ], + "score": 0.87, + "content": "\\mathbf { M } _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 471, + 143, + 506, + 157 + ], + "score": 1.0, + "content": ". Hence", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 107, + 154, + 506, + 168 + ], + "spans": [ + { + "bbox": [ + 107, + 155, + 186, + 167 + ], + "score": 0.91, + "content": "\\mathbf { P } _ { \\mathrm { b r } } \\mathbf { M } _ { 1 } ^ { * } = \\mathbf { M } _ { 2 } \\mathbf { P } _ { \\mathrm { b r } } ,", + "type": "inline_equation" + }, + { + "bbox": [ + 186, + 154, + 227, + 168 + ], + "score": 1.0, + "content": ", and thus", + "type": "text" + }, + { + "bbox": [ + 228, + 155, + 306, + 167 + ], + "score": 0.92, + "content": "\\mathbf { P } _ { \\mathrm { b r } } \\mathbf { M } _ { 1 } ^ { * } \\mathbf { P } _ { \\mathrm { b r } } = \\mathbf { M } _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 307, + 154, + 469, + 168 + ], + "score": 1.0, + "content": "(for these particular butterfly matrices", + "type": "text" + }, + { + "bbox": [ + 469, + 155, + 486, + 167 + ], + "score": 0.89, + "content": "{ { \\bf { M } } _ { 1 } }", + "type": "inline_equation" + }, + { + "bbox": [ + 486, + 154, + 506, + 168 + ], + "score": 1.0, + "content": "and", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 166, + 505, + 179 + ], + "spans": [ + { + "bbox": [ + 106, + 167, + 123, + 177 + ], + "score": 0.79, + "content": "{ { \\bf { M } } _ { 2 } }", + "type": "inline_equation" + }, + { + "bbox": [ + 124, + 166, + 383, + 179 + ], + "score": 1.0, + "content": "). Note that this yields another decomposition of the DFT matrix,", + "type": "text" + }, + { + "bbox": [ + 383, + 166, + 435, + 178 + ], + "score": 0.93, + "content": "\\mathbf { F } = \\mathbf { P } _ { \\mathrm { b r } } \\mathbf { M } _ { 2 } ^ { * }", + "type": "inline_equation" + }, + { + "bbox": [ + 436, + 166, + 505, + 179 + ], + "score": 1.0, + "content": ", which is exactly", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 177, + 285, + 190 + ], + "spans": [ + { + "bbox": [ + 105, + 177, + 285, + 190 + ], + "score": 1.0, + "content": "the decimation-in-frequency FFT algorithm.", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 5.5 + }, + { + "type": "text", + "bbox": [ + 106, + 198, + 314, + 210 + ], + "lines": [ + { + "bbox": [ + 105, + 197, + 315, + 211 + ], + "spans": [ + { + "bbox": [ + 105, + 197, + 291, + 211 + ], + "score": 1.0, + "content": "We are ready to show that permutations are in", + "type": "text" + }, + { + "bbox": [ + 291, + 199, + 311, + 209 + ], + "score": 0.89, + "content": "B ^ { * } B", + "type": "inline_equation" + }, + { + "bbox": [ + 311, + 197, + 315, + 211 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 9 + }, + { + "type": "text", + "bbox": [ + 107, + 213, + 504, + 237 + ], + "lines": [ + { + "bbox": [ + 105, + 212, + 505, + 227 + ], + "spans": [ + { + "bbox": [ + 105, + 212, + 179, + 227 + ], + "score": 1.0, + "content": "Lemma G.6. Let", + "type": "text" + }, + { + "bbox": [ + 180, + 214, + 189, + 224 + ], + "score": 0.29, + "content": "\\mathbf { P }", + "type": "inline_equation" + }, + { + "bbox": [ + 189, + 212, + 214, + 227 + ], + "score": 1.0, + "content": "be an", + "type": "text" + }, + { + "bbox": [ + 214, + 216, + 239, + 225 + ], + "score": 0.89, + "content": "n \\times n", + "type": "inline_equation" + }, + { + "bbox": [ + 240, + 212, + 505, + 227 + ], + "score": 1.0, + "content": "permutation matrix (with n a power of 2). Then there are butterfly", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 223, + 295, + 238 + ], + "spans": [ + { + "bbox": [ + 105, + 223, + 143, + 238 + ], + "score": 1.0, + "content": "matrices", + "type": "text" + }, + { + "bbox": [ + 144, + 225, + 199, + 236 + ], + "score": 0.89, + "content": "\\mathbf { M } _ { 1 } , \\mathbf { M } _ { 2 } \\in B", + "type": "inline_equation" + }, + { + "bbox": [ + 199, + 223, + 239, + 238 + ], + "score": 1.0, + "content": "such that", + "type": "text" + }, + { + "bbox": [ + 239, + 225, + 291, + 237 + ], + "score": 0.93, + "content": "\\mathbf { P } = \\mathbf { M } _ { 1 } ^ { * } \\mathbf { M } _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 291, + 223, + 295, + 238 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 10.5 + }, + { + "type": "text", + "bbox": [ + 106, + 252, + 505, + 289 + ], + "lines": [ + { + "bbox": [ + 106, + 252, + 505, + 266 + ], + "spans": [ + { + "bbox": [ + 106, + 252, + 241, + 266 + ], + "score": 1.0, + "content": "Proof. Consider the permutation", + "type": "text" + }, + { + "bbox": [ + 241, + 252, + 303, + 265 + ], + "score": 0.92, + "content": "\\tilde { \\mathbf { P } } = \\mathbf { P } _ { \\mathrm { b r } } \\mathbf { P } \\mathbf { P } _ { \\mathrm { b r } }", + "type": "inline_equation" + }, + { + "bbox": [ + 303, + 252, + 505, + 266 + ], + "score": 1.0, + "content": ". By Theorem 2, there are some butterfly matrices", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 107, + 263, + 504, + 280 + ], + "spans": [ + { + "bbox": [ + 107, + 264, + 162, + 278 + ], + "score": 0.89, + "content": "\\tilde { \\mathbf { M } } _ { 1 } , \\tilde { \\mathbf { M } } _ { 2 } \\in B", + "type": "inline_equation" + }, + { + "bbox": [ + 163, + 263, + 200, + 280 + ], + "score": 1.0, + "content": "such that", + "type": "text" + }, + { + "bbox": [ + 201, + 264, + 257, + 277 + ], + "score": 0.92, + "content": "\\tilde { \\mathbf { P } } = \\tilde { \\mathbf { M } } _ { 1 } \\tilde { \\mathbf { M } } _ { 2 } ^ { * }", + "type": "inline_equation" + }, + { + "bbox": [ + 258, + 263, + 413, + 280 + ], + "score": 1.0, + "content": ". Applying Lemma G.4, we can replace", + "type": "text" + }, + { + "bbox": [ + 414, + 264, + 434, + 278 + ], + "score": 0.91, + "content": "{ { \\tilde { \\mathbf { M } } } _ { 2 } } ^ { * }", + "type": "inline_equation" + }, + { + "bbox": [ + 435, + 263, + 455, + 280 + ], + "score": 1.0, + "content": "with", + "type": "text" + }, + { + "bbox": [ + 456, + 266, + 504, + 279 + ], + "score": 0.91, + "content": "\\mathbf { P } _ { \\mathrm { b r } } \\mathbf { M } _ { 2 } \\mathbf { P } _ { \\mathrm { b r } }", + "type": "inline_equation" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 277, + 307, + 290 + ], + "spans": [ + { + "bbox": [ + 105, + 277, + 208, + 290 + ], + "score": 1.0, + "content": "for some butterfly matrix", + "type": "text" + }, + { + "bbox": [ + 209, + 278, + 244, + 289 + ], + "score": 0.91, + "content": "{ \\bf M } _ { 2 } \\in { B }", + "type": "inline_equation" + }, + { + "bbox": [ + 245, + 277, + 307, + 290 + ], + "score": 1.0, + "content": ". We thus have:", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 13 + }, + { + "type": "interline_equation", + "bbox": [ + 245, + 294, + 366, + 310 + ], + "lines": [ + { + "bbox": [ + 245, + 294, + 366, + 310 + ], + "spans": [ + { + "bbox": [ + 245, + 294, + 366, + 310 + ], + "score": 0.91, + "content": "\\mathbf { P } _ { \\mathrm { b r } } \\mathbf { P } \\mathbf { P } _ { \\mathrm { b r } } = \\tilde { \\mathbf { M } } _ { 1 } \\mathbf { P } _ { \\mathrm { b r } } \\mathbf { M } _ { 2 } \\mathbf { P } _ { \\mathrm { b r } } .", + "type": "interline_equation", + "image_path": "e1d5e6180591a144282515a08a0ca3670cb09d6428bd006e8d56f6dce90801d8.jpg" + } + ] + } + ], + "index": 15, + "virtual_lines": [ + { + "bbox": [ + 245, + 294, + 366, + 310 + ], + "spans": [], + "index": 15 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 316, + 379, + 329 + ], + "lines": [ + { + "bbox": [ + 105, + 315, + 379, + 330 + ], + "spans": [ + { + "bbox": [ + 105, + 315, + 253, + 330 + ], + "score": 1.0, + "content": "Pre- and post-multiply both sides by", + "type": "text" + }, + { + "bbox": [ + 254, + 317, + 271, + 328 + ], + "score": 0.9, + "content": "\\mathbf { P } _ { \\mathrm { b r } }", + "type": "inline_equation" + }, + { + "bbox": [ + 271, + 315, + 379, + 330 + ], + "score": 1.0, + "content": "(which is its own inverse):", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 16 + }, + { + "type": "interline_equation", + "bbox": [ + 261, + 334, + 350, + 349 + ], + "lines": [ + { + "bbox": [ + 261, + 334, + 350, + 349 + ], + "spans": [ + { + "bbox": [ + 261, + 334, + 350, + 349 + ], + "score": 0.91, + "content": "\\mathbf { P } = \\mathbf { P } _ { \\mathrm { b r } } \\tilde { \\mathbf { M } } _ { 1 } \\mathbf { P } _ { \\mathrm { b r } } \\mathbf { M } _ { 2 } .", + "type": "interline_equation", + "image_path": "d2342287854341664e25674d5d9a093021eccd317b4596e1e4ab86275967cc29.jpg" + } + ] + } + ], + "index": 17, + "virtual_lines": [ + { + "bbox": [ + 261, + 334, + 350, + 349 + ], + "spans": [], + "index": 17 + } + ] + }, + { + "type": "text", + "bbox": [ + 105, + 355, + 507, + 378 + ], + "lines": [ + { + "bbox": [ + 105, + 354, + 506, + 369 + ], + "spans": [ + { + "bbox": [ + 105, + 354, + 280, + 369 + ], + "score": 1.0, + "content": "Applying Lemma G.4 again, we can replace", + "type": "text" + }, + { + "bbox": [ + 280, + 354, + 329, + 367 + ], + "score": 0.93, + "content": "\\mathbf { P } _ { \\mathrm { b r } } \\mathbf { \\tilde { M } } _ { 1 } \\mathbf { P } _ { \\mathrm { b r } }", + "type": "inline_equation" + }, + { + "bbox": [ + 329, + 354, + 349, + 369 + ], + "score": 1.0, + "content": "with", + "type": "text" + }, + { + "bbox": [ + 350, + 356, + 366, + 368 + ], + "score": 0.9, + "content": "\\mathbf { M } _ { 1 } ^ { * }", + "type": "inline_equation" + }, + { + "bbox": [ + 367, + 354, + 467, + 369 + ], + "score": 1.0, + "content": "for some butterfly matrix", + "type": "text" + }, + { + "bbox": [ + 467, + 356, + 503, + 367 + ], + "score": 0.92, + "content": "{ \\bf M } _ { 1 } \\in { B }", + "type": "inline_equation" + }, + { + "bbox": [ + 504, + 354, + 506, + 369 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 104, + 365, + 133, + 380 + ], + "spans": [ + { + "bbox": [ + 104, + 365, + 133, + 380 + ], + "score": 1.0, + "content": "Thus:", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 18.5 + }, + { + "type": "interline_equation", + "bbox": [ + 276, + 378, + 334, + 392 + ], + "lines": [ + { + "bbox": [ + 276, + 378, + 334, + 392 + ], + "spans": [ + { + "bbox": [ + 276, + 378, + 334, + 392 + ], + "score": 0.89, + "content": "\\mathbf { P } = \\mathbf { M } _ { 1 } ^ { * } \\mathbf { M } _ { 2 } .", + "type": "interline_equation", + "image_path": "e6a6be9d6c2fb0debc91d029fa4f36fdb17253558d89dd8a2f6a78ebf0c52fac.jpg" + } + ] + } + ], + "index": 20, + "virtual_lines": [ + { + "bbox": [ + 276, + 378, + 334, + 392 + ], + "spans": [], + "index": 20 + } + ] + }, + { + "type": "title", + "bbox": [ + 107, + 424, + 252, + 437 + ], + "lines": [ + { + "bbox": [ + 105, + 422, + 253, + 439 + ], + "spans": [ + { + "bbox": [ + 105, + 422, + 127, + 439 + ], + "score": 1.0, + "content": "H", + "type": "text" + }, + { + "bbox": [ + 127, + 424, + 150, + 437 + ], + "score": 0.82, + "content": "B B ^ { * }", + "type": "inline_equation" + }, + { + "bbox": [ + 151, + 422, + 253, + 439 + ], + "score": 1.0, + "content": "CLOSURE LEMMAS", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 21 + }, + { + "type": "text", + "bbox": [ + 107, + 449, + 504, + 483 + ], + "lines": [ + { + "bbox": [ + 106, + 449, + 506, + 461 + ], + "spans": [ + { + "bbox": [ + 106, + 449, + 266, + 461 + ], + "score": 1.0, + "content": "Here, we present some basic facts of the", + "type": "text" + }, + { + "bbox": [ + 266, + 450, + 285, + 460 + ], + "score": 0.88, + "content": "B B ^ { * }", + "type": "inline_equation" + }, + { + "bbox": [ + 285, + 449, + 506, + 461 + ], + "score": 1.0, + "content": "hierarchy that will be useful for later constructions. For", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 461, + 506, + 473 + ], + "spans": [ + { + "bbox": [ + 106, + 461, + 506, + 473 + ], + "score": 1.0, + "content": "simplicity, we assume (WLOG via 0-padding) that all matrices are square matrices with size that is a", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 104, + 472, + 155, + 483 + ], + "spans": [ + { + "bbox": [ + 104, + 472, + 155, + 483 + ], + "score": 1.0, + "content": "power of 2.", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 23 + }, + { + "type": "text", + "bbox": [ + 107, + 486, + 500, + 499 + ], + "lines": [ + { + "bbox": [ + 105, + 486, + 501, + 501 + ], + "spans": [ + { + "bbox": [ + 105, + 486, + 173, + 501 + ], + "score": 1.0, + "content": "Lemma H.1. If", + "type": "text" + }, + { + "bbox": [ + 173, + 487, + 204, + 498 + ], + "score": 0.87, + "content": "\\mathbf { M } \\in B", + "type": "inline_equation" + }, + { + "bbox": [ + 205, + 486, + 219, + 501 + ], + "score": 1.0, + "content": "(or", + "type": "text" + }, + { + "bbox": [ + 220, + 487, + 256, + 498 + ], + "score": 0.85, + "content": "\\mathbf { M } \\in B ^ { * }", + "type": "inline_equation" + }, + { + "bbox": [ + 256, + 486, + 306, + 501 + ], + "score": 1.0, + "content": "), then DM,", + "type": "text" + }, + { + "bbox": [ + 307, + 487, + 347, + 498 + ], + "score": 0.73, + "content": "\\mathbf { M D } \\in B", + "type": "inline_equation" + }, + { + "bbox": [ + 347, + 486, + 351, + 501 + ], + "score": 1.0, + "content": "(", + "type": "text" + }, + { + "bbox": [ + 351, + 487, + 363, + 498 + ], + "score": 0.8, + "content": "B ^ { * }", + "type": "inline_equation" + }, + { + "bbox": [ + 364, + 486, + 501, + 501 + ], + "score": 1.0, + "content": "resp.) for any diagonal matrix D.", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 25 + }, + { + "type": "text", + "bbox": [ + 106, + 513, + 505, + 559 + ], + "lines": [ + { + "bbox": [ + 105, + 513, + 506, + 527 + ], + "spans": [ + { + "bbox": [ + 105, + 513, + 506, + 527 + ], + "score": 1.0, + "content": "Proof. Left multiplication by a diagonal matrix scales the rows of M by the corresponding diagonal", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 524, + 506, + 538 + ], + "spans": [ + { + "bbox": [ + 105, + 524, + 506, + 538 + ], + "score": 1.0, + "content": "entries. The same can be achieved by scaling all entries the leftmost butterfly factor matrix. Similarly,", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 536, + 505, + 549 + ], + "spans": [ + { + "bbox": [ + 105, + 536, + 355, + 549 + ], + "score": 1.0, + "content": "right multiplication by a diagonal matrix scales the columns of", + "type": "text" + }, + { + "bbox": [ + 356, + 536, + 367, + 546 + ], + "score": 0.42, + "content": "\\mathbf { M }", + "type": "inline_equation" + }, + { + "bbox": [ + 368, + 536, + 505, + 549 + ], + "score": 1.0, + "content": ", which can be achieved by scaling", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 547, + 502, + 559 + ], + "spans": [ + { + "bbox": [ + 106, + 547, + 368, + 559 + ], + "score": 1.0, + "content": "all entries in the columns of the rightmost butterfly factor matrix.", + "type": "text" + }, + { + "bbox": [ + 497, + 550, + 502, + 556 + ], + "score": 0.0, + "content": "", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 27.5 + }, + { + "type": "text", + "bbox": [ + 105, + 567, + 491, + 581 + ], + "lines": [ + { + "bbox": [ + 102, + 565, + 488, + 584 + ], + "spans": [ + { + "bbox": [ + 102, + 565, + 191, + 584 + ], + "score": 1.0, + "content": "Lemma H.2. Let A,", + "type": "text" + }, + { + "bbox": [ + 191, + 568, + 236, + 579 + ], + "score": 0.85, + "content": "\\mathbf { B } \\in \\mathbb { F } ^ { n \\times n }", + "type": "inline_equation" + }, + { + "bbox": [ + 237, + 565, + 248, + 584 + ], + "score": 1.0, + "content": ". If", + "type": "text" + }, + { + "bbox": [ + 249, + 568, + 307, + 580 + ], + "score": 0.89, + "content": "\\mathbf { \\dot { A } } \\in ( B B ^ { * } ) _ { e } ^ { w _ { 1 } }", + "type": "inline_equation" + }, + { + "bbox": [ + 308, + 565, + 326, + 584 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 327, + 568, + 384, + 580 + ], + "score": 0.85, + "content": "\\mathbf { B } \\in ( B B ^ { * } ) _ { e } ^ { w _ { 2 } }", + "type": "inline_equation" + }, + { + "bbox": [ + 384, + 565, + 405, + 584 + ], + "score": 1.0, + "content": "then", + "type": "text" + }, + { + "bbox": [ + 406, + 568, + 488, + 581 + ], + "score": 0.84, + "content": "\\mathbf { A B } \\in ( B B ^ { * } ) _ { e } ^ { w _ { 1 } + w _ { 2 } } .", + "type": "inline_equation" + } + ], + "index": 30 + } + ], + "index": 30 + }, + { + "type": "text", + "bbox": [ + 107, + 594, + 502, + 617 + ], + "lines": [ + { + "bbox": [ + 105, + 592, + 505, + 608 + ], + "spans": [ + { + "bbox": [ + 105, + 592, + 153, + 608 + ], + "score": 1.0, + "content": "Proof. Let", + "type": "text" + }, + { + "bbox": [ + 154, + 595, + 234, + 606 + ], + "score": 0.9, + "content": "\\mathbf { E _ { A } } , \\mathbf { E _ { B } } \\in \\mathbb { F } ^ { e n \\times e n }", + "type": "inline_equation" + }, + { + "bbox": [ + 234, + 592, + 322, + 608 + ], + "score": 1.0, + "content": "be defined such that", + "type": "text" + }, + { + "bbox": [ + 322, + 594, + 382, + 606 + ], + "score": 0.9, + "content": "\\mathbf { A } = \\mathbf { S } \\mathbf { E } _ { \\mathbf { A } } \\mathbf { S } ^ { T }", + "type": "inline_equation" + }, + { + "bbox": [ + 382, + 592, + 387, + 608 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 387, + 594, + 446, + 606 + ], + "score": 0.9, + "content": "{ \\bf B } = { \\bf S } { \\bf E } _ { \\bf B } { \\bf S } ^ { T }", + "type": "inline_equation" + }, + { + "bbox": [ + 446, + 592, + 505, + 608 + ], + "score": 1.0, + "content": "(with S as in", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 605, + 195, + 618 + ], + "spans": [ + { + "bbox": [ + 106, + 605, + 195, + 618 + ], + "score": 1.0, + "content": "Definition 2.4). Then", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 31.5 + }, + { + "type": "interline_equation", + "bbox": [ + 225, + 616, + 387, + 660 + ], + "lines": [ + { + "bbox": [ + 225, + 616, + 387, + 660 + ], + "spans": [ + { + "bbox": [ + 225, + 616, + 387, + 660 + ], + "score": 0.93, + "content": "\\mathbf { A } \\mathbf { B } = \\mathbf { S } \\underbrace { \\left[ \\mathbf { I } _ { n } \\mathbf { \\Gamma } _ { 0 } \\right] } _ { e n \\times e n } \\mathbf { E } _ { \\mathbf { A } } \\underbrace { \\left[ \\mathbf { I } _ { n } \\mathbf { \\Gamma } _ { 0 } \\right] } _ { e n \\times e n } \\mathbf { E } _ { \\mathbf { B } } \\mathbf { S } ^ { T }", + "type": "interline_equation", + "image_path": "163a4b39e4da26f297e8d3db683a356062efc3cb8eaa5dafffac0f17c76a7784.jpg" + } + ] + } + ], + "index": 34, + "virtual_lines": [ + { + "bbox": [ + 225, + 616, + 387, + 630.6666666666666 + ], + "spans": [], + "index": 33 + }, + { + "bbox": [ + 225, + 630.6666666666666, + 387, + 645.3333333333333 + ], + "spans": [], + "index": 34 + }, + { + "bbox": [ + 225, + 645.3333333333333, + 387, + 659.9999999999999 + ], + "spans": [], + "index": 35 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 664, + 505, + 700 + ], + "lines": [ + { + "bbox": [ + 107, + 659, + 508, + 694 + ], + "spans": [ + { + "bbox": [ + 107, + 664, + 312, + 691 + ], + "score": 0.76, + "content": "\\begin{array} { r l } { \\bigg [ \\mathbf { I } _ { n } } & { { } 0 \\bigg ] \\mathbf { E _ { A } } \\in ( B B ^ { * } ) ^ { w _ { 1 } } , \\bigg [ \\mathbf { I } _ { n } \\quad 0 \\bigg ] \\mathbf { E _ { B } } \\in ( B B ^ { * } ) ^ { w _ { 2 } } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 312, + 659, + 410, + 694 + ], + "score": 1.0, + "content": "by Lemma H.1. Hence,", + "type": "text" + }, + { + "bbox": [ + 410, + 671, + 491, + 684 + ], + "score": 0.93, + "content": "\\mathbf { A B } \\in ( B B ^ { * } ) _ { e } ^ { w _ { 1 } + w _ { 2 } }", + "type": "inline_equation" + }, + { + "bbox": [ + 491, + 659, + 508, + 694 + ], + "score": 1.0, + "content": "by", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 687, + 168, + 700 + ], + "spans": [ + { + "bbox": [ + 105, + 687, + 168, + 700 + ], + "score": 1.0, + "content": "Definition 2.4.", + "type": "text" + } + ], + "index": 37 + } + ], + "index": 36.5 + }, + { + "type": "text", + "bbox": [ + 105, + 708, + 505, + 733 + ], + "lines": [ + { + "bbox": [ + 104, + 706, + 505, + 724 + ], + "spans": [ + { + "bbox": [ + 104, + 706, + 182, + 724 + ], + "score": 1.0, + "content": "Lemma H.3. Let", + "type": "text" + }, + { + "bbox": [ + 182, + 709, + 271, + 721 + ], + "score": 0.9, + "content": "\\mathbf { A } _ { 1 } , \\ldots , \\mathbf { A } _ { m } \\in \\mathbb { F } ^ { k \\times k }", + "type": "inline_equation" + }, + { + "bbox": [ + 272, + 706, + 288, + 724 + ], + "score": 1.0, + "content": ". If", + "type": "text" + }, + { + "bbox": [ + 289, + 709, + 388, + 722 + ], + "score": 0.89, + "content": "\\mathbf { A } _ { 1 } , \\ldots , \\mathbf { A } _ { m } \\in ( B B ^ { * } ) _ { e } ^ { w }", + "type": "inline_equation" + }, + { + "bbox": [ + 389, + 706, + 412, + 724 + ], + "score": 1.0, + "content": "then", + "type": "text" + }, + { + "bbox": [ + 412, + 709, + 505, + 722 + ], + "score": 0.79, + "content": "\\mathrm { D i a g } ( \\mathbf { A } _ { 1 } , \\ldots , \\mathbf { A } _ { m } ) \\in", + "type": "inline_equation" + } + ], + "index": 38 + }, + { + "bbox": [ + 106, + 717, + 154, + 735 + ], + "spans": [ + { + "bbox": [ + 106, + 720, + 148, + 733 + ], + "score": 0.91, + "content": "( B B ^ { * } ) _ { e } ^ { w + 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 149, + 717, + 154, + 735 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 39 + } + ], + "index": 38.5 + } + ], + "page_idx": 27, + "page_size": [ + 612, + 792 + ], + 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[ + { + "bbox": [ + 496, + 691, + 504, + 699 + ], + "score": 0.997, + "content": "□", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 104, + 81, + 505, + 105 + ], + "lines": [ + { + "bbox": [ + 105, + 81, + 506, + 96 + ], + "spans": [ + { + "bbox": [ + 105, + 81, + 255, + 96 + ], + "score": 1.0, + "content": "FFT, we can write the DFT matrix", + "type": "text" + }, + { + "bbox": [ + 256, + 83, + 264, + 92 + ], + "score": 0.72, + "content": "\\mathbf { F }", + "type": "inline_equation" + }, + { + "bbox": [ + 265, + 81, + 401, + 96 + ], + "score": 1.0, + "content": "as product of a butterfly matrix", + "type": "text" + }, + { + "bbox": [ + 402, + 83, + 418, + 93 + ], + "score": 0.88, + "content": "{ { \\bf { M } } _ { 1 } }", + "type": "inline_equation" + }, + { + "bbox": [ + 419, + 81, + 506, + 96 + ], + "score": 1.0, + "content": "and the bit-reversal", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 104, + 93, + 222, + 106 + ], + "spans": [ + { + "bbox": [ + 104, + 93, + 222, + 106 + ], + "score": 1.0, + "content": "permutation (see Section J):", + "type": "text" + } + ], + "index": 1 + } + ], + "index": 0.5, + "bbox_fs": [ + 104, + 81, + 506, + 106 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 277, + 105, + 333, + 118 + ], + "lines": [ + { + "bbox": [ + 277, + 105, + 333, + 118 + ], + "spans": [ + { + "bbox": [ + 277, + 105, + 333, + 118 + ], + "score": 0.89, + "content": "\\mathbf { F } = \\mathbf { M } _ { 1 } \\mathbf { P } _ { \\mathrm { b r } } .", + "type": "interline_equation", + "image_path": "57737368eaabd7699e4a2c05595918e0b33408dadc98eb87c72f1ea99064baae.jpg" + } + ] + } + ], + "index": 2, + "virtual_lines": [ + { + "bbox": [ + 277, + 105, + 333, + 118 + ], + "spans": [], + "index": 2 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 121, + 505, + 189 + ], + "lines": [ + { + "bbox": [ + 104, + 120, + 506, + 136 + ], + "spans": [ + { + "bbox": [ + 104, + 120, + 263, + 136 + ], + "score": 1.0, + "content": "Taking conjugate transpose, we obtain", + "type": "text" + }, + { + "bbox": [ + 264, + 122, + 321, + 134 + ], + "score": 0.93, + "content": "\\mathbf { F } ^ { * } = \\mathbf { P } _ { \\mathrm { b r } } \\mathbf { M } _ { 1 } ^ { * }", + "type": "inline_equation" + }, + { + "bbox": [ + 321, + 120, + 369, + 136 + ], + "score": 1.0, + "content": "(recall that", + "type": "text" + }, + { + "bbox": [ + 369, + 123, + 386, + 133 + ], + "score": 0.89, + "content": "\\mathbf { P } _ { \\mathrm { b r } }", + "type": "inline_equation" + }, + { + "bbox": [ + 387, + 120, + 506, + 136 + ], + "score": 1.0, + "content": "is its own transpose/inverse).", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 132, + 505, + 147 + ], + "spans": [ + { + "bbox": [ + 105, + 132, + 182, + 147 + ], + "score": 1.0, + "content": "On the other hand,", + "type": "text" + }, + { + "bbox": [ + 182, + 133, + 195, + 144 + ], + "score": 0.85, + "content": "\\mathbf { F } ^ { * }", + "type": "inline_equation" + }, + { + "bbox": [ + 195, + 132, + 505, + 147 + ], + "score": 1.0, + "content": "is just a scaled version of the inverse DFT matrix, so apply decimation-in-time", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 104, + 143, + 506, + 157 + ], + "spans": [ + { + "bbox": [ + 104, + 143, + 264, + 157 + ], + "score": 1.0, + "content": "FFT to the inverse DFT, we can write", + "type": "text" + }, + { + "bbox": [ + 264, + 144, + 324, + 155 + ], + "score": 0.91, + "content": "\\mathbf { F } ^ { * } = \\mathbf { M } _ { 2 } \\mathbf { P } _ { \\mathrm { b r } }", + "type": "inline_equation" + }, + { + "bbox": [ + 324, + 143, + 454, + 157 + ], + "score": 1.0, + "content": "for some other butterfly matrix", + "type": "text" + }, + { + "bbox": [ + 454, + 144, + 470, + 155 + ], + "score": 0.87, + "content": "\\mathbf { M } _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 471, + 143, + 506, + 157 + ], + "score": 1.0, + "content": ". Hence", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 107, + 154, + 506, + 168 + ], + "spans": [ + { + "bbox": [ + 107, + 155, + 186, + 167 + ], + "score": 0.91, + "content": "\\mathbf { P } _ { \\mathrm { b r } } \\mathbf { M } _ { 1 } ^ { * } = \\mathbf { M } _ { 2 } \\mathbf { P } _ { \\mathrm { b r } } ,", + "type": "inline_equation" + }, + { + "bbox": [ + 186, + 154, + 227, + 168 + ], + "score": 1.0, + "content": ", and thus", + "type": "text" + }, + { + "bbox": [ + 228, + 155, + 306, + 167 + ], + "score": 0.92, + "content": "\\mathbf { P } _ { \\mathrm { b r } } \\mathbf { M } _ { 1 } ^ { * } \\mathbf { P } _ { \\mathrm { b r } } = \\mathbf { M } _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 307, + 154, + 469, + 168 + ], + "score": 1.0, + "content": "(for these particular butterfly matrices", + "type": "text" + }, + { + "bbox": [ + 469, + 155, + 486, + 167 + ], + "score": 0.89, + "content": "{ { \\bf { M } } _ { 1 } }", + "type": "inline_equation" + }, + { + "bbox": [ + 486, + 154, + 506, + 168 + ], + "score": 1.0, + "content": "and", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 166, + 505, + 179 + ], + "spans": [ + { + "bbox": [ + 106, + 167, + 123, + 177 + ], + "score": 0.79, + "content": "{ { \\bf { M } } _ { 2 } }", + "type": "inline_equation" + }, + { + "bbox": [ + 124, + 166, + 383, + 179 + ], + "score": 1.0, + "content": "). Note that this yields another decomposition of the DFT matrix,", + "type": "text" + }, + { + "bbox": [ + 383, + 166, + 435, + 178 + ], + "score": 0.93, + "content": "\\mathbf { F } = \\mathbf { P } _ { \\mathrm { b r } } \\mathbf { M } _ { 2 } ^ { * }", + "type": "inline_equation" + }, + { + "bbox": [ + 436, + 166, + 505, + 179 + ], + "score": 1.0, + "content": ", which is exactly", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 177, + 285, + 190 + ], + "spans": [ + { + "bbox": [ + 105, + 177, + 285, + 190 + ], + "score": 1.0, + "content": "the decimation-in-frequency FFT algorithm.", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 5.5, + "bbox_fs": [ + 104, + 120, + 506, + 190 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 198, + 314, + 210 + ], + "lines": [ + { + "bbox": [ + 105, + 197, + 315, + 211 + ], + "spans": [ + { + "bbox": [ + 105, + 197, + 291, + 211 + ], + "score": 1.0, + "content": "We are ready to show that permutations are in", + "type": "text" + }, + { + "bbox": [ + 291, + 199, + 311, + 209 + ], + "score": 0.89, + "content": "B ^ { * } B", + "type": "inline_equation" + }, + { + "bbox": [ + 311, + 197, + 315, + 211 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 9, + "bbox_fs": [ + 105, + 197, + 315, + 211 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 213, + 504, + 237 + ], + "lines": [ + { + "bbox": [ + 105, + 212, + 505, + 227 + ], + "spans": [ + { + "bbox": [ + 105, + 212, + 179, + 227 + ], + "score": 1.0, + "content": "Lemma G.6. Let", + "type": "text" + }, + { + "bbox": [ + 180, + 214, + 189, + 224 + ], + "score": 0.29, + "content": "\\mathbf { P }", + "type": "inline_equation" + }, + { + "bbox": [ + 189, + 212, + 214, + 227 + ], + "score": 1.0, + "content": "be an", + "type": "text" + }, + { + "bbox": [ + 214, + 216, + 239, + 225 + ], + "score": 0.89, + "content": "n \\times n", + "type": "inline_equation" + }, + { + "bbox": [ + 240, + 212, + 505, + 227 + ], + "score": 1.0, + "content": "permutation matrix (with n a power of 2). Then there are butterfly", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 223, + 295, + 238 + ], + "spans": [ + { + "bbox": [ + 105, + 223, + 143, + 238 + ], + "score": 1.0, + "content": "matrices", + "type": "text" + }, + { + "bbox": [ + 144, + 225, + 199, + 236 + ], + "score": 0.89, + "content": "\\mathbf { M } _ { 1 } , \\mathbf { M } _ { 2 } \\in B", + "type": "inline_equation" + }, + { + "bbox": [ + 199, + 223, + 239, + 238 + ], + "score": 1.0, + "content": "such that", + "type": "text" + }, + { + "bbox": [ + 239, + 225, + 291, + 237 + ], + "score": 0.93, + "content": "\\mathbf { P } = \\mathbf { M } _ { 1 } ^ { * } \\mathbf { M } _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 291, + 223, + 295, + 238 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 10.5, + "bbox_fs": [ + 105, + 212, + 505, + 238 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 252, + 505, + 289 + ], + "lines": [ + { + "bbox": [ + 106, + 252, + 505, + 266 + ], + "spans": [ + { + "bbox": [ + 106, + 252, + 241, + 266 + ], + "score": 1.0, + "content": "Proof. Consider the permutation", + "type": "text" + }, + { + "bbox": [ + 241, + 252, + 303, + 265 + ], + "score": 0.92, + "content": "\\tilde { \\mathbf { P } } = \\mathbf { P } _ { \\mathrm { b r } } \\mathbf { P } \\mathbf { P } _ { \\mathrm { b r } }", + "type": "inline_equation" + }, + { + "bbox": [ + 303, + 252, + 505, + 266 + ], + "score": 1.0, + "content": ". By Theorem 2, there are some butterfly matrices", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 107, + 263, + 504, + 280 + ], + "spans": [ + { + "bbox": [ + 107, + 264, + 162, + 278 + ], + "score": 0.89, + "content": "\\tilde { \\mathbf { M } } _ { 1 } , \\tilde { \\mathbf { M } } _ { 2 } \\in B", + "type": "inline_equation" + }, + { + "bbox": [ + 163, + 263, + 200, + 280 + ], + "score": 1.0, + "content": "such that", + "type": "text" + }, + { + "bbox": [ + 201, + 264, + 257, + 277 + ], + "score": 0.92, + "content": "\\tilde { \\mathbf { P } } = \\tilde { \\mathbf { M } } _ { 1 } \\tilde { \\mathbf { M } } _ { 2 } ^ { * }", + "type": "inline_equation" + }, + { + "bbox": [ + 258, + 263, + 413, + 280 + ], + "score": 1.0, + "content": ". Applying Lemma G.4, we can replace", + "type": "text" + }, + { + "bbox": [ + 414, + 264, + 434, + 278 + ], + "score": 0.91, + "content": "{ { \\tilde { \\mathbf { M } } } _ { 2 } } ^ { * }", + "type": "inline_equation" + }, + { + "bbox": [ + 435, + 263, + 455, + 280 + ], + "score": 1.0, + "content": "with", + "type": "text" + }, + { + "bbox": [ + 456, + 266, + 504, + 279 + ], + "score": 0.91, + "content": "\\mathbf { P } _ { \\mathrm { b r } } \\mathbf { M } _ { 2 } \\mathbf { P } _ { \\mathrm { b r } }", + "type": "inline_equation" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 277, + 307, + 290 + ], + "spans": [ + { + "bbox": [ + 105, + 277, + 208, + 290 + ], + "score": 1.0, + "content": "for some butterfly matrix", + "type": "text" + }, + { + "bbox": [ + 209, + 278, + 244, + 289 + ], + "score": 0.91, + "content": "{ \\bf M } _ { 2 } \\in { B }", + "type": "inline_equation" + }, + { + "bbox": [ + 245, + 277, + 307, + 290 + ], + "score": 1.0, + "content": ". We thus have:", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 13, + "bbox_fs": [ + 105, + 252, + 505, + 290 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 245, + 294, + 366, + 310 + ], + "lines": [ + { + "bbox": [ + 245, + 294, + 366, + 310 + ], + "spans": [ + { + "bbox": [ + 245, + 294, + 366, + 310 + ], + "score": 0.91, + "content": "\\mathbf { P } _ { \\mathrm { b r } } \\mathbf { P } \\mathbf { P } _ { \\mathrm { b r } } = \\tilde { \\mathbf { M } } _ { 1 } \\mathbf { P } _ { \\mathrm { b r } } \\mathbf { M } _ { 2 } \\mathbf { P } _ { \\mathrm { b r } } .", + "type": "interline_equation", + "image_path": "e1d5e6180591a144282515a08a0ca3670cb09d6428bd006e8d56f6dce90801d8.jpg" + } + ] + } + ], + "index": 15, + "virtual_lines": [ + { + "bbox": [ + 245, + 294, + 366, + 310 + ], + "spans": [], + "index": 15 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 316, + 379, + 329 + ], + "lines": [ + { + "bbox": [ + 105, + 315, + 379, + 330 + ], + "spans": [ + { + "bbox": [ + 105, + 315, + 253, + 330 + ], + "score": 1.0, + "content": "Pre- and post-multiply both sides by", + "type": "text" + }, + { + "bbox": [ + 254, + 317, + 271, + 328 + ], + "score": 0.9, + "content": "\\mathbf { P } _ { \\mathrm { b r } }", + "type": "inline_equation" + }, + { + "bbox": [ + 271, + 315, + 379, + 330 + ], + "score": 1.0, + "content": "(which is its own inverse):", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 16, + "bbox_fs": [ + 105, + 315, + 379, + 330 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 261, + 334, + 350, + 349 + ], + "lines": [ + { + "bbox": [ + 261, + 334, + 350, + 349 + ], + "spans": [ + { + "bbox": [ + 261, + 334, + 350, + 349 + ], + "score": 0.91, + "content": "\\mathbf { P } = \\mathbf { P } _ { \\mathrm { b r } } \\tilde { \\mathbf { M } } _ { 1 } \\mathbf { P } _ { \\mathrm { b r } } \\mathbf { M } _ { 2 } .", + "type": "interline_equation", + "image_path": "d2342287854341664e25674d5d9a093021eccd317b4596e1e4ab86275967cc29.jpg" + } + ] + } + ], + "index": 17, + "virtual_lines": [ + { + "bbox": [ + 261, + 334, + 350, + 349 + ], + "spans": [], + "index": 17 + } + ] + }, + { + "type": "text", + "bbox": [ + 105, + 355, + 507, + 378 + ], + "lines": [ + { + "bbox": [ + 105, + 354, + 506, + 369 + ], + "spans": [ + { + "bbox": [ + 105, + 354, + 280, + 369 + ], + "score": 1.0, + "content": "Applying Lemma G.4 again, we can replace", + "type": "text" + }, + { + "bbox": [ + 280, + 354, + 329, + 367 + ], + "score": 0.93, + "content": "\\mathbf { P } _ { \\mathrm { b r } } \\mathbf { \\tilde { M } } _ { 1 } \\mathbf { P } _ { \\mathrm { b r } }", + "type": "inline_equation" + }, + { + "bbox": [ + 329, + 354, + 349, + 369 + ], + "score": 1.0, + "content": "with", + "type": "text" + }, + { + "bbox": [ + 350, + 356, + 366, + 368 + ], + "score": 0.9, + "content": "\\mathbf { M } _ { 1 } ^ { * }", + "type": "inline_equation" + }, + { + "bbox": [ + 367, + 354, + 467, + 369 + ], + "score": 1.0, + "content": "for some butterfly matrix", + "type": "text" + }, + { + "bbox": [ + 467, + 356, + 503, + 367 + ], + "score": 0.92, + "content": "{ \\bf M } _ { 1 } \\in { B }", + "type": "inline_equation" + }, + { + "bbox": [ + 504, + 354, + 506, + 369 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 104, + 365, + 133, + 380 + ], + "spans": [ + { + "bbox": [ + 104, + 365, + 133, + 380 + ], + "score": 1.0, + "content": "Thus:", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 18.5, + "bbox_fs": [ + 104, + 354, + 506, + 380 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 276, + 378, + 334, + 392 + ], + "lines": [ + { + "bbox": [ + 276, + 378, + 334, + 392 + ], + "spans": [ + { + "bbox": [ + 276, + 378, + 334, + 392 + ], + "score": 0.89, + "content": "\\mathbf { P } = \\mathbf { M } _ { 1 } ^ { * } \\mathbf { M } _ { 2 } .", + "type": "interline_equation", + "image_path": "e6a6be9d6c2fb0debc91d029fa4f36fdb17253558d89dd8a2f6a78ebf0c52fac.jpg" + } + ] + } + ], + "index": 20, + "virtual_lines": [ + { + "bbox": [ + 276, + 378, + 334, + 392 + ], + "spans": [], + "index": 20 + } + ] + }, + { + "type": "title", + "bbox": [ + 107, + 424, + 252, + 437 + ], + "lines": [ + { + "bbox": [ + 105, + 422, + 253, + 439 + ], + "spans": [ + { + "bbox": [ + 105, + 422, + 127, + 439 + ], + "score": 1.0, + "content": "H", + "type": "text" + }, + { + "bbox": [ + 127, + 424, + 150, + 437 + ], + "score": 0.82, + "content": "B B ^ { * }", + "type": "inline_equation" + }, + { + "bbox": [ + 151, + 422, + 253, + 439 + ], + "score": 1.0, + "content": "CLOSURE LEMMAS", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 21 + }, + { + "type": "text", + "bbox": [ + 107, + 449, + 504, + 483 + ], + "lines": [ + { + "bbox": [ + 106, + 449, + 506, + 461 + ], + "spans": [ + { + "bbox": [ + 106, + 449, + 266, + 461 + ], + "score": 1.0, + "content": "Here, we present some basic facts of the", + "type": "text" + }, + { + "bbox": [ + 266, + 450, + 285, + 460 + ], + "score": 0.88, + "content": "B B ^ { * }", + "type": "inline_equation" + }, + { + "bbox": [ + 285, + 449, + 506, + 461 + ], + "score": 1.0, + "content": "hierarchy that will be useful for later constructions. For", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 461, + 506, + 473 + ], + "spans": [ + { + "bbox": [ + 106, + 461, + 506, + 473 + ], + "score": 1.0, + "content": "simplicity, we assume (WLOG via 0-padding) that all matrices are square matrices with size that is a", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 104, + 472, + 155, + 483 + ], + "spans": [ + { + "bbox": [ + 104, + 472, + 155, + 483 + ], + "score": 1.0, + "content": "power of 2.", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 23, + "bbox_fs": [ + 104, + 449, + 506, + 483 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 486, + 500, + 499 + ], + "lines": [ + { + "bbox": [ + 105, + 486, + 501, + 501 + ], + "spans": [ + { + "bbox": [ + 105, + 486, + 173, + 501 + ], + "score": 1.0, + "content": "Lemma H.1. If", + "type": "text" + }, + { + "bbox": [ + 173, + 487, + 204, + 498 + ], + "score": 0.87, + "content": "\\mathbf { M } \\in B", + "type": "inline_equation" + }, + { + "bbox": [ + 205, + 486, + 219, + 501 + ], + "score": 1.0, + "content": "(or", + "type": "text" + }, + { + "bbox": [ + 220, + 487, + 256, + 498 + ], + "score": 0.85, + "content": "\\mathbf { M } \\in B ^ { * }", + "type": "inline_equation" + }, + { + "bbox": [ + 256, + 486, + 306, + 501 + ], + "score": 1.0, + "content": "), then DM,", + "type": "text" + }, + { + "bbox": [ + 307, + 487, + 347, + 498 + ], + "score": 0.73, + "content": "\\mathbf { M D } \\in B", + "type": "inline_equation" + }, + { + "bbox": [ + 347, + 486, + 351, + 501 + ], + "score": 1.0, + "content": "(", + "type": "text" + }, + { + "bbox": [ + 351, + 487, + 363, + 498 + ], + "score": 0.8, + "content": "B ^ { * }", + "type": "inline_equation" + }, + { + "bbox": [ + 364, + 486, + 501, + 501 + ], + "score": 1.0, + "content": "resp.) for any diagonal matrix D.", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 25, + "bbox_fs": [ + 105, + 486, + 501, + 501 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 513, + 505, + 559 + ], + "lines": [ + { + "bbox": [ + 105, + 513, + 506, + 527 + ], + "spans": [ + { + "bbox": [ + 105, + 513, + 506, + 527 + ], + "score": 1.0, + "content": "Proof. Left multiplication by a diagonal matrix scales the rows of M by the corresponding diagonal", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 524, + 506, + 538 + ], + "spans": [ + { + "bbox": [ + 105, + 524, + 506, + 538 + ], + "score": 1.0, + "content": "entries. The same can be achieved by scaling all entries the leftmost butterfly factor matrix. Similarly,", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 536, + 505, + 549 + ], + "spans": [ + { + "bbox": [ + 105, + 536, + 355, + 549 + ], + "score": 1.0, + "content": "right multiplication by a diagonal matrix scales the columns of", + "type": "text" + }, + { + "bbox": [ + 356, + 536, + 367, + 546 + ], + "score": 0.42, + "content": "\\mathbf { M }", + "type": "inline_equation" + }, + { + "bbox": [ + 368, + 536, + 505, + 549 + ], + "score": 1.0, + "content": ", which can be achieved by scaling", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 547, + 502, + 559 + ], + "spans": [ + { + "bbox": [ + 106, + 547, + 368, + 559 + ], + "score": 1.0, + "content": "all entries in the columns of the rightmost butterfly factor matrix.", + "type": "text" + }, + { + "bbox": [ + 497, + 550, + 502, + 556 + ], + "score": 0.0, + "content": "", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 27.5, + "bbox_fs": [ + 105, + 513, + 506, + 559 + ] + }, + { + "type": "text", + "bbox": [ + 105, + 567, + 491, + 581 + ], + "lines": [ + { + "bbox": [ + 102, + 565, + 488, + 584 + ], + "spans": [ + { + "bbox": [ + 102, + 565, + 191, + 584 + ], + "score": 1.0, + "content": "Lemma H.2. Let A,", + "type": "text" + }, + { + "bbox": [ + 191, + 568, + 236, + 579 + ], + "score": 0.85, + "content": "\\mathbf { B } \\in \\mathbb { F } ^ { n \\times n }", + "type": "inline_equation" + }, + { + "bbox": [ + 237, + 565, + 248, + 584 + ], + "score": 1.0, + "content": ". If", + "type": "text" + }, + { + "bbox": [ + 249, + 568, + 307, + 580 + ], + "score": 0.89, + "content": "\\mathbf { \\dot { A } } \\in ( B B ^ { * } ) _ { e } ^ { w _ { 1 } }", + "type": "inline_equation" + }, + { + "bbox": [ + 308, + 565, + 326, + 584 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 327, + 568, + 384, + 580 + ], + "score": 0.85, + "content": "\\mathbf { B } \\in ( B B ^ { * } ) _ { e } ^ { w _ { 2 } }", + "type": "inline_equation" + }, + { + "bbox": [ + 384, + 565, + 405, + 584 + ], + "score": 1.0, + "content": "then", + "type": "text" + }, + { + "bbox": [ + 406, + 568, + 488, + 581 + ], + "score": 0.84, + "content": "\\mathbf { A B } \\in ( B B ^ { * } ) _ { e } ^ { w _ { 1 } + w _ { 2 } } .", + "type": "inline_equation" + } + ], + "index": 30 + } + ], + "index": 30, + "bbox_fs": [ + 102, + 565, + 488, + 584 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 594, + 502, + 617 + ], + "lines": [ + { + "bbox": [ + 105, + 592, + 505, + 608 + ], + "spans": [ + { + "bbox": [ + 105, + 592, + 153, + 608 + ], + "score": 1.0, + "content": "Proof. Let", + "type": "text" + }, + { + "bbox": [ + 154, + 595, + 234, + 606 + ], + "score": 0.9, + "content": "\\mathbf { E _ { A } } , \\mathbf { E _ { B } } \\in \\mathbb { F } ^ { e n \\times e n }", + "type": "inline_equation" + }, + { + "bbox": [ + 234, + 592, + 322, + 608 + ], + "score": 1.0, + "content": "be defined such that", + "type": "text" + }, + { + "bbox": [ + 322, + 594, + 382, + 606 + ], + "score": 0.9, + "content": "\\mathbf { A } = \\mathbf { S } \\mathbf { E } _ { \\mathbf { A } } \\mathbf { S } ^ { T }", + "type": "inline_equation" + }, + { + "bbox": [ + 382, + 592, + 387, + 608 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 387, + 594, + 446, + 606 + ], + "score": 0.9, + "content": "{ \\bf B } = { \\bf S } { \\bf E } _ { \\bf B } { \\bf S } ^ { T }", + "type": "inline_equation" + }, + { + "bbox": [ + 446, + 592, + 505, + 608 + ], + "score": 1.0, + "content": "(with S as in", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 605, + 195, + 618 + ], + "spans": [ + { + "bbox": [ + 106, + 605, + 195, + 618 + ], + "score": 1.0, + "content": "Definition 2.4). Then", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 31.5, + "bbox_fs": [ + 105, + 592, + 505, + 618 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 225, + 616, + 387, + 660 + ], + "lines": [ + { + "bbox": [ + 225, + 616, + 387, + 660 + ], + "spans": [ + { + "bbox": [ + 225, + 616, + 387, + 660 + ], + "score": 0.93, + "content": "\\mathbf { A } \\mathbf { B } = \\mathbf { S } \\underbrace { \\left[ \\mathbf { I } _ { n } \\mathbf { \\Gamma } _ { 0 } \\right] } _ { e n \\times e n } \\mathbf { E } _ { \\mathbf { A } } \\underbrace { \\left[ \\mathbf { I } _ { n } \\mathbf { \\Gamma } _ { 0 } \\right] } _ { e n \\times e n } \\mathbf { E } _ { \\mathbf { B } } \\mathbf { S } ^ { T }", + "type": "interline_equation", + "image_path": "163a4b39e4da26f297e8d3db683a356062efc3cb8eaa5dafffac0f17c76a7784.jpg" + } + ] + } + ], + "index": 34, + "virtual_lines": [ + { + "bbox": [ + 225, + 616, + 387, + 630.6666666666666 + ], + "spans": [], + "index": 33 + }, + { + "bbox": [ + 225, + 630.6666666666666, + 387, + 645.3333333333333 + ], + "spans": [], + "index": 34 + }, + { + "bbox": [ + 225, + 645.3333333333333, + 387, + 659.9999999999999 + ], + "spans": [], + "index": 35 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 664, + 505, + 700 + ], + "lines": [ + { + "bbox": [ + 107, + 659, + 508, + 694 + ], + "spans": [ + { + "bbox": [ + 107, + 664, + 312, + 691 + ], + "score": 0.76, + "content": "\\begin{array} { r l } { \\bigg [ \\mathbf { I } _ { n } } & { { } 0 \\bigg ] \\mathbf { E _ { A } } \\in ( B B ^ { * } ) ^ { w _ { 1 } } , \\bigg [ \\mathbf { I } _ { n } \\quad 0 \\bigg ] \\mathbf { E _ { B } } \\in ( B B ^ { * } ) ^ { w _ { 2 } } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 312, + 659, + 410, + 694 + ], + "score": 1.0, + "content": "by Lemma H.1. Hence,", + "type": "text" + }, + { + "bbox": [ + 410, + 671, + 491, + 684 + ], + "score": 0.93, + "content": "\\mathbf { A B } \\in ( B B ^ { * } ) _ { e } ^ { w _ { 1 } + w _ { 2 } }", + "type": "inline_equation" + }, + { + "bbox": [ + 491, + 659, + 508, + 694 + ], + "score": 1.0, + "content": "by", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 687, + 168, + 700 + ], + "spans": [ + { + "bbox": [ + 105, + 687, + 168, + 700 + ], + "score": 1.0, + "content": "Definition 2.4.", + "type": "text" + } + ], + "index": 37 + } + ], + "index": 36.5, + "bbox_fs": [ + 105, + 659, + 508, + 700 + ] + }, + { + "type": "text", + "bbox": [ + 105, + 708, + 505, + 733 + ], + "lines": [ + { + "bbox": [ + 104, + 706, + 505, + 724 + ], + "spans": [ + { + "bbox": [ + 104, + 706, + 182, + 724 + ], + "score": 1.0, + "content": "Lemma H.3. Let", + "type": "text" + }, + { + "bbox": [ + 182, + 709, + 271, + 721 + ], + "score": 0.9, + "content": "\\mathbf { A } _ { 1 } , \\ldots , \\mathbf { A } _ { m } \\in \\mathbb { F } ^ { k \\times k }", + "type": "inline_equation" + }, + { + "bbox": [ + 272, + 706, + 288, + 724 + ], + "score": 1.0, + "content": ". 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If", + "type": "text" + }, + { + "bbox": [ + 178, + 222, + 208, + 232 + ], + "score": 0.88, + "content": "\\textit { e } = \\ 1", + "type": "inline_equation" + }, + { + "bbox": [ + 208, + 219, + 297, + 235 + ], + "score": 1.0, + "content": "in Lemma H.3, then", + "type": "text" + }, + { + "bbox": [ + 298, + 222, + 307, + 231 + ], + "score": 0.64, + "content": "\\mathbf { P }", + "type": "inline_equation" + }, + { + "bbox": [ + 307, + 219, + 411, + 235 + ], + "score": 1.0, + "content": "is unnecessary. 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Let", + "type": "text" + }, + { + "bbox": [ + 180, + 248, + 232, + 259 + ], + "score": 0.88, + "content": "\\mathbf { A } _ { 1 } , \\ldots , \\mathbf { A } _ { m }", + "type": "inline_equation" + }, + { + "bbox": [ + 232, + 241, + 245, + 266 + ], + "score": 1.0, + "content": "be", + "type": "text" + }, + { + "bbox": [ + 245, + 248, + 270, + 258 + ], + "score": 0.87, + "content": "k \\times k", + "type": "inline_equation" + }, + { + "bbox": [ + 270, + 241, + 318, + 266 + ], + "score": 1.0, + "content": "matrices in", + "type": "text" + }, + { + "bbox": [ + 319, + 247, + 351, + 260 + ], + "score": 0.9, + "content": "( B B ^ { * } ) _ { e } ^ { w }", + "type": "inline_equation" + }, + { + "bbox": [ + 352, + 241, + 372, + 266 + ], + "score": 1.0, + "content": "then", + "type": "text" + }, + { + "bbox": [ + 373, + 246, + 464, + 260 + ], + "score": 0.83, + "content": "\\begin{array} { r } { \\sum _ { i = 1 } ^ { m } { \\bf A } _ { i } \\in ( B B ^ { * } ) _ { 4 e } ^ { m w } . } \\end{array}", + "type": "inline_equation" + } + ], + "index": 11 + } + ], + "index": 11, + "bbox_fs": [ + 102, + 241, + 464, + 266 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 272, + 504, + 297 + ], + "lines": [ + { + "bbox": [ + 105, + 271, + 506, + 287 + ], + "spans": [ + { + "bbox": [ + 105, + 271, + 174, + 287 + ], + "score": 1.0, + "content": "Proof. 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Note that", + "type": "text" + }, + { + "bbox": [ + 210, + 285, + 273, + 297 + ], + "score": 0.92, + "content": "\\mathbf { E } _ { \\mathbf { A } _ { i } } \\in ( B B ^ { * } ) ^ { w }", + "type": "inline_equation" + }, + { + "bbox": [ + 274, + 283, + 403, + 297 + ], + "score": 1.0, + "content": ". 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"inline_equation" + }, + { + "bbox": [ + 243, + 609, + 461, + 621 + ], + "score": 1.0, + "content": "columns of the second block-column of M to the left.", + "type": "text" + }, + { + "bbox": [ + 462, + 610, + 475, + 621 + ], + "score": 0.88, + "content": "\\mathbf { P } _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 476, + 609, + 505, + 621 + ], + "score": 1.0, + "content": "can be", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 620, + 325, + 632 + ], + "spans": [ + { + "bbox": [ + 106, + 620, + 259, + 632 + ], + "score": 1.0, + "content": "folded into the final summation factor", + "type": "text" + }, + { + "bbox": [ + 260, + 621, + 278, + 632 + ], + "score": 0.9, + "content": "M _ { m }", + "type": "inline_equation" + }, + { + "bbox": [ + 279, + 620, + 325, + 632 + ], + "score": 1.0, + "content": "as follows:", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 32, + "bbox_fs": [ + 102, + 593, + 509, + 632 + ] + }, + { + "type": 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} } \\\\ { \\mathbf { I } _ { e k } } & { \\mathbf { 0 } } & { \\mathbf { 0 } } & { \\mathbf { 0 } } \\\\ { \\mathbf { 0 } } & { \\mathbf { 0 } } & { \\mathbf { I } _ { e k } } & { \\mathbf { 0 } } \\\\ { \\mathbf { 0 } } & { \\mathbf { 0 } } & { \\mathbf { 0 } } & { \\mathbf { I } _ { e k } } \\end{array} \\right] } _ { \\mathbf { P } _ { 2 } } = \\underbrace { \\left[ \\begin{array} { c c c c } { \\mathbf { 0 } } & { \\mathbf { I } _ { e k } } & { \\mathbf { 0 } } & { \\mathbf { 0 } } \\\\ { \\mathbf { I } _ { e k } } & { \\mathbf { 0 } } & { \\mathbf { 0 } } & { \\mathbf { 0 } } \\\\ { \\mathbf { 0 } } & { \\mathbf { 0 } } & { \\mathbf { I } _ { e k } } & { \\mathbf { 0 } } \\\\ { \\mathbf { 0 } } & { \\mathbf { 0 } } & { \\mathbf { 0 } } & { \\mathbf { I } _ { e k } } \\end{array} \\right] } _ { \\mathbf { P } _ { 1 } ^ { \\prime } } \\underbrace { \\left[ \\mathbf { I } _ { 2 e k } & { \\mathbf { 0 } } \\\\ & & { \\mathbf { R } } \\end{array} \\right] } _ { \\mathbf { R } } \\underbrace { \\left[ \\begin{array} { c c c c } { \\mathbf { I } _ { e k } } & { \\mathbf { 0 } } & { \\mathbf { I } _ { e k } } & { \\mathbf { 0 } } \\\\ & & { \\mathbf { I } _ { e k } } & { \\mathbf { 0 } } \\\\ & & { \\mathbf { P } _ { 2 } ^ { \\prime } } & { \\mathbf { R } } \\end{array} \\right] } _ \\mathbf ", + "type": "interline_equation", + "image_path": "342929852fdaea4b80c280f12ec2473f9213e0a974c11eeae3518f01525b3f35.jpg" + } + ] + } + ], + "index": 35, + "virtual_lines": [ + { + "bbox": [ + 115, + 636, + 484, + 660.3333333333334 + ], + "spans": [], + "index": 34 + }, + { + "bbox": [ + 115, + 660.3333333333334, + 484, + 684.6666666666667 + ], + "spans": [], + "index": 35 + }, + { + "bbox": [ + 115, + 684.6666666666667, + 484, + 709.0000000000001 + ], + "spans": [], + "index": 36 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 719, + 275, + 734 + ], + "lines": [ + { + "bbox": [ + 105, + 716, + 277, + 736 + ], + "spans": [ + { + "bbox": [ + 105, + 716, + 137, + 736 + ], + "score": 1.0, + "content": "Hence,", + "type": "text" + }, + { + "bbox": [ + 137, + 720, + 227, + 734 + ], + "score": 0.92, + "content": "\\begin{array} { r } { \\sum _ { i = 1 } ^ { m } \\mathbf { A } _ { i } \\in ( B B ^ { * } ) _ { 4 e } ^ { m w } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 227, + 716, + 277, + 736 + ], + "score": 1.0, + "content": ", as desired.", + "type": "text" + } + ], + "index": 37 + } + ], + "index": 37, + "bbox_fs": [ + 105, + 716, + 277, + 736 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 106, + 81, + 457, + 94 + ], + "lines": [ + { + "bbox": [ + 105, + 81, + 455, + 95 + ], + "spans": [ + { + "bbox": [ + 105, + 81, + 257, + 95 + ], + "score": 1.0, + "content": "Lemma H.6. Let M be an invertible", + "type": "text" + }, + { + "bbox": [ + 258, + 84, + 283, + 93 + ], + "score": 0.86, + "content": "n \\times n", + "type": "inline_equation" + }, + { + "bbox": [ + 283, + 81, + 351, + 95 + ], + "score": 1.0, + "content": "matrix such that", + "type": "text" + }, + { + "bbox": [ + 352, + 82, + 382, + 93 + ], + "score": 0.89, + "content": "M \\in B", + "type": "inline_equation" + }, + { + "bbox": [ + 383, + 81, + 409, + 95 + ], + "score": 1.0, + "content": ". Then", + "type": "text" + }, + { + "bbox": [ + 409, + 82, + 455, + 93 + ], + "score": 0.91, + "content": "\\mathbf { M } ^ { - 1 } \\in { \\mathfrak { B } } ^ { * }", + "type": "inline_equation" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "text", + "bbox": [ + 107, + 109, + 270, + 122 + ], + "lines": [ + { + "bbox": [ + 105, + 108, + 272, + 124 + ], + "spans": [ + { + "bbox": [ + 105, + 108, + 272, + 124 + ], + "score": 1.0, + "content": "Proof. We prove this in a series of steps.", + "type": "text" + } + ], + "index": 1 + } + ], + "index": 1 + }, + { + "type": "text", + "bbox": [ + 107, + 126, + 505, + 175 + ], + "lines": [ + { + "bbox": [ + 105, + 125, + 505, + 140 + ], + "spans": [ + { + "bbox": [ + 105, + 125, + 142, + 140 + ], + "score": 1.0, + "content": "First, let", + "type": "text" + }, + { + "bbox": [ + 143, + 127, + 157, + 138 + ], + "score": 0.89, + "content": "\\mathbf { B } _ { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 158, + 125, + 316, + 140 + ], + "score": 1.0, + "content": "be an invertible butterfly factor of size", + "type": "text" + }, + { + "bbox": [ + 316, + 127, + 322, + 137 + ], + "score": 0.76, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 323, + 125, + 471, + 140 + ], + "score": 1.0, + "content": ". Consider the method of computing", + "type": "text" + }, + { + "bbox": [ + 471, + 126, + 491, + 140 + ], + "score": 0.92, + "content": "\\mathbf { B } _ { k } ^ { - 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 491, + 125, + 505, + 140 + ], + "score": 1.0, + "content": "by", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 138, + 505, + 153 + ], + "spans": [ + { + "bbox": [ + 105, + 138, + 296, + 153 + ], + "score": 1.0, + "content": "performing Gaussian elimination on the matrix", + "type": "text" + }, + { + "bbox": [ + 296, + 139, + 328, + 151 + ], + "score": 0.93, + "content": "[ \\mathbf { B } _ { k } | \\mathbf { I } _ { k } ]", + "type": "inline_equation" + }, + { + "bbox": [ + 328, + 138, + 410, + 153 + ], + "score": 1.0, + "content": "to obtain the matrix", + "type": "text" + }, + { + "bbox": [ + 411, + 138, + 450, + 153 + ], + "score": 0.92, + "content": "\\left[ \\mathbf { I } _ { k } | \\mathbf { B } _ { k } ^ { - 1 } \\right]", + "type": "inline_equation" + }, + { + "bbox": [ + 450, + 138, + 505, + 153 + ], + "score": 1.0, + "content": ". By the form", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 151, + 505, + 167 + ], + "spans": [ + { + "bbox": [ + 105, + 151, + 372, + 167 + ], + "score": 1.0, + "content": "of B, non-zero entries within a row or column are always exactly", + "type": "text" + }, + { + "bbox": [ + 372, + 151, + 380, + 165 + ], + "score": 0.86, + "content": "\\frac { k } { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 380, + 151, + 505, + 167 + ], + "score": 1.0, + "content": "positions apart. Therefore, the", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 163, + 355, + 176 + ], + "spans": [ + { + "bbox": [ + 105, + 163, + 355, + 176 + ], + "score": 1.0, + "content": "only row operations needed for this Gaussian elimination are:", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 3.5 + }, + { + "type": "text", + "bbox": [ + 131, + 189, + 361, + 222 + ], + "lines": [ + { + "bbox": [ + 132, + 189, + 304, + 202 + ], + "spans": [ + { + "bbox": [ + 132, + 189, + 280, + 202 + ], + "score": 1.0, + "content": "• Scaling a row by a constant factor", + "type": "text" + }, + { + "bbox": [ + 281, + 190, + 304, + 201 + ], + "score": 0.87, + "content": "c \\neq 0", + "type": "inline_equation" + } + ], + "index": 6 + }, + { + "bbox": [ + 131, + 206, + 362, + 224 + ], + "spans": [ + { + "bbox": [ + 131, + 206, + 307, + 224 + ], + "score": 1.0, + "content": "• Addition of a row to another row exactly", + "type": "text" + }, + { + "bbox": [ + 308, + 208, + 315, + 222 + ], + "score": 0.86, + "content": "\\frac { k } { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 316, + 206, + 362, + 224 + ], + "score": 1.0, + "content": "rows apart", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 6.5 + }, + { + "type": "text", + "bbox": [ + 106, + 237, + 505, + 263 + ], + "lines": [ + { + "bbox": [ + 105, + 235, + 506, + 252 + ], + "spans": [ + { + "bbox": [ + 105, + 235, + 234, + 252 + ], + "score": 1.0, + "content": "Performing these operations on", + "type": "text" + }, + { + "bbox": [ + 234, + 238, + 245, + 249 + ], + "score": 0.87, + "content": "\\mathbf { I } _ { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 245, + 235, + 455, + 252 + ], + "score": 1.0, + "content": "will only allow non-zeros on the main diagonal and", + "type": "text" + }, + { + "bbox": [ + 455, + 236, + 463, + 250 + ], + "score": 0.85, + "content": "\\frac { k } { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 463, + 235, + 506, + 252 + ], + "score": 1.0, + "content": "diagonals", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 249, + 416, + 264 + ], + "spans": [ + { + "bbox": [ + 106, + 249, + 258, + 264 + ], + "score": 1.0, + "content": "away from the main diagonal. Hence,", + "type": "text" + }, + { + "bbox": [ + 259, + 249, + 279, + 263 + ], + "score": 0.92, + "content": "\\mathbf { B } _ { k } ^ { - 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 279, + 249, + 405, + 264 + ], + "score": 1.0, + "content": "is also a butterfly factor of size", + "type": "text" + }, + { + "bbox": [ + 405, + 251, + 412, + 261 + ], + "score": 0.81, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 412, + 249, + 416, + 264 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 8.5 + }, + { + "type": "text", + "bbox": [ + 106, + 270, + 506, + 315 + ], + "lines": [ + { + "bbox": [ + 104, + 267, + 506, + 284 + ], + "spans": [ + { + "bbox": [ + 104, + 267, + 144, + 284 + ], + "score": 1.0, + "content": "Next, let", + "type": "text" + }, + { + "bbox": [ + 144, + 268, + 165, + 283 + ], + "score": 0.92, + "content": "\\mathbf { B } _ { k } ^ { \\left( n \\right) }", + "type": "inline_equation" + }, + { + "bbox": [ + 165, + 267, + 354, + 284 + ], + "score": 1.0, + "content": "be an invertible butterfly factor matrix of size", + "type": "text" + }, + { + "bbox": [ + 354, + 273, + 362, + 281 + ], + "score": 0.76, + "content": "n", + "type": "inline_equation" + }, + { + "bbox": [ + 362, + 267, + 424, + 284 + ], + "score": 1.0, + "content": "and block size", + "type": "text" + }, + { + "bbox": [ + 424, + 272, + 431, + 281 + ], + "score": 0.79, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 432, + 267, + 506, + 284 + ], + "score": 1.0, + "content": ". Its inverse is the", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 281, + 507, + 295 + ], + "spans": [ + { + "bbox": [ + 105, + 281, + 507, + 295 + ], + "score": 1.0, + "content": "block diagonal matrix formed by the inverses of each of its constituent butterfly factors. From above,", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 107, + 291, + 385, + 318 + ], + "spans": [ + { + "bbox": [ + 107, + 293, + 149, + 315 + ], + "score": 0.92, + "content": "\\left( \\mathbf { B } _ { k } ^ { \\left( n \\right) } \\right) ^ { - 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 150, + 291, + 304, + 318 + ], + "score": 1.0, + "content": "is also a butterfly factor matrix of size", + "type": "text" + }, + { + "bbox": [ + 305, + 301, + 312, + 309 + ], + "score": 0.77, + "content": "n", + "type": "inline_equation" + }, + { + "bbox": [ + 313, + 291, + 372, + 318 + ], + "score": 1.0, + "content": "and block size", + "type": "text" + }, + { + "bbox": [ + 373, + 299, + 379, + 309 + ], + "score": 0.82, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 380, + 291, + 385, + 318 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 11 + }, + { + "type": "text", + "bbox": [ + 107, + 318, + 209, + 331 + ], + "lines": [ + { + "bbox": [ + 105, + 318, + 210, + 331 + ], + "spans": [ + { + "bbox": [ + 105, + 318, + 175, + 331 + ], + "score": 1.0, + "content": "Finally, consider", + "type": "text" + }, + { + "bbox": [ + 175, + 319, + 207, + 329 + ], + "score": 0.89, + "content": "\\mathbf { M } \\in B", + "type": "inline_equation" + }, + { + "bbox": [ + 207, + 318, + 210, + 331 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 13 + }, + { + "type": "interline_equation", + "bbox": [ + 111, + 336, + 514, + 361 + ], + "lines": [ + { + "bbox": [ + 111, + 336, + 514, + 361 + ], + "spans": [ + { + "bbox": [ + 111, + 336, + 514, + 361 + ], + "score": 0.87, + "content": "\\mathbf { { \\cal { M } } } ^ { - 1 } = \\left( \\mathbf { B } _ { n } ^ { ( n ) } \\mathbf { B } _ { \\frac { n } { 2 } } ^ { ( n ) } \\dots \\mathbf { B } _ { 2 } ^ { ( n ) } \\right) ^ { - 1 } = \\left( \\mathbf { B } _ { 2 } ^ { ( n ) } \\right) ^ { - 1 } \\left( \\mathbf { B } _ { 4 } ^ { ( n ) } \\right) ^ { - 1 } \\dots \\left( \\mathbf { B } _ { n } ^ { ( n ) } \\right) ^ { - 1 } = \\mathbf { B } _ { 2 } ^ { \\prime { } ^ { ( n ) } } \\mathbf { B } _ { 4 } ^ { \\prime { } ^ { ( n ) } } \\dots \\mathbf { B } _ { n } ^ { \\prime { } ^ { ( n ) } } \\in \\mathcal { B } _ { n } ^ { ( n ) } .", + "type": "interline_equation", + "image_path": "7e97d956a109d454e63c54b713f6d44ce6988d1399ea49191fff34622a6596bb.jpg" + } + ] + } + ], + "index": 14, + "virtual_lines": [ + { + "bbox": [ + 111, + 336, + 514, + 361 + ], + "spans": [], + "index": 14 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 392, + 505, + 438 + ], + "lines": [ + { + "bbox": [ + 106, + 393, + 505, + 405 + ], + "spans": [ + { + "bbox": [ + 106, + 393, + 505, + 405 + ], + "score": 1.0, + "content": "Finally, we include a closure result for the Kronecker product, another common matrix composition", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 404, + 505, + 417 + ], + "spans": [ + { + "bbox": [ + 105, + 404, + 505, + 417 + ], + "score": 1.0, + "content": "operation. Although Lemma H.7 is not directly used in the subsequent proofs, it allows for examples", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 416, + 505, + 426 + ], + "spans": [ + { + "bbox": [ + 106, + 416, + 505, + 426 + ], + "score": 1.0, + "content": "the results for the DFT to be lifted to higher-dimensional Fourier transforms. We also note that the", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 426, + 416, + 438 + ], + "spans": [ + { + "bbox": [ + 106, + 426, + 416, + 438 + ], + "score": 1.0, + "content": "closure bound in Lemma H.7 can be tightened in such cases (cfRemark H.4). ˙", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 16.5 + }, + { + "type": "text", + "bbox": [ + 105, + 441, + 504, + 455 + ], + "lines": [ + { + "bbox": [ + 104, + 438, + 502, + 456 + ], + "spans": [ + { + "bbox": [ + 104, + 438, + 190, + 456 + ], + "score": 1.0, + "content": "Lemma H.7. Let A,", + "type": "text" + }, + { + "bbox": [ + 190, + 441, + 235, + 453 + ], + "score": 0.85, + "content": "\\mathbf { B } \\in \\mathbb { F } ^ { n \\times n }", + "type": "inline_equation" + }, + { + "bbox": [ + 235, + 438, + 245, + 456 + ], + "score": 1.0, + "content": ". I", + "type": "text" + }, + { + "bbox": [ + 246, + 441, + 305, + 454 + ], + "score": 0.9, + "content": "^ { c } \\mathbf { A } \\in ( B B ^ { * } ) _ { e } ^ { w _ { 1 } }", + "type": "inline_equation" + }, + { + "bbox": [ + 306, + 438, + 324, + 456 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 324, + 441, + 381, + 454 + ], + "score": 0.93, + "content": "\\mathbf { B } \\in ( B B ^ { * } ) _ { e } ^ { w _ { 2 } }", + "type": "inline_equation" + }, + { + "bbox": [ + 381, + 438, + 401, + 456 + ], + "score": 1.0, + "content": "then", + "type": "text" + }, + { + "bbox": [ + 401, + 441, + 502, + 454 + ], + "score": 0.89, + "content": "\\mathbf { A } \\otimes \\mathbf { B } \\in ( B B ^ { * } ) _ { e } ^ { w _ { 1 } + w _ { 2 } + 6 }", + "type": "inline_equation" + } + ], + "index": 19 + } + ], + "index": 19 + }, + { + "type": "text", + "bbox": [ + 107, + 469, + 175, + 480 + ], + "lines": [ + { + "bbox": [ + 106, + 469, + 176, + 482 + ], + "spans": [ + { + "bbox": [ + 106, + 469, + 176, + 482 + ], + "score": 1.0, + "content": "Proof. Note that", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 20 + }, + { + "type": "interline_equation", + "bbox": [ + 196, + 479, + 413, + 494 + ], + "lines": [ + { + "bbox": [ + 196, + 479, + 413, + 494 + ], + "spans": [ + { + "bbox": [ + 196, + 479, + 413, + 494 + ], + "score": 0.88, + "content": "\\mathbf { A } \\otimes \\mathbf { B } = ( \\mathbf { A } \\otimes \\mathbf { I } ) ( \\mathbf { I } \\otimes \\mathbf { B } ) = \\mathbf { P } ^ { - 1 } ( \\mathbf { I } \\otimes \\mathbf { A } ) \\mathbf { P } ( \\mathbf { I } \\otimes \\mathbf { B } ) ,", + "type": "interline_equation", + "image_path": "0c1a34063ecb12b25aafe14a72e3df94b6b00236c59aab6a836d70d94dd574d8.jpg" + } + ] + } + ], + "index": 21, + "virtual_lines": [ + { + "bbox": [ + 196, + 479, + 413, + 494 + ], + "spans": [], + "index": 21 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 497, + 504, + 521 + ], + "lines": [ + { + "bbox": [ + 104, + 494, + 509, + 515 + ], + "spans": [ + { + "bbox": [ + 104, + 494, + 191, + 515 + ], + "score": 1.0, + "content": "for some permutation", + "type": "text" + }, + { + "bbox": [ + 192, + 498, + 201, + 508 + ], + "score": 0.3, + "content": "\\mathbf { P }", + "type": "inline_equation" + }, + { + "bbox": [ + 201, + 494, + 269, + 515 + ], + "score": 1.0, + "content": ". By Lemma H.3,", + "type": "text" + }, + { + "bbox": [ + 269, + 498, + 293, + 509 + ], + "score": 0.74, + "content": "\\mathbf { I } \\otimes \\mathbf { A }", + "type": "inline_equation" + }, + { + "bbox": [ + 293, + 494, + 309, + 515 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 309, + 498, + 332, + 509 + ], + "score": 0.88, + "content": "\\mathbf { I } \\otimes \\mathbf { B }", + "type": "inline_equation" + }, + { + "bbox": [ + 333, + 494, + 357, + 515 + ], + "score": 1.0, + "content": "are in", + "type": "text" + }, + { + "bbox": [ + 357, + 497, + 455, + 510 + ], + "score": 0.83, + "content": "( B B ^ { * } ) _ { e } ^ { w _ { 1 } + 2 } , ( B B ^ { * } ) _ { e } ^ { w _ { 2 } + 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 456, + 494, + 509, + 515 + ], + "score": 1.0, + "content": "respectively.", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 509, + 506, + 520 + ], + "spans": [ + { + "bbox": [ + 106, + 509, + 268, + 520 + ], + "score": 1.0, + "content": "The result follows from combining with", + "type": "text" + }, + { + "bbox": [ + 268, + 509, + 307, + 519 + ], + "score": 0.91, + "content": "\\mathbf { P } \\in B B ^ { * }", + "type": "inline_equation" + }, + { + "bbox": [ + 308, + 509, + 379, + 520 + ], + "score": 1.0, + "content": "and Lemma H.2.", + "type": "text" + }, + { + "bbox": [ + 494, + 509, + 506, + 520 + ], + "score": 0.999, + "content": "□", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 22.5 + }, + { + "type": "title", + "bbox": [ + 107, + 537, + 325, + 551 + ], + "lines": [ + { + "bbox": [ + 104, + 536, + 326, + 552 + ], + "spans": [ + { + "bbox": [ + 104, + 536, + 236, + 552 + ], + "score": 1.0, + "content": "I SPARSE MATRICES IN", + "type": "text" + }, + { + "bbox": [ + 236, + 538, + 259, + 550 + ], + "score": 0.84, + "content": "B B ^ { * }", + "type": "inline_equation" + }, + { + "bbox": [ + 259, + 536, + 326, + 552 + ], + "score": 1.0, + "content": "HIERARCHY", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 24 + }, + { + "type": "text", + "bbox": [ + 106, + 563, + 457, + 575 + ], + "lines": [ + { + "bbox": [ + 105, + 563, + 459, + 576 + ], + "spans": [ + { + "bbox": [ + 105, + 563, + 425, + 576 + ], + "score": 1.0, + "content": "In this appendix, we prove Theorem 3. First, we consider matrices with at most", + "type": "text" + }, + { + "bbox": [ + 425, + 565, + 433, + 573 + ], + "score": 0.72, + "content": "n", + "type": "inline_equation" + }, + { + "bbox": [ + 433, + 563, + 459, + 576 + ], + "score": 1.0, + "content": "NNZ.", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 25 + }, + { + "type": "text", + "bbox": [ + 107, + 578, + 424, + 591 + ], + "lines": [ + { + "bbox": [ + 106, + 578, + 423, + 592 + ], + "spans": [ + { + "bbox": [ + 106, + 578, + 206, + 592 + ], + "score": 1.0, + "content": "Lemma I.1. let S be an", + "type": "text" + }, + { + "bbox": [ + 206, + 580, + 231, + 590 + ], + "score": 0.86, + "content": "n \\times n", + "type": "inline_equation" + }, + { + "bbox": [ + 232, + 578, + 369, + 592 + ], + "score": 1.0, + "content": "matrix with at most n NNZ. Then,", + "type": "text" + }, + { + "bbox": [ + 370, + 578, + 419, + 591 + ], + "score": 0.92, + "content": "\\mathbf { S } \\in ( B B ^ { * } ) ^ { 4 }", + "type": "inline_equation" + }, + { + "bbox": [ + 420, + 578, + 423, + 592 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 26 + }, + { + "type": "text", + "bbox": [ + 106, + 600, + 394, + 612 + ], + "lines": [ + { + "bbox": [ + 105, + 599, + 394, + 614 + ], + "spans": [ + { + "bbox": [ + 105, + 599, + 394, + 614 + ], + "score": 1.0, + "content": "We use this lemma and the addition closure lemma to prove Theorem 3.", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 27 + }, + { + "type": "text", + "bbox": [ + 106, + 626, + 505, + 650 + ], + "lines": [ + { + "bbox": [ + 105, + 627, + 507, + 641 + ], + "spans": [ + { + "bbox": [ + 105, + 627, + 260, + 641 + ], + "score": 1.0, + "content": "Proof of Theorem 3. We note that any", + "type": "text" + }, + { + "bbox": [ + 260, + 631, + 266, + 638 + ], + "score": 0.72, + "content": "s", + "type": "inline_equation" + }, + { + "bbox": [ + 266, + 627, + 376, + 641 + ], + "score": 1.0, + "content": "sparse matrix is the sum of", + "type": "text" + }, + { + "bbox": [ + 376, + 627, + 393, + 641 + ], + "score": 0.9, + "content": "\\textstyle { \\left\\lceil { \\frac { s } { n } } \\right\\rceil }", + "type": "inline_equation" + }, + { + "bbox": [ + 393, + 627, + 472, + 641 + ], + "score": 1.0, + "content": "matrices of at most", + "type": "text" + }, + { + "bbox": [ + 473, + 630, + 480, + 638 + ], + "score": 0.71, + "content": "n", + "type": "inline_equation" + }, + { + "bbox": [ + 480, + 627, + 507, + 641 + ], + "score": 1.0, + "content": "NNZ,", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 639, + 504, + 651 + ], + "spans": [ + { + "bbox": [ + 105, + 639, + 229, + 651 + ], + "score": 1.0, + "content": "and we appeal to Lemma H.5.", + "type": "text" + }, + { + "bbox": [ + 496, + 641, + 504, + 648 + ], + "score": 0.999, + "content": "□", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 28.5 + }, + { + "type": "text", + "bbox": [ + 106, + 665, + 504, + 689 + ], + "lines": [ + { + "bbox": [ + 105, + 665, + 506, + 679 + ], + "spans": [ + { + "bbox": [ + 105, + 665, + 506, + 679 + ], + "score": 1.0, + "content": "In the rest of the section we will prove Lemma I.1. We begin by defining two classes of matrices that", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 677, + 246, + 689 + ], + "spans": [ + { + "bbox": [ + 106, + 677, + 246, + 689 + ], + "score": 1.0, + "content": "will be used in our decomposition.", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 30.5 + }, + { + "type": "text", + "bbox": [ + 105, + 692, + 504, + 716 + ], + "lines": [ + { + "bbox": [ + 105, + 691, + 506, + 705 + ], + "spans": [ + { + "bbox": [ + 105, + 691, + 186, + 705 + ], + "score": 1.0, + "content": "Definition I.1. An", + "type": "text" + }, + { + "bbox": [ + 187, + 694, + 214, + 703 + ], + "score": 0.87, + "content": "n \\times n", + "type": "inline_equation" + }, + { + "bbox": [ + 214, + 691, + 246, + 705 + ], + "score": 1.0, + "content": "matrix", + "type": "text" + }, + { + "bbox": [ + 246, + 693, + 257, + 703 + ], + "score": 0.55, + "content": "\\mathbf { H }", + "type": "inline_equation" + }, + { + "bbox": [ + 257, + 691, + 424, + 705 + ], + "score": 1.0, + "content": "is a horizontal step matrix if for every", + "type": "text" + }, + { + "bbox": [ + 424, + 693, + 485, + 704 + ], + "score": 0.91, + "content": "0 \\leq i , i ^ { \\prime } < n", + "type": "inline_equation" + }, + { + "bbox": [ + 486, + 691, + 506, + 705 + ], + "score": 1.0, + "content": "and", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 107, + 702, + 431, + 717 + ], + "spans": [ + { + "bbox": [ + 107, + 703, + 171, + 716 + ], + "score": 0.9, + "content": "0 \\leq j \\leq j ^ { \\prime } < n ,", + "type": "inline_equation" + }, + { + "bbox": [ + 171, + 702, + 176, + 717 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 177, + 704, + 229, + 716 + ], + "score": 0.87, + "content": "i f \\mathbf { H } [ i , j ] \\neq 0", + "type": "inline_equation" + }, + { + "bbox": [ + 230, + 702, + 248, + 717 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 248, + 703, + 300, + 716 + ], + "score": 0.92, + "content": "\\mathbf { H } [ i ^ { \\prime } , j ^ { \\prime } ] \\neq 0", + "type": "inline_equation" + }, + { + "bbox": [ + 300, + 702, + 323, + 717 + ], + "score": 1.0, + "content": ", then", + "type": "text" + }, + { + "bbox": [ + 324, + 704, + 392, + 716 + ], + "score": 0.92, + "content": "j ^ { \\prime } - j \\geq ( i ^ { \\prime } - i )", + "type": "inline_equation" + }, + { + "bbox": [ + 392, + 702, + 419, + 717 + ], + "score": 1.0, + "content": "mod", + "type": "text" + }, + { + "bbox": [ + 420, + 706, + 426, + 713 + ], + "score": 0.71, + "content": "n", + "type": "inline_equation" + }, + { + "bbox": [ + 426, + 702, + 431, + 717 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 32.5 + }, + { + "type": "text", + "bbox": [ + 106, + 720, + 416, + 732 + ], + "lines": [ + { + "bbox": [ + 104, + 720, + 416, + 734 + ], + "spans": [ + { + "bbox": [ + 104, + 720, + 119, + 734 + ], + "score": 1.0, + "content": "An", + "type": "text" + }, + { + "bbox": [ + 119, + 722, + 145, + 731 + ], + "score": 0.82, + "content": "n \\times n", + "type": "inline_equation" + }, + { + "bbox": [ + 145, + 720, + 174, + 734 + ], + "score": 1.0, + "content": "matrix", + "type": "text" + }, + { + "bbox": [ + 174, + 721, + 185, + 731 + ], + "score": 0.59, + "content": "\\mathbf { V }", + "type": "inline_equation" + }, + { + "bbox": [ + 185, + 720, + 291, + 734 + ], + "score": 1.0, + "content": "is a vertical step matrix if", + "type": "text" + }, + { + "bbox": [ + 292, + 721, + 306, + 731 + ], + "score": 0.7, + "content": "\\mathbf { V } ^ { * }", + "type": "inline_equation" + }, + { + "bbox": [ + 306, + 720, + 416, + 734 + ], + "score": 1.0, + "content": "is a horizontal step matrix.", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 34 + } + ], + "page_idx": 29, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 107, + 26, + 294, + 38 + ], + "lines": [ + { + "bbox": [ + 106, + 25, + 294, + 39 + ], + "spans": [ + { + "bbox": [ + 106, + 25, + 294, + 39 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2020", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 300, + 751, + 311, + 760 + ], + "lines": [ + { + "bbox": [ + 298, + 750, + 313, + 763 + ], + "spans": [ + { + "bbox": [ + 298, + 750, + 313, + 763 + ], + "score": 1.0, + "content": "30", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 495, + 366, + 504, + 376 + ], + "lines": [ + { + "bbox": [ + 496, + 368, + 504, + 376 + ], + "spans": [ + { + "bbox": [ + 496, + 368, + 504, + 376 + ], + "score": 0.998, + "content": "□", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 106, + 81, + 457, + 94 + ], + "lines": [ + { + "bbox": [ + 105, + 81, + 455, + 95 + ], + "spans": [ + { + "bbox": [ + 105, + 81, + 257, + 95 + ], + "score": 1.0, + "content": "Lemma H.6. Let M be an invertible", + "type": "text" + }, + { + "bbox": [ + 258, + 84, + 283, + 93 + ], + "score": 0.86, + "content": "n \\times n", + "type": "inline_equation" + }, + { + "bbox": [ + 283, + 81, + 351, + 95 + ], + "score": 1.0, + "content": "matrix such that", + "type": "text" + }, + { + "bbox": [ + 352, + 82, + 382, + 93 + ], + "score": 0.89, + "content": "M \\in B", + "type": "inline_equation" + }, + { + "bbox": [ + 383, + 81, + 409, + 95 + ], + "score": 1.0, + "content": ". Then", + "type": "text" + }, + { + "bbox": [ + 409, + 82, + 455, + 93 + ], + "score": 0.91, + "content": "\\mathbf { M } ^ { - 1 } \\in { \\mathfrak { B } } ^ { * }", + "type": "inline_equation" + } + ], + "index": 0 + } + ], + "index": 0, + "bbox_fs": [ + 105, + 81, + 455, + 95 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 109, + 270, + 122 + ], + "lines": [ + { + "bbox": [ + 105, + 108, + 272, + 124 + ], + "spans": [ + { + "bbox": [ + 105, + 108, + 272, + 124 + ], + "score": 1.0, + "content": "Proof. We prove this in a series of steps.", + "type": "text" + } + ], + "index": 1 + } + ], + "index": 1, + "bbox_fs": [ + 105, + 108, + 272, + 124 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 126, + 505, + 175 + ], + "lines": [ + { + "bbox": [ + 105, + 125, + 505, + 140 + ], + "spans": [ + { + "bbox": [ + 105, + 125, + 142, + 140 + ], + "score": 1.0, + "content": "First, let", + "type": "text" + }, + { + "bbox": [ + 143, + 127, + 157, + 138 + ], + "score": 0.89, + "content": "\\mathbf { B } _ { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 158, + 125, + 316, + 140 + ], + "score": 1.0, + "content": "be an invertible butterfly factor of size", + "type": "text" + }, + { + "bbox": [ + 316, + 127, + 322, + 137 + ], + "score": 0.76, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 323, + 125, + 471, + 140 + ], + "score": 1.0, + "content": ". Consider the method of computing", + "type": "text" + }, + { + "bbox": [ + 471, + 126, + 491, + 140 + ], + "score": 0.92, + "content": "\\mathbf { B } _ { k } ^ { - 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 491, + 125, + 505, + 140 + ], + "score": 1.0, + "content": "by", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 138, + 505, + 153 + ], + "spans": [ + { + "bbox": [ + 105, + 138, + 296, + 153 + ], + "score": 1.0, + "content": "performing Gaussian elimination on the matrix", + "type": "text" + }, + { + "bbox": [ + 296, + 139, + 328, + 151 + ], + "score": 0.93, + "content": "[ \\mathbf { B } _ { k } | \\mathbf { I } _ { k } ]", + "type": "inline_equation" + }, + { + "bbox": [ + 328, + 138, + 410, + 153 + ], + "score": 1.0, + "content": "to obtain the matrix", + "type": "text" + }, + { + "bbox": [ + 411, + 138, + 450, + 153 + ], + "score": 0.92, + "content": "\\left[ \\mathbf { I } _ { k } | \\mathbf { B } _ { k } ^ { - 1 } \\right]", + "type": "inline_equation" + }, + { + "bbox": [ + 450, + 138, + 505, + 153 + ], + "score": 1.0, + "content": ". By the form", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 151, + 505, + 167 + ], + "spans": [ + { + "bbox": [ + 105, + 151, + 372, + 167 + ], + "score": 1.0, + "content": "of B, non-zero entries within a row or column are always exactly", + "type": "text" + }, + { + "bbox": [ + 372, + 151, + 380, + 165 + ], + "score": 0.86, + "content": "\\frac { k } { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 380, + 151, + 505, + 167 + ], + "score": 1.0, + "content": "positions apart. Therefore, the", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 163, + 355, + 176 + ], + "spans": [ + { + "bbox": [ + 105, + 163, + 355, + 176 + ], + "score": 1.0, + "content": "only row operations needed for this Gaussian elimination are:", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 3.5, + "bbox_fs": [ + 105, + 125, + 505, + 176 + ] + }, + { + "type": "text", + "bbox": [ + 131, + 189, + 361, + 222 + ], + "lines": [ + { + "bbox": [ + 132, + 189, + 304, + 202 + ], + "spans": [ + { + "bbox": [ + 132, + 189, + 280, + 202 + ], + "score": 1.0, + "content": "• Scaling a row by a constant factor", + "type": "text" + }, + { + "bbox": [ + 281, + 190, + 304, + 201 + ], + "score": 0.87, + "content": "c \\neq 0", + "type": "inline_equation" + } + ], + "index": 6 + }, + { + "bbox": [ + 131, + 206, + 362, + 224 + ], + "spans": [ + { + "bbox": [ + 131, + 206, + 307, + 224 + ], + "score": 1.0, + "content": "• Addition of a row to another row exactly", + "type": "text" + }, + { + "bbox": [ + 308, + 208, + 315, + 222 + ], + "score": 0.86, + "content": "\\frac { k } { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 316, + 206, + 362, + 224 + ], + "score": 1.0, + "content": "rows apart", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 6.5, + "bbox_fs": [ + 131, + 189, + 362, + 224 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 237, + 505, + 263 + ], + "lines": [ + { + "bbox": [ + 105, + 235, + 506, + 252 + ], + "spans": [ + { + "bbox": [ + 105, + 235, + 234, + 252 + ], + "score": 1.0, + "content": "Performing these operations on", + "type": "text" + }, + { + "bbox": [ + 234, + 238, + 245, + 249 + ], + "score": 0.87, + "content": "\\mathbf { I } _ { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 245, + 235, + 455, + 252 + ], + "score": 1.0, + "content": "will only allow non-zeros on the main diagonal and", + "type": "text" + }, + { + "bbox": [ + 455, + 236, + 463, + 250 + ], + "score": 0.85, + "content": "\\frac { k } { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 463, + 235, + 506, + 252 + ], + "score": 1.0, + "content": "diagonals", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 249, + 416, + 264 + ], + "spans": [ + { + "bbox": [ + 106, + 249, + 258, + 264 + ], + "score": 1.0, + "content": "away from the main diagonal. Hence,", + "type": "text" + }, + { + "bbox": [ + 259, + 249, + 279, + 263 + ], + "score": 0.92, + "content": "\\mathbf { B } _ { k } ^ { - 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 279, + 249, + 405, + 264 + ], + "score": 1.0, + "content": "is also a butterfly factor of size", + "type": "text" + }, + { + "bbox": [ + 405, + 251, + 412, + 261 + ], + "score": 0.81, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 412, + 249, + 416, + 264 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 8.5, + "bbox_fs": [ + 105, + 235, + 506, + 264 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 270, + 506, + 315 + ], + "lines": [ + { + "bbox": [ + 104, + 267, + 506, + 284 + ], + "spans": [ + { + "bbox": [ + 104, + 267, + 144, + 284 + ], + "score": 1.0, + "content": "Next, let", + "type": "text" + }, + { + "bbox": [ + 144, + 268, + 165, + 283 + ], + "score": 0.92, + "content": "\\mathbf { B } _ { k } ^ { \\left( n \\right) }", + "type": "inline_equation" + }, + { + "bbox": [ + 165, + 267, + 354, + 284 + ], + "score": 1.0, + "content": "be an invertible butterfly factor matrix of size", + "type": "text" + }, + { + "bbox": [ + 354, + 273, + 362, + 281 + ], + "score": 0.76, + "content": "n", + "type": "inline_equation" + }, + { + "bbox": [ + 362, + 267, + 424, + 284 + ], + "score": 1.0, + "content": "and block size", + "type": "text" + }, + { + "bbox": [ + 424, + 272, + 431, + 281 + ], + "score": 0.79, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 432, + 267, + 506, + 284 + ], + "score": 1.0, + "content": ". Its inverse is the", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 281, + 507, + 295 + ], + "spans": [ + { + "bbox": [ + 105, + 281, + 507, + 295 + ], + "score": 1.0, + "content": "block diagonal matrix formed by the inverses of each of its constituent butterfly factors. From above,", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 107, + 291, + 385, + 318 + ], + "spans": [ + { + "bbox": [ + 107, + 293, + 149, + 315 + ], + "score": 0.92, + "content": "\\left( \\mathbf { B } _ { k } ^ { \\left( n \\right) } \\right) ^ { - 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 150, + 291, + 304, + 318 + ], + "score": 1.0, + "content": "is also a butterfly factor matrix of size", + "type": "text" + }, + { + "bbox": [ + 305, + 301, + 312, + 309 + ], + "score": 0.77, + "content": "n", + "type": "inline_equation" + }, + { + "bbox": [ + 313, + 291, + 372, + 318 + ], + "score": 1.0, + "content": "and block size", + "type": "text" + }, + { + "bbox": [ + 373, + 299, + 379, + 309 + ], + "score": 0.82, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 380, + 291, + 385, + 318 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 11, + "bbox_fs": [ + 104, + 267, + 507, + 318 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 318, + 209, + 331 + ], + "lines": [ + { + "bbox": [ + 105, + 318, + 210, + 331 + ], + "spans": [ + { + "bbox": [ + 105, + 318, + 175, + 331 + ], + "score": 1.0, + "content": "Finally, consider", + "type": "text" + }, + { + "bbox": [ + 175, + 319, + 207, + 329 + ], + "score": 0.89, + "content": "\\mathbf { M } \\in B", + "type": "inline_equation" + }, + { + "bbox": [ + 207, + 318, + 210, + 331 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 13, + "bbox_fs": [ + 105, + 318, + 210, + 331 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 111, + 336, + 514, + 361 + ], + "lines": [ + { + "bbox": [ + 111, + 336, + 514, + 361 + ], + "spans": [ + { + "bbox": [ + 111, + 336, + 514, + 361 + ], + "score": 0.87, + "content": "\\mathbf { { \\cal { M } } } ^ { - 1 } = \\left( \\mathbf { B } _ { n } ^ { ( n ) } \\mathbf { B } _ { \\frac { n } { 2 } } ^ { ( n ) } \\dots \\mathbf { B } _ { 2 } ^ { ( n ) } \\right) ^ { - 1 } = \\left( \\mathbf { B } _ { 2 } ^ { ( n ) } \\right) ^ { - 1 } \\left( \\mathbf { B } _ { 4 } ^ { ( n ) } \\right) ^ { - 1 } \\dots \\left( \\mathbf { B } _ { n } ^ { ( n ) } \\right) ^ { - 1 } = \\mathbf { B } _ { 2 } ^ { \\prime { } ^ { ( n ) } } \\mathbf { B } _ { 4 } ^ { \\prime { } ^ { ( n ) } } \\dots \\mathbf { B } _ { n } ^ { \\prime { } ^ { ( n ) } } \\in \\mathcal { B } _ { n } ^ { ( n ) } .", + "type": "interline_equation", + "image_path": "7e97d956a109d454e63c54b713f6d44ce6988d1399ea49191fff34622a6596bb.jpg" + } + ] + } + ], + "index": 14, + "virtual_lines": [ + { + "bbox": [ + 111, + 336, + 514, + 361 + ], + "spans": [], + "index": 14 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 392, + 505, + 438 + ], + "lines": [ + { + "bbox": [ + 106, + 393, + 505, + 405 + ], + "spans": [ + { + "bbox": [ + 106, + 393, + 505, + 405 + ], + "score": 1.0, + "content": "Finally, we include a closure result for the Kronecker product, another common matrix composition", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 404, + 505, + 417 + ], + "spans": [ + { + "bbox": [ + 105, + 404, + 505, + 417 + ], + "score": 1.0, + "content": "operation. Although Lemma H.7 is not directly used in the subsequent proofs, it allows for examples", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 416, + 505, + 426 + ], + "spans": [ + { + "bbox": [ + 106, + 416, + 505, + 426 + ], + "score": 1.0, + "content": "the results for the DFT to be lifted to higher-dimensional Fourier transforms. We also note that the", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 426, + 416, + 438 + ], + "spans": [ + { + "bbox": [ + 106, + 426, + 416, + 438 + ], + "score": 1.0, + "content": "closure bound in Lemma H.7 can be tightened in such cases (cfRemark H.4). ˙", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 16.5, + "bbox_fs": [ + 105, + 393, + 505, + 438 + ] + }, + { + "type": "text", + "bbox": [ + 105, + 441, + 504, + 455 + ], + "lines": [ + { + "bbox": [ + 104, + 438, + 502, + 456 + ], + "spans": [ + { + "bbox": [ + 104, + 438, + 190, + 456 + ], + "score": 1.0, + "content": "Lemma H.7. Let A,", + "type": "text" + }, + { + "bbox": [ + 190, + 441, + 235, + 453 + ], + "score": 0.85, + "content": "\\mathbf { B } \\in \\mathbb { F } ^ { n \\times n }", + "type": "inline_equation" + }, + { + "bbox": [ + 235, + 438, + 245, + 456 + ], + "score": 1.0, + "content": ". I", + "type": "text" + }, + { + "bbox": [ + 246, + 441, + 305, + 454 + ], + "score": 0.9, + "content": "^ { c } \\mathbf { A } \\in ( B B ^ { * } ) _ { e } ^ { w _ { 1 } }", + "type": "inline_equation" + }, + { + "bbox": [ + 306, + 438, + 324, + 456 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 324, + 441, + 381, + 454 + ], + "score": 0.93, + "content": "\\mathbf { B } \\in ( B B ^ { * } ) _ { e } ^ { w _ { 2 } }", + "type": "inline_equation" + }, + { + "bbox": [ + 381, + 438, + 401, + 456 + ], + "score": 1.0, + "content": "then", + "type": "text" + }, + { + "bbox": [ + 401, + 441, + 502, + 454 + ], + "score": 0.89, + "content": "\\mathbf { A } \\otimes \\mathbf { B } \\in ( B B ^ { * } ) _ { e } ^ { w _ { 1 } + w _ { 2 } + 6 }", + "type": "inline_equation" + } + ], + "index": 19 + } + ], + "index": 19, + "bbox_fs": [ + 104, + 438, + 502, + 456 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 469, + 175, + 480 + ], + "lines": [ + { + "bbox": [ + 106, + 469, + 176, + 482 + ], + "spans": [ + { + "bbox": [ + 106, + 469, + 176, + 482 + ], + "score": 1.0, + "content": "Proof. Note that", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 20, + "bbox_fs": [ + 106, + 469, + 176, + 482 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 196, + 479, + 413, + 494 + ], + "lines": [ + { + "bbox": [ + 196, + 479, + 413, + 494 + ], + "spans": [ + { + "bbox": [ + 196, + 479, + 413, + 494 + ], + "score": 0.88, + "content": "\\mathbf { A } \\otimes \\mathbf { B } = ( \\mathbf { A } \\otimes \\mathbf { I } ) ( \\mathbf { I } \\otimes \\mathbf { B } ) = \\mathbf { P } ^ { - 1 } ( \\mathbf { I } \\otimes \\mathbf { A } ) \\mathbf { P } ( \\mathbf { I } \\otimes \\mathbf { B } ) ,", + "type": "interline_equation", + "image_path": "0c1a34063ecb12b25aafe14a72e3df94b6b00236c59aab6a836d70d94dd574d8.jpg" + } + ] + } + ], + "index": 21, + "virtual_lines": [ + { + "bbox": [ + 196, + 479, + 413, + 494 + ], + "spans": [], + "index": 21 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 497, + 504, + 521 + ], + "lines": [ + { + "bbox": [ + 104, + 494, + 509, + 515 + ], + "spans": [ + { + "bbox": [ + 104, + 494, + 191, + 515 + ], + "score": 1.0, + "content": "for some permutation", + "type": "text" + }, + { + "bbox": [ + 192, + 498, + 201, + 508 + ], + "score": 0.3, + "content": "\\mathbf { P }", + "type": "inline_equation" + }, + { + "bbox": [ + 201, + 494, + 269, + 515 + ], + "score": 1.0, + "content": ". By Lemma H.3,", + "type": "text" + }, + { + "bbox": [ + 269, + 498, + 293, + 509 + ], + "score": 0.74, + "content": "\\mathbf { I } \\otimes \\mathbf { A }", + "type": "inline_equation" + }, + { + "bbox": [ + 293, + 494, + 309, + 515 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 309, + 498, + 332, + 509 + ], + "score": 0.88, + "content": "\\mathbf { I } \\otimes \\mathbf { B }", + "type": "inline_equation" + }, + { + "bbox": [ + 333, + 494, + 357, + 515 + ], + "score": 1.0, + "content": "are in", + "type": "text" + }, + { + "bbox": [ + 357, + 497, + 455, + 510 + ], + "score": 0.83, + "content": "( B B ^ { * } ) _ { e } ^ { w _ { 1 } + 2 } , ( B B ^ { * } ) _ { e } ^ { w _ { 2 } + 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 456, + 494, + 509, + 515 + ], + "score": 1.0, + "content": "respectively.", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 509, + 506, + 520 + ], + "spans": [ + { + "bbox": [ + 106, + 509, + 268, + 520 + ], + "score": 1.0, + "content": "The result follows from combining with", + "type": "text" + }, + { + "bbox": [ + 268, + 509, + 307, + 519 + ], + "score": 0.91, + "content": "\\mathbf { P } \\in B B ^ { * }", + "type": "inline_equation" + }, + { + "bbox": [ + 308, + 509, + 379, + 520 + ], + "score": 1.0, + "content": "and Lemma H.2.", + "type": "text" + }, + { + "bbox": [ + 494, + 509, + 506, + 520 + ], + "score": 0.999, + "content": "□", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 22.5, + "bbox_fs": [ + 104, + 494, + 509, + 520 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 537, + 325, + 551 + ], + "lines": [ + { + "bbox": [ + 104, + 536, + 326, + 552 + ], + "spans": [ + { + "bbox": [ + 104, + 536, + 236, + 552 + ], + "score": 1.0, + "content": "I SPARSE MATRICES IN", + "type": "text" + }, + { + "bbox": [ + 236, + 538, + 259, + 550 + ], + "score": 0.84, + "content": "B B ^ { * }", + "type": "inline_equation" + }, + { + "bbox": [ + 259, + 536, + 326, + 552 + ], + "score": 1.0, + "content": "HIERARCHY", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 24 + }, + { + "type": "text", + "bbox": [ + 106, + 563, + 457, + 575 + ], + "lines": [ + { + "bbox": [ + 105, + 563, + 459, + 576 + ], + "spans": [ + { + "bbox": [ + 105, + 563, + 425, + 576 + ], + "score": 1.0, + "content": "In this appendix, we prove Theorem 3. First, we consider matrices with at most", + "type": "text" + }, + { + "bbox": [ + 425, + 565, + 433, + 573 + ], + "score": 0.72, + "content": "n", + "type": "inline_equation" + }, + { + "bbox": [ + 433, + 563, + 459, + 576 + ], + "score": 1.0, + "content": "NNZ.", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 25, + "bbox_fs": [ + 105, + 563, + 459, + 576 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 578, + 424, + 591 + ], + "lines": [ + { + "bbox": [ + 106, + 578, + 423, + 592 + ], + "spans": [ + { + "bbox": [ + 106, + 578, + 206, + 592 + ], + "score": 1.0, + "content": "Lemma I.1. let S be an", + "type": "text" + }, + { + "bbox": [ + 206, + 580, + 231, + 590 + ], + "score": 0.86, + "content": "n \\times n", + "type": "inline_equation" + }, + { + "bbox": [ + 232, + 578, + 369, + 592 + ], + "score": 1.0, + "content": "matrix with at most n NNZ. Then,", + "type": "text" + }, + { + "bbox": [ + 370, + 578, + 419, + 591 + ], + "score": 0.92, + "content": "\\mathbf { S } \\in ( B B ^ { * } ) ^ { 4 }", + "type": "inline_equation" + }, + { + "bbox": [ + 420, + 578, + 423, + 592 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 26, + "bbox_fs": [ + 106, + 578, + 423, + 592 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 600, + 394, + 612 + ], + "lines": [ + { + "bbox": [ + 105, + 599, + 394, + 614 + ], + "spans": [ + { + "bbox": [ + 105, + 599, + 394, + 614 + ], + "score": 1.0, + "content": "We use this lemma and the addition closure lemma to prove Theorem 3.", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 27, + "bbox_fs": [ + 105, + 599, + 394, + 614 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 626, + 505, + 650 + ], + "lines": [ + { + "bbox": [ + 105, + 627, + 507, + 641 + ], + "spans": [ + { + "bbox": [ + 105, + 627, + 260, + 641 + ], + "score": 1.0, + "content": "Proof of Theorem 3. We note that any", + "type": "text" + }, + { + "bbox": [ + 260, + 631, + 266, + 638 + ], + "score": 0.72, + "content": "s", + "type": "inline_equation" + }, + { + "bbox": [ + 266, + 627, + 376, + 641 + ], + "score": 1.0, + "content": "sparse matrix is the sum of", + "type": "text" + }, + { + "bbox": [ + 376, + 627, + 393, + 641 + ], + "score": 0.9, + "content": "\\textstyle { \\left\\lceil { \\frac { s } { n } } \\right\\rceil }", + "type": "inline_equation" + }, + { + "bbox": [ + 393, + 627, + 472, + 641 + ], + "score": 1.0, + "content": "matrices of at most", + "type": "text" + }, + { + "bbox": [ + 473, + 630, + 480, + 638 + ], + "score": 0.71, + "content": "n", + "type": "inline_equation" + }, + { + "bbox": [ + 480, + 627, + 507, + 641 + ], + "score": 1.0, + "content": "NNZ,", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 639, + 504, + 651 + ], + "spans": [ + { + "bbox": [ + 105, + 639, + 229, + 651 + ], + "score": 1.0, + "content": "and we appeal to Lemma H.5.", + "type": "text" + }, + { + "bbox": [ + 496, + 641, + 504, + 648 + ], + "score": 0.999, + "content": "□", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 28.5, + "bbox_fs": [ + 105, + 627, + 507, + 651 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 665, + 504, + 689 + ], + "lines": [ + { + "bbox": [ + 105, + 665, + 506, + 679 + ], + "spans": [ + { + "bbox": [ + 105, + 665, + 506, + 679 + ], + "score": 1.0, + "content": "In the rest of the section we will prove Lemma I.1. We begin by defining two classes of matrices that", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 677, + 246, + 689 + ], + "spans": [ + { + "bbox": [ + 106, + 677, + 246, + 689 + ], + "score": 1.0, + "content": "will be used in our decomposition.", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 30.5, + "bbox_fs": [ + 105, + 665, + 506, + 689 + ] + }, + { + "type": "text", + "bbox": [ + 105, + 692, + 504, + 716 + ], + "lines": [ + { + "bbox": [ + 105, + 691, + 506, + 705 + ], + "spans": [ + { + "bbox": [ + 105, + 691, + 186, + 705 + ], + "score": 1.0, + "content": "Definition I.1. An", + "type": "text" + }, + { + "bbox": [ + 187, + 694, + 214, + 703 + ], + "score": 0.87, + "content": "n \\times n", + "type": "inline_equation" + }, + { + "bbox": [ + 214, + 691, + 246, + 705 + ], + "score": 1.0, + "content": "matrix", + "type": "text" + }, + { + "bbox": [ + 246, + 693, + 257, + 703 + ], + "score": 0.55, + "content": "\\mathbf { H }", + "type": "inline_equation" + }, + { + "bbox": [ + 257, + 691, + 424, + 705 + ], + "score": 1.0, + "content": "is a horizontal step matrix if for every", + "type": "text" + }, + { + "bbox": [ + 424, + 693, + 485, + 704 + ], + "score": 0.91, + "content": "0 \\leq i , i ^ { \\prime } < n", + "type": "inline_equation" + }, + { + "bbox": [ + 486, + 691, + 506, + 705 + ], + "score": 1.0, + "content": "and", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 107, + 702, + 431, + 717 + ], + "spans": [ + { + "bbox": [ + 107, + 703, + 171, + 716 + ], + "score": 0.9, + "content": "0 \\leq j \\leq j ^ { \\prime } < n ,", + "type": "inline_equation" + }, + { + "bbox": [ + 171, + 702, + 176, + 717 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 177, + 704, + 229, + 716 + ], + "score": 0.87, + "content": "i f \\mathbf { H } [ i , j ] \\neq 0", + "type": "inline_equation" + }, + { + "bbox": [ + 230, + 702, + 248, + 717 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 248, + 703, + 300, + 716 + ], + "score": 0.92, + "content": "\\mathbf { H } [ i ^ { \\prime } , j ^ { \\prime } ] \\neq 0", + "type": "inline_equation" + }, + { + "bbox": [ + 300, + 702, + 323, + 717 + ], + "score": 1.0, + "content": ", then", + "type": "text" + }, + { + "bbox": [ + 324, + 704, + 392, + 716 + ], + "score": 0.92, + "content": "j ^ { \\prime } - j \\geq ( i ^ { \\prime } - i )", + "type": "inline_equation" + }, + { + "bbox": [ + 392, + 702, + 419, + 717 + ], + "score": 1.0, + "content": "mod", + "type": "text" + }, + { + "bbox": [ + 420, + 706, + 426, + 713 + ], + "score": 0.71, + "content": "n", + "type": "inline_equation" + }, + { + "bbox": [ + 426, + 702, + 431, + 717 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 32.5, + "bbox_fs": [ + 105, + 691, + 506, + 717 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 720, + 416, + 732 + ], + "lines": [ + { + "bbox": [ + 104, + 720, + 416, + 734 + ], + "spans": [ + { + "bbox": [ + 104, + 720, + 119, + 734 + ], + "score": 1.0, + "content": "An", + "type": "text" + }, + { + "bbox": [ + 119, + 722, + 145, + 731 + ], + "score": 0.82, + "content": "n \\times n", + "type": "inline_equation" + }, + { + "bbox": [ + 145, + 720, + 174, + 734 + ], + "score": 1.0, + "content": "matrix", + "type": "text" + }, + { + "bbox": [ + 174, + 721, + 185, + 731 + ], + "score": 0.59, + "content": "\\mathbf { V }", + "type": "inline_equation" + }, + { + "bbox": [ + 185, + 720, + 291, + 734 + ], + "score": 1.0, + "content": "is a vertical step matrix if", + "type": "text" + }, + { + "bbox": [ + 292, + 721, + 306, + 731 + ], + "score": 0.7, + "content": "\\mathbf { V } ^ { * }", + "type": "inline_equation" + }, + { + "bbox": [ + 306, + 720, + 416, + 734 + ], + "score": 1.0, + "content": "is a horizontal step matrix.", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 34, + "bbox_fs": [ + 104, + 720, + 416, + 734 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 106, + 82, + 505, + 138 + ], + "lines": [ + { + "bbox": [ + 106, + 82, + 505, + 95 + ], + "spans": [ + { + "bbox": [ + 106, + 82, + 505, + 95 + ], + "score": 1.0, + "content": "With this definition, the horizontal step matrix obeys a “Lipschitz-like\" condition. Each column of a", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 94, + 506, + 106 + ], + "spans": [ + { + "bbox": [ + 106, + 94, + 473, + 106 + ], + "score": 1.0, + "content": "horizontal step matrix can have at most one non-zero entry, and given two non-zero columns", + "type": "text" + }, + { + "bbox": [ + 474, + 94, + 480, + 104 + ], + "score": 0.77, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 481, + 94, + 506, + 106 + ], + "score": 1.0, + "content": "apart,", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 106, + 105, + 505, + 117 + ], + "spans": [ + { + "bbox": [ + 106, + 105, + 350, + 117 + ], + "score": 1.0, + "content": "the non-zero entry in the right column must be between 0 and", + "type": "text" + }, + { + "bbox": [ + 351, + 105, + 358, + 114 + ], + "score": 0.8, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 358, + 105, + 505, + 117 + ], + "score": 1.0, + "content": "rows below the non-zero entry in the", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 115, + 505, + 127 + ], + "spans": [ + { + "bbox": [ + 105, + 115, + 505, + 127 + ], + "score": 1.0, + "content": "left column. Note that to show that a matrix is a horizontal step matrix, it is sufficient to argue that", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 106, + 126, + 378, + 139 + ], + "spans": [ + { + "bbox": [ + 106, + 126, + 378, + 139 + ], + "score": 1.0, + "content": "this condition holds for each pair of neighboring non-zero columns.", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 2 + }, + { + "type": "text", + "bbox": [ + 108, + 143, + 503, + 177 + ], + "lines": [ + { + "bbox": [ + 106, + 144, + 505, + 156 + ], + "spans": [ + { + "bbox": [ + 106, + 144, + 505, + 156 + ], + "score": 1.0, + "content": "Similarly, each row of a vertical step matrix can have at most one non-zero entry, and given two", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 154, + 505, + 167 + ], + "spans": [ + { + "bbox": [ + 106, + 154, + 166, + 167 + ], + "score": 1.0, + "content": "non-zero rows", + "type": "text" + }, + { + "bbox": [ + 166, + 155, + 173, + 164 + ], + "score": 0.81, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 173, + 154, + 435, + 167 + ], + "score": 1.0, + "content": "apart, the non-zero entry in the lower row must be between 0 and", + "type": "text" + }, + { + "bbox": [ + 436, + 155, + 443, + 164 + ], + "score": 0.81, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 443, + 154, + 505, + 167 + ], + "score": 1.0, + "content": "columns to the", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 166, + 283, + 178 + ], + "spans": [ + { + "bbox": [ + 106, + 166, + 283, + 178 + ], + "score": 1.0, + "content": "right of the non-zero entry in the upper row.", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 6 + }, + { + "type": "text", + "bbox": [ + 106, + 180, + 386, + 192 + ], + "lines": [ + { + "bbox": [ + 106, + 180, + 386, + 193 + ], + "spans": [ + { + "bbox": [ + 106, + 180, + 176, + 193 + ], + "score": 1.0, + "content": "Lemma I.2. Let", + "type": "text" + }, + { + "bbox": [ + 176, + 181, + 186, + 190 + ], + "score": 0.4, + "content": "\\mathbf { H }", + "type": "inline_equation" + }, + { + "bbox": [ + 186, + 180, + 211, + 193 + ], + "score": 1.0, + "content": "be an", + "type": "text" + }, + { + "bbox": [ + 212, + 181, + 237, + 191 + ], + "score": 0.88, + "content": "n \\times n", + "type": "inline_equation" + }, + { + "bbox": [ + 237, + 180, + 354, + 193 + ], + "score": 1.0, + "content": "horizontal step matrix. Then", + "type": "text" + }, + { + "bbox": [ + 354, + 181, + 383, + 191 + ], + "score": 0.87, + "content": "\\mathbf { H } \\in B", + "type": "inline_equation" + }, + { + "bbox": [ + 384, + 180, + 386, + 193 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 8 + }, + { + "type": "text", + "bbox": [ + 107, + 205, + 506, + 239 + ], + "lines": [ + { + "bbox": [ + 106, + 206, + 506, + 218 + ], + "spans": [ + { + "bbox": [ + 106, + 206, + 252, + 218 + ], + "score": 1.0, + "content": "Proof. We proceed by induction on", + "type": "text" + }, + { + "bbox": [ + 253, + 208, + 259, + 216 + ], + "score": 0.67, + "content": "n", + "type": "inline_equation" + }, + { + "bbox": [ + 260, + 206, + 322, + 218 + ], + "score": 1.0, + "content": ". The base case", + "type": "text" + }, + { + "bbox": [ + 323, + 207, + 348, + 216 + ], + "score": 0.9, + "content": "n = 2", + "type": "inline_equation" + }, + { + "bbox": [ + 349, + 206, + 506, + 218 + ], + "score": 1.0, + "content": "is trivial. 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Note that to show that a matrix is a horizontal step matrix, it is sufficient to argue that", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 106, + 126, + 378, + 139 + ], + "spans": [ + { + "bbox": [ + 106, + 126, + 378, + 139 + ], + "score": 1.0, + "content": "this condition holds for each pair of neighboring non-zero columns.", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 2, + "bbox_fs": [ + 105, + 82, + 506, + 139 + ] + }, + { + "type": "text", + "bbox": [ + 108, + 143, + 503, + 177 + ], + "lines": [ + { + "bbox": [ + 106, + 144, + 505, + 156 + ], + "spans": [ + { + "bbox": [ + 106, + 144, + 505, + 156 + ], + "score": 1.0, + "content": "Similarly, each row of a vertical step matrix can have at most one non-zero entry, and given two", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 154, + 505, + 167 + ], + "spans": [ + { + "bbox": [ + 106, + 154, + 166, + 167 + ], + "score": 1.0, + "content": "non-zero rows", + "type": "text" + }, + { + "bbox": [ + 166, + 155, + 173, + 164 + ], + "score": 0.81, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 173, + 154, + 435, + 167 + ], + "score": 1.0, + "content": "apart, the non-zero entry in the lower row must be between 0 and", + "type": "text" + }, + { + "bbox": [ + 436, + 155, + 443, + 164 + ], + "score": 0.81, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 443, + 154, + 505, + 167 + ], + "score": 1.0, + "content": "columns to the", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 166, + 283, + 178 + ], + "spans": [ + { + "bbox": [ + 106, + 166, + 283, + 178 + ], + "score": 1.0, + "content": "right of the non-zero entry in the upper row.", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 6, + "bbox_fs": [ + 106, + 144, + 505, + 178 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 180, + 386, + 192 + ], + "lines": [ + { + "bbox": [ + 106, + 180, + 386, + 193 + ], + "spans": [ + { + "bbox": [ + 106, + 180, + 176, + 193 + ], + "score": 1.0, + "content": "Lemma I.2. Let", + "type": "text" + }, + { + "bbox": [ + 176, + 181, + 186, + 190 + ], + "score": 0.4, + "content": "\\mathbf { H }", + "type": "inline_equation" + }, + { + "bbox": [ + 186, + 180, + 211, + 193 + ], + "score": 1.0, + "content": "be an", + "type": "text" + }, + { + "bbox": [ + 212, + 181, + 237, + 191 + ], + "score": 0.88, + "content": "n \\times n", + "type": "inline_equation" + }, + { + "bbox": [ + 237, + 180, + 354, + 193 + ], + "score": 1.0, + "content": "horizontal step matrix. Then", + "type": "text" + }, + { + "bbox": [ + 354, + 181, + 383, + 191 + ], + "score": 0.87, + "content": "\\mathbf { H } \\in B", + "type": "inline_equation" + }, + { + "bbox": [ + 384, + 180, + 386, + 193 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 8, + "bbox_fs": [ + 106, + 180, + 386, + 193 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 205, + 506, + 239 + ], + "lines": [ + { + "bbox": [ + 106, + 206, + 506, + 218 + ], + "spans": [ + { + "bbox": [ + 106, + 206, + 252, + 218 + ], + "score": 1.0, + "content": "Proof. We proceed by induction on", + "type": "text" + }, + { + "bbox": [ + 253, + 208, + 259, + 216 + ], + "score": 0.67, + "content": "n", + "type": "inline_equation" + }, + { + "bbox": [ + 260, + 206, + 322, + 218 + ], + "score": 1.0, + "content": ". 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This contradicts", + "type": "text" + }, + { + "bbox": [ + 469, + 420, + 480, + 430 + ], + "score": 0.65, + "content": "\\mathbf { H }", + "type": "inline_equation" + }, + { + "bbox": [ + 480, + 418, + 506, + 433 + ], + "score": 1.0, + "content": "being", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 104, + 429, + 505, + 444 + ], + "spans": [ + { + "bbox": [ + 104, + 429, + 234, + 444 + ], + "score": 1.0, + "content": "a horizontal step matrix. Hence,", + "type": "text" + }, + { + "bbox": [ + 235, + 431, + 249, + 442 + ], + "score": 0.89, + "content": "\\mathbf { H } _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 249, + 429, + 414, + 444 + ], + "score": 1.0, + "content": "must be a horizontal step matrix, as must", + "type": "text" + }, + { + "bbox": [ + 414, + 431, + 428, + 442 + ], + "score": 0.89, + "content": "\\mathbf { H } _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 429, + 429, + 505, + 444 + ], + "score": 1.0, + "content": "from an analogous", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 442, + 149, + 454 + ], + "spans": [ + { + "bbox": [ + 105, + 442, + 149, + 454 + ], + "score": 1.0, + "content": "argument.", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 22.5, + "bbox_fs": [ + 104, + 380, + 507, + 454 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 457, + 255, + 470 + ], + "lines": [ + { + "bbox": [ + 105, + 455, + 256, + 473 + ], + "spans": [ + { + "bbox": [ + 105, + 455, + 171, + 473 + ], + "score": 1.0, + "content": "Next, we define", + "type": "text" + }, + { + "bbox": [ + 172, + 458, + 239, + 470 + ], + "score": 0.9, + "content": "\\mathbf { D } _ { 1 } , \\mathbf { D } _ { 2 } , \\mathbf { D } _ { 3 } , \\mathbf { D } _ { 4 }", + "type": "inline_equation" + }, + { + "bbox": [ + 239, + 455, + 256, + 473 + ], + "score": 1.0, + "content": "by:", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 26, + "bbox_fs": [ + 105, + 455, + 256, + 473 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 161, + 476, + 449, + 541 + ], + "lines": [ + { + "bbox": [ + 161, + 476, + 449, + 541 + ], + "spans": [ + { + "bbox": [ + 161, + 476, + 449, + 541 + ], + "score": 0.91, + "content": "\\begin{array} { r l r l } & { \\mathbf { D } _ { 1 } [ k , k ] = \\left\\{ \\begin{array} { l l } { 1 } & { \\mathbf { H } _ { 2 1 } [ k , : ] = \\mathbf { 0 } } \\\\ { 0 } & { \\mathrm { o t h e r w i s e } } \\end{array} \\right. } & & { \\mathbf { D } _ { 2 } [ k , k ] = \\left\\{ \\begin{array} { l l } { 1 } & { \\mathbf { H } _ { 2 2 } [ k , : ] = \\mathbf { 0 } } \\\\ { 0 } & { \\mathrm { o t h e r w i s e } } \\end{array} \\right. } \\\\ & { \\mathbf { D } _ { 3 } [ k , k ] = \\left\\{ \\begin{array} { l l } { 1 } & { \\mathbf { H } _ { 1 1 } [ k , : ] = \\mathbf { 0 } } \\\\ { 0 } & { \\mathrm { o t h e r w i s e } . } \\end{array} \\right. } & & { \\mathbf { D } _ { 4 } [ k , k ] = \\left\\{ \\begin{array} { l l } { 1 } & { \\mathbf { H } _ { 1 2 } [ k , : ] = \\mathbf { 0 } } \\\\ { 0 } & { \\mathrm { o t h e r w i s e } . } \\end{array} \\right. } \\end{array}", + "type": "interline_equation", + "image_path": "2c99e29e47b32aab9acdbf1b8bdf0f32c95db2c94933646bad236e9aec4c80cd.jpg" + } + ] + } + ], + "index": 28, + "virtual_lines": [ + { + "bbox": [ + 161, + 476, + 449, + 497.6666666666667 + ], + "spans": [], + "index": 27 + }, + { + "bbox": [ + 161, + 497.6666666666667, + 449, + 519.3333333333334 + ], + "spans": [], + "index": 28 + }, + { + "bbox": [ + 161, + 519.3333333333334, + 449, + 541.0 + ], + "spans": [], + "index": 29 + } + ] + }, + { + "type": "text", + "bbox": [ + 105, + 549, + 505, + 572 + ], + "lines": [ + { + "bbox": [ + 106, + 549, + 505, + 562 + ], + "spans": [ + { + "bbox": [ + 106, + 549, + 505, + 562 + ], + "score": 1.0, + "content": "To finish the proof, we argue the correctness of the decomposition by equating arbitrary entries of", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 560, + 402, + 573 + ], + "spans": [ + { + "bbox": [ + 105, + 560, + 402, + 573 + ], + "score": 1.0, + "content": "each of the 4 corner submatrices. We begin with the upper left submatrix.", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 30.5, + "bbox_fs": [ + 105, + 549, + 505, + 573 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 119, + 577, + 499, + 645 + ], + "lines": [ + { + "bbox": [ + 119, + 577, + 499, + 645 + ], + "spans": [ + { + "bbox": [ + 119, + 577, + 499, + 645 + ], + "score": 0.89, + "content": "\\begin{array} { r l r } { \\mathbf { D } _ { 1 } \\mathbf { H } _ { 1 } \\big [ i , j \\big ] = \\displaystyle \\sum _ { k = 0 } ^ { \\frac { n } { 2 } } \\mathbf { D } _ { 1 } [ i , k ] \\cdot \\mathbf { H } _ { 1 } \\big [ k , j \\big ] \\quad } & { \\mathrm { b y ~ d e f i n i t i o n ~ o f ~ m a t r i x ~ m u l t i p l i c a t i o n } } \\\\ { = \\mathbf { D } _ { 1 } \\big [ i , i \\big ] \\cdot \\mathbf { H } _ { 1 } \\big [ i , j \\big ] \\quad } & { \\mathbf { D } _ { 1 } \\mathrm { ~ i s ~ a d i a g o n a l ~ m a t r i x } } \\\\ { = \\mathbb { 1 } _ { ( \\mathbf { H } _ { 2 1 } [ i , : ] = 0 ) } \\cdot \\left( \\mathbf { H } _ { 1 1 } [ i , j ] + \\mathbf { H } _ { 2 1 } [ i , j ] \\right) \\quad } & { \\mathrm { b y ~ d e f i n i t i o n ~ o f ~ \\mathbf { D } _ 1 ~ a n d ~ \\mathbf { H } _ 1 ~ } } \\end{array}", + "type": "interline_equation", + "image_path": "009086f621b094da4ef14bfe8e5b883b0e60b84b6a2ca6fadf45b409a63056ab.jpg" + } + ] + } + ], + "index": 33, + "virtual_lines": [ + { + "bbox": [ + 119, + 577, + 499, + 599.6666666666666 + ], + "spans": [], + "index": 32 + }, + { + "bbox": [ + 119, + 599.6666666666666, + 499, + 622.3333333333333 + ], + "spans": [], + "index": 33 + }, + { + "bbox": [ + 119, + 622.3333333333333, + 499, + 644.9999999999999 + ], + "spans": [], + "index": 34 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 648, + 224, + 659 + ], + "lines": [ + { + "bbox": [ + 106, + 648, + 225, + 660 + ], + "spans": [ + { + "bbox": [ + 106, + 648, + 225, + 660 + ], + "score": 1.0, + "content": "Here, we consider two cases:", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 35, + "bbox_fs": [ + 106, + 648, + 225, + 660 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 663, + 200, + 677 + ], + "lines": [ + { + "bbox": [ + 106, + 663, + 200, + 678 + ], + "spans": [ + { + "bbox": [ + 106, + 663, + 132, + 678 + ], + "score": 1.0, + "content": "Case", + "type": "text" + }, + { + "bbox": [ + 133, + 663, + 200, + 677 + ], + "score": 0.76, + "content": "1 : \\mathbf { H } _ { 2 1 } [ i , j ] \\neq 0", + "type": "inline_equation", + "image_path": "bd81ce042b2862671f20296669b41480d8a0173aefcdd1e5c9c35039ae76f36f.jpg" + } + ], + "index": 36 + } + ], + "index": 36, + "bbox_fs": [ + 106, + 663, + 200, + 678 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 682, + 505, + 715 + ], + "lines": [ + { + "bbox": [ + 105, + 681, + 505, + 694 + ], + "spans": [ + { + "bbox": [ + 105, + 681, + 131, + 694 + ], + "score": 1.0, + "content": "Since", + "type": "text" + }, + { + "bbox": [ + 131, + 682, + 141, + 692 + ], + "score": 0.57, + "content": "\\mathbf { H }", + "type": "inline_equation" + }, + { + "bbox": [ + 142, + 681, + 505, + 694 + ], + "score": 1.0, + "content": "is a horizontal step matrix (and hence may have at most one non-zero entry per column), it", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 106, + 693, + 505, + 705 + ], + "spans": [ + { + "bbox": [ + 106, + 693, + 157, + 704 + ], + "score": 1.0, + "content": "follows that", + "type": "text" + }, + { + "bbox": [ + 158, + 693, + 212, + 705 + ], + "score": 0.93, + "content": "\\mathbf { H } _ { 1 1 } [ i , j ] = 0", + "type": "inline_equation" + }, + { + "bbox": [ + 213, + 693, + 429, + 704 + ], + "score": 1.0, + "content": ". 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Otherwise, for sake of contradiction, suppose", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 92, + 507, + 107 + ], + "spans": [ + { + "bbox": [ + 105, + 92, + 124, + 107 + ], + "score": 1.0, + "content": "that", + "type": "text" + }, + { + "bbox": [ + 124, + 94, + 178, + 105 + ], + "score": 0.91, + "content": "\\mathbf { H } _ { 2 1 } [ i , : ] \\neq \\mathbf { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 178, + 92, + 268, + 107 + ], + "score": 1.0, + "content": ". Then, two of the first", + "type": "text" + }, + { + "bbox": [ + 268, + 95, + 276, + 106 + ], + "score": 0.86, + "content": "\\frac { n } { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 277, + 92, + 324, + 107 + ], + "score": 1.0, + "content": "columns of", + "type": "text" + }, + { + "bbox": [ + 324, + 94, + 335, + 104 + ], + "score": 0.47, + "content": "\\mathbf { H }", + "type": "inline_equation" + }, + { + "bbox": [ + 335, + 92, + 450, + 107 + ], + "score": 1.0, + "content": "would have non-zero entries", + "type": "text" + }, + { + "bbox": [ + 451, + 94, + 459, + 105 + ], + "score": 0.8, + "content": "\\frac { n } { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 459, + 92, + 507, + 107 + ], + "score": 1.0, + "content": "rows apart,", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 104, + 506, + 117 + ], + "spans": [ + { + "bbox": [ + 105, + 104, + 161, + 117 + ], + "score": 1.0, + "content": "contradicting", + "type": "text" + }, + { + "bbox": [ + 161, + 106, + 172, + 115 + ], + "score": 0.38, + "content": "\\mathbf { H }", + "type": "inline_equation" + }, + { + "bbox": [ + 172, + 104, + 327, + 117 + ], + "score": 1.0, + "content": "being a horizontal step matrix. Hence,", + "type": "text" + }, + { + "bbox": [ + 327, + 104, + 381, + 116 + ], + "score": 0.92, + "content": "\\mathbf { H } _ { 2 1 } [ i , : ] = \\mathbf { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 381, + 104, + 396, + 117 + ], + "score": 1.0, + "content": ", so", + "type": "text" + }, + { + "bbox": [ + 396, + 104, + 491, + 117 + ], + "score": 0.91, + "content": "\\mathbf D _ { 1 } \\mathbf H _ { 1 } [ i , j ] = \\mathbf { \\bar { H } } _ { 1 1 } [ i , j ]", + "type": "inline_equation" + }, + { + "bbox": [ + 491, + 104, + 506, + 117 + ], + "score": 1.0, + "content": ", as", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 115, + 141, + 127 + ], + "spans": [ + { + "bbox": [ + 105, + 115, + 141, + 127 + ], + "score": 1.0, + "content": "desired.", + "type": "text" + } + ], + "index": 3 + } + ], + "index": 1.5 + }, + { + "type": "text", + "bbox": [ + 106, + 131, + 506, + 165 + ], + "lines": [ + { + "bbox": [ + 104, + 131, + 506, + 145 + ], + "spans": [ + { + "bbox": [ + 104, + 131, + 155, + 145 + ], + "score": 1.0, + "content": "In all cases,", + "type": "text" + }, + { + "bbox": [ + 155, + 132, + 250, + 144 + ], + "score": 0.92, + "content": "\\mathbf { D } _ { 1 } \\mathbf { H } _ { 1 } [ i , j ] = \\mathbf { H } _ { 1 1 } [ i , j ]", + "type": "inline_equation" + }, + { + "bbox": [ + 250, + 131, + 506, + 145 + ], + "score": 1.0, + "content": ", so our decomposition correctly recovers the upper left corner of", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 106, + 142, + 506, + 157 + ], + "spans": [ + { + "bbox": [ + 106, + 144, + 117, + 154 + ], + "score": 0.34, + "content": "\\mathbf { H }", + "type": "inline_equation" + }, + { + "bbox": [ + 117, + 142, + 506, + 157 + ], + "score": 1.0, + "content": ". Analogous arguments show that the other three corners are also correctly recovered. Hence, our", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 154, + 505, + 166 + ], + "spans": [ + { + "bbox": [ + 106, + 154, + 281, + 166 + ], + "score": 1.0, + "content": "decomposition is correct, and by induction,", + "type": "text" + }, + { + "bbox": [ + 281, + 154, + 311, + 164 + ], + "score": 0.9, + "content": "\\mathbf { H } \\in B", + "type": "inline_equation" + }, + { + "bbox": [ + 311, + 154, + 315, + 166 + ], + "score": 1.0, + "content": ".", + "type": "text" + }, + { + "bbox": [ + 494, + 155, + 505, + 165 + ], + "score": 0.999, + "content": "□", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 5 + }, + { + "type": "text", + "bbox": [ + 106, + 172, + 357, + 184 + ], + "lines": [ + { + "bbox": [ + 106, + 171, + 358, + 185 + ], + "spans": [ + { + "bbox": [ + 106, + 171, + 184, + 185 + ], + "score": 1.0, + "content": "Corollary I.3. Let", + "type": "text" + }, + { + "bbox": [ + 184, + 172, + 194, + 182 + ], + "score": 0.36, + "content": "\\mathbf { V }", + "type": "inline_equation" + }, + { + "bbox": [ + 195, + 171, + 320, + 185 + ], + "score": 1.0, + "content": "be a vertical step matrix. 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Given S, we decompose it as", + "type": "text" + }, + { + "bbox": [ + 304, + 560, + 380, + 571 + ], + "score": 0.92, + "content": "\\mathbf { S } = \\mathbf { P } _ { 1 } \\mathbf { H } \\mathbf { P } _ { 2 } \\mathbf { V } \\mathbf { P } _ { 3 }", + "type": "inline_equation" + }, + { + "bbox": [ + 380, + 559, + 428, + 573 + ], + "score": 1.0, + "content": ", where each", + "type": "text" + }, + { + "bbox": [ + 429, + 560, + 440, + 571 + ], + "score": 0.87, + "content": "P _ { \\ell }", + "type": "inline_equation" + }, + { + "bbox": [ + 440, + 559, + 505, + 573 + ], + "score": 1.0, + "content": "is a permutation", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 570, + 505, + 584 + ], + "spans": [ + { + "bbox": [ + 105, + 570, + 138, + 584 + ], + "score": 1.0, + "content": "matrix,", + "type": "text" + }, + { + "bbox": [ + 138, + 571, + 149, + 581 + ], + "score": 0.48, + "content": "\\mathbf { H }", + "type": "inline_equation" + }, + { + "bbox": [ + 149, + 570, + 279, + 584 + ], + "score": 1.0, + "content": "is a horizontal step matrix, and", + "type": "text" + }, + { + "bbox": [ + 279, + 571, + 289, + 581 + ], + "score": 0.62, + "content": "\\mathbf { V }", + "type": "inline_equation" + }, + { + "bbox": [ + 289, + 570, + 505, + 584 + ], + "score": 1.0, + "content": "is a vertical step matrix. For an example of this, see", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 581, + 147, + 595 + ], + "spans": [ + { + "bbox": [ + 105, + 581, + 147, + 595 + ], + "score": 1.0, + "content": "Figure 9.", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 31 + }, + { + "type": "text", + "bbox": [ + 105, + 598, + 504, + 622 + ], + "lines": [ + { + "bbox": [ + 106, + 598, + 505, + 611 + ], + "spans": [ + { + "bbox": [ + 106, + 598, + 208, + 611 + ], + "score": 1.0, + "content": "We first decompose S as", + "type": "text" + }, + { + "bbox": [ + 209, + 599, + 263, + 610 + ], + "score": 0.92, + "content": "\\mathbf { S } = \\mathbf { P } _ { 1 } \\mathbf { S } ^ { \\prime } \\mathbf { P } _ { 3 }", + "type": "inline_equation" + }, + { + "bbox": [ + 263, + 598, + 294, + 611 + ], + "score": 1.0, + "content": ", where", + "type": "text" + }, + { + "bbox": [ + 295, + 599, + 308, + 610 + ], + "score": 0.89, + "content": "{ \\bf P } _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 308, + 598, + 505, + 611 + ], + "score": 1.0, + "content": "is the permutation that moves all 0 rows of S to", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 609, + 428, + 622 + ], + "spans": [ + { + "bbox": [ + 106, + 609, + 168, + 622 + ], + "score": 1.0, + "content": "the bottom and", + "type": "text" + }, + { + "bbox": [ + 169, + 610, + 182, + 621 + ], + "score": 0.91, + "content": "\\mathbf { P } _ { 3 }", + "type": "inline_equation" + }, + { + "bbox": [ + 183, + 609, + 428, + 622 + ], + "score": 1.0, + "content": "is the permutation that moves all 0 columns of S to the right.", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 33.5 + }, + { + "type": "text", + "bbox": [ + 107, + 626, + 504, + 660 + ], + "lines": [ + { + "bbox": [ + 105, + 625, + 506, + 640 + ], + "spans": [ + { + "bbox": [ + 105, + 625, + 227, + 640 + ], + "score": 1.0, + "content": "Next, we further decompose", + "type": "text" + }, + { + "bbox": [ + 228, + 627, + 238, + 637 + ], + "score": 0.85, + "content": "\\mathbf { S } ^ { \\prime }", + "type": "inline_equation" + }, + { + "bbox": [ + 239, + 625, + 259, + 640 + ], + "score": 1.0, + "content": "into", + "type": "text" + }, + { + "bbox": [ + 259, + 627, + 307, + 637 + ], + "score": 0.9, + "content": "\\mathbf { S } ^ { \\prime } = \\mathbf { H } \\mathbf { V } ^ { \\prime }", + "type": "inline_equation" + }, + { + "bbox": [ + 308, + 625, + 385, + 640 + ], + "score": 1.0, + "content": "as follows. 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Analogous arguments show that the other three corners are also correctly recovered. Hence, our", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 154, + 505, + 166 + ], + "spans": [ + { + "bbox": [ + 106, + 154, + 281, + 166 + ], + "score": 1.0, + "content": "decomposition is correct, and by induction,", + "type": "text" + }, + { + "bbox": [ + 281, + 154, + 311, + 164 + ], + "score": 0.9, + "content": "\\mathbf { H } \\in B", + "type": "inline_equation" + }, + { + "bbox": [ + 311, + 154, + 315, + 166 + ], + "score": 1.0, + "content": ".", + "type": "text" + }, + { + "bbox": [ + 494, + 155, + 505, + 165 + ], + "score": 0.999, + "content": "□", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 5, + "bbox_fs": [ + 104, + 131, + 506, + 166 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 172, + 357, + 184 + ], + "lines": [ + { + "bbox": [ + 106, + 171, + 358, + 185 + ], + "spans": [ + { + "bbox": [ + 106, + 171, + 184, + 185 + ], + "score": 1.0, + "content": "Corollary I.3. Let", + "type": "text" + }, + { + "bbox": [ + 184, + 172, + 194, + 182 + ], + "score": 0.36, + "content": "\\mathbf { V }", + "type": "inline_equation" + }, + { + "bbox": [ + 195, + 171, + 320, + 185 + ], + "score": 1.0, + "content": "be a vertical step matrix. Then", + "type": "text" + }, + { + "bbox": [ + 321, + 172, + 353, + 182 + ], + "score": 0.89, + "content": "\\mathbf { V } \\in B ^ { * }", + "type": "inline_equation" + }, + { + "bbox": [ + 354, + 171, + 358, + 185 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 7, + "bbox_fs": [ + 106, + 171, + 358, + 185 + ] + }, + { + "type": "image", + "bbox": [ + 171, + 200, + 439, + 489 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 171, + 200, + 439, + 489 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 171, + 200, + 439, + 489 + ], + "spans": [ + { + "bbox": [ + 171, + 200, + 439, + 489 + ], + "score": 0.967, + "type": "image", + "image_path": "d4daf5b9090c82cf9774fe7121e9c63dac335d55f5330daff1bf763ab1367e88.jpg" + } + ] + } + ], + "index": 17.5, + "virtual_lines": [ + { + "bbox": [ + 171, + 200, + 439, + 214.45 + ], + "spans": [], + "index": 8 + }, + { + "bbox": [ + 171, + 214.45, + 439, + 228.89999999999998 + ], + "spans": [], + "index": 9 + }, + { + "bbox": [ + 171, + 228.89999999999998, + 439, + 243.34999999999997 + ], + "spans": [], + "index": 10 + }, + { + "bbox": [ + 171, + 243.34999999999997, + 439, + 257.79999999999995 + ], + "spans": [], + "index": 11 + }, + { + "bbox": [ + 171, + 257.79999999999995, + 439, + 272.24999999999994 + ], + "spans": [], + "index": 12 + }, + { + "bbox": [ + 171, + 272.24999999999994, + 439, + 286.69999999999993 + ], + "spans": [], + "index": 13 + }, + { + "bbox": [ + 171, + 286.69999999999993, + 439, + 301.1499999999999 + ], + "spans": [], + "index": 14 + }, + { + "bbox": [ + 171, + 301.1499999999999, + 439, + 315.5999999999999 + ], + "spans": [], + "index": 15 + }, + { + "bbox": [ + 171, + 315.5999999999999, + 439, + 330.0499999999999 + ], + "spans": [], + "index": 16 + }, + { + "bbox": [ + 171, + 330.0499999999999, + 439, + 344.4999999999999 + ], + "spans": [], + "index": 17 + }, + { + "bbox": [ + 171, + 344.4999999999999, + 439, + 358.9499999999999 + ], + "spans": [], + "index": 18 + }, + { + "bbox": [ + 171, + 358.9499999999999, + 439, + 373.39999999999986 + ], + "spans": [], + "index": 19 + }, + { + "bbox": [ + 171, + 373.39999999999986, + 439, + 387.84999999999985 + ], + "spans": [], + "index": 20 + }, + { + "bbox": [ + 171, + 387.84999999999985, + 439, + 402.29999999999984 + ], + "spans": [], + "index": 21 + }, + { + "bbox": [ + 171, + 402.29999999999984, + 439, + 416.74999999999983 + ], + "spans": [], + "index": 22 + }, + { + "bbox": [ + 171, + 416.74999999999983, + 439, + 431.1999999999998 + ], + "spans": [], + "index": 23 + }, + { + "bbox": [ + 171, + 431.1999999999998, + 439, + 445.6499999999998 + ], + "spans": [], + "index": 24 + }, + { + "bbox": [ + 171, + 445.6499999999998, + 439, + 460.0999999999998 + ], + "spans": [], + "index": 25 + }, + { + "bbox": [ + 171, + 460.0999999999998, + 439, + 474.5499999999998 + ], + "spans": [], + "index": 26 + }, + { + "bbox": [ + 171, + 474.5499999999998, + 439, + 488.9999999999998 + ], + "spans": [], + "index": 27 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 166, + 504, + 444, + 517 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 166, + 503, + 444, + 519 + ], + "spans": [ + { + "bbox": [ + 166, + 503, + 280, + 519 + ], + "score": 1.0, + "content": "Figure 9: Decomposition of", + "type": "text" + }, + { + "bbox": [ + 281, + 505, + 304, + 515 + ], + "score": 0.89, + "content": "4 \\times 4", + "type": "inline_equation" + }, + { + "bbox": [ + 304, + 503, + 388, + 519 + ], + "score": 1.0, + "content": "sparse matrix S into", + "type": "text" + }, + { + "bbox": [ + 388, + 505, + 444, + 517 + ], + "score": 0.89, + "content": "\\mathbf { P } _ { 1 } \\mathbf { H } \\mathbf { P } _ { 2 } \\mathbf { V } \\mathbf { P } _ { 3 }", + "type": "inline_equation" + } + ], + "index": 28 + } + ], + "index": 28 + } + ], + "index": 22.75 + }, + { + "type": "text", + "bbox": [ + 107, + 536, + 297, + 548 + ], + "lines": [ + { + "bbox": [ + 106, + 535, + 298, + 549 + ], + "spans": [ + { + "bbox": [ + 106, + 535, + 298, + 549 + ], + "score": 1.0, + "content": "Now, we use step matrices to prove Lemma I.1.", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 29, + "bbox_fs": [ + 106, + 535, + 298, + 549 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 559, + 505, + 593 + ], + "lines": [ + { + "bbox": [ + 105, + 559, + 505, + 573 + ], + "spans": [ + { + "bbox": [ + 105, + 559, + 304, + 573 + ], + "score": 1.0, + "content": "Proof of Lemma I.1. Given S, we decompose it as", + "type": "text" + }, + { + "bbox": [ + 304, + 560, + 380, + 571 + ], + "score": 0.92, + "content": "\\mathbf { S } = \\mathbf { P } _ { 1 } \\mathbf { H } \\mathbf { P } _ { 2 } \\mathbf { V } \\mathbf { P } _ { 3 }", + "type": "inline_equation" + }, + { + "bbox": [ + 380, + 559, + 428, + 573 + ], + "score": 1.0, + "content": ", where each", + "type": "text" + }, + { + "bbox": [ + 429, + 560, + 440, + 571 + ], + "score": 0.87, + "content": "P _ { \\ell }", + "type": "inline_equation" + }, + { + "bbox": [ + 440, + 559, + 505, + 573 + ], + "score": 1.0, + "content": "is a permutation", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 570, + 505, + 584 + ], + "spans": [ + { + "bbox": [ + 105, + 570, + 138, + 584 + ], + "score": 1.0, + "content": "matrix,", + "type": "text" + }, + { + "bbox": [ + 138, + 571, + 149, + 581 + ], + "score": 0.48, + "content": "\\mathbf { H }", + "type": "inline_equation" + }, + { + "bbox": [ + 149, + 570, + 279, + 584 + ], + "score": 1.0, + "content": "is a horizontal step matrix, and", + "type": "text" + }, + { + "bbox": [ + 279, + 571, + 289, + 581 + ], + "score": 0.62, + "content": "\\mathbf { V }", + "type": "inline_equation" + }, + { + "bbox": [ + 289, + 570, + 505, + 584 + ], + "score": 1.0, + "content": "is a vertical step matrix. For an example of this, see", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 581, + 147, + 595 + ], + "spans": [ + { + "bbox": [ + 105, + 581, + 147, + 595 + ], + "score": 1.0, + "content": "Figure 9.", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 31, + "bbox_fs": [ + 105, + 559, + 505, + 595 + ] + }, + { + "type": "text", + "bbox": [ + 105, + 598, + 504, + 622 + ], + "lines": [ + { + "bbox": [ + 106, + 598, + 505, + 611 + ], + "spans": [ + { + "bbox": [ + 106, + 598, + 208, + 611 + ], + "score": 1.0, + "content": "We first decompose S as", + "type": "text" + }, + { + "bbox": [ + 209, + 599, + 263, + 610 + ], + "score": 0.92, + "content": "\\mathbf { S } = \\mathbf { P } _ { 1 } \\mathbf { S } ^ { \\prime } \\mathbf { P } _ { 3 }", + "type": "inline_equation" + }, + { + "bbox": [ + 263, + 598, + 294, + 611 + ], + "score": 1.0, + "content": ", where", + "type": "text" + }, + { + "bbox": [ + 295, + 599, + 308, + 610 + ], + "score": 0.89, + "content": "{ \\bf P } _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 308, + 598, + 505, + 611 + ], + "score": 1.0, + "content": "is the permutation that moves all 0 rows of S to", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 609, + 428, + 622 + ], + "spans": [ + { + "bbox": [ + 106, + 609, + 168, + 622 + ], + "score": 1.0, + "content": "the bottom and", + "type": "text" + }, + { + "bbox": [ + 169, + 610, + 182, + 621 + ], + "score": 0.91, + "content": "\\mathbf { P } _ { 3 }", + "type": "inline_equation" + }, + { + "bbox": [ + 183, + 609, + 428, + 622 + ], + "score": 1.0, + "content": "is the permutation that moves all 0 columns of S to the right.", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 33.5, + "bbox_fs": [ + 106, + 598, + 505, + 622 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 626, + 504, + 660 + ], + "lines": [ + { + "bbox": [ + 105, + 625, + 506, + 640 + ], + "spans": [ + { + "bbox": [ + 105, + 625, + 227, + 640 + ], + "score": 1.0, + "content": "Next, we further decompose", + "type": "text" + }, + { + "bbox": [ + 228, + 627, + 238, + 637 + ], + "score": 0.85, + "content": "\\mathbf { S } ^ { \\prime }", + "type": "inline_equation" + }, + { + "bbox": [ + 239, + 625, + 259, + 640 + ], + "score": 1.0, + "content": "into", + "type": "text" + }, + { + "bbox": [ + 259, + 627, + 307, + 637 + ], + "score": 0.9, + "content": "\\mathbf { S } ^ { \\prime } = \\mathbf { H } \\mathbf { V } ^ { \\prime }", + "type": "inline_equation" + }, + { + "bbox": [ + 308, + 625, + 385, + 640 + ], + "score": 1.0, + "content": "as follows. 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"inline_equation" + }, + { + "bbox": [ + 204, + 79, + 334, + 97 + ], + "score": 1.0, + "content": ", we consider an arbitrary entry:", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "interline_equation", + "bbox": [ + 141, + 100, + 476, + 201 + ], + "lines": [ + { + "bbox": [ + 141, + 100, + 476, + 201 + ], + "spans": [ + { + "bbox": [ + 141, + 100, + 476, + 201 + ], + "score": 0.89, + "content": "\\begin{array} { r l } { \\mathbf { H } \\mathbf { V } ^ { \\prime } [ i , j ] = \\displaystyle \\sum _ { k = 0 } ^ { n } \\mathbf { H } [ i , k ] \\cdot \\mathbf { V } ^ { \\prime } [ k , j ] \\qquad } & { \\mathrm { ~ b y ~ d e f i n i t i o n ~ o f ~ m a t r i x ~ m u l t i p l i c a t i o n } } \\\\ { = \\displaystyle \\sum _ { k = 0 } ^ { s } \\mathbf { H } [ i , k ] \\cdot \\mathbf { V } ^ { \\prime } [ k , j ] \\qquad } & { \\mathrm { ~ H ~ i s ~ 0 ~ i n ~ a l l ~ b u t ~ f i r s t ~ s ~ c o l u m n s } } \\\\ { = \\displaystyle \\sum _ { k = 0 } ^ { s } c _ { k } \\cdot \\mathbb { I } _ { i = i _ { k } } \\cdot \\mathbb { I } _ { j = j _ { k } } \\qquad } & { \\mathrm { ~ b y ~ d e f i n i t i o n ~ o f ~ } \\mathbf { H } \\mathrm { ~ a n d ~ } \\mathbf { V } ^ { \\prime } } \\end{array}", + "type": "interline_equation", + "image_path": "9a0ee6522ceafb8907365528743cc60af391240dfdb2856dd136af9d3fa79b76.jpg" + } + ] + } + ], + "index": 2, + "virtual_lines": [ + { + "bbox": [ + 141, + 100, + 476, + 133.66666666666666 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 141, + 133.66666666666666, + 476, + 167.33333333333331 + ], + "spans": [], + "index": 2 + }, + { + "bbox": [ + 141, + 167.33333333333331, + 476, + 200.99999999999997 + ], + "spans": [], + "index": 3 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 206, + 505, + 240 + ], + "lines": [ + { + "bbox": [ + 105, + 205, + 505, + 219 + ], + "spans": [ + { + "bbox": [ + 105, + 205, + 185, + 219 + ], + "score": 1.0, + "content": "Here, we note that", + "type": "text" + }, 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Hence,", + "type": "text" + }, + { + "bbox": [ + 169, + 218, + 231, + 229 + ], + "score": 0.92, + "content": "\\mathbf { H } \\mathbf { V } ^ { \\prime } [ i , j ] = c _ { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 231, + 217, + 261, + 230 + ], + "score": 1.0, + "content": "only if", + "type": "text" + }, + { + "bbox": [ + 261, + 218, + 325, + 230 + ], + "score": 0.93, + "content": "( i , j ) = ( i _ { k } , j _ { k } )", + "type": "inline_equation" + }, + { + "bbox": [ + 325, + 217, + 363, + 230 + ], + "score": 1.0, + "content": "for some", + "type": "text" + }, + { + "bbox": [ + 363, + 218, + 370, + 227 + ], + "score": 0.81, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 370, + 217, + 506, + 230 + ], + "score": 1.0, + "content": ", which is exactly the definition of", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 228, + 200, + 240 + ], + "spans": [ + { + "bbox": [ + 106, + 228, + 117, + 239 + ], + "score": 0.81, + "content": "\\mathbf { S } ^ { \\prime }", + "type": "inline_equation" + }, + { + "bbox": [ + 117, + 228, + 151, + 240 + ], + "score": 1.0, + "content": ". Hence,", + "type": "text" + }, + { + "bbox": [ + 152, + 228, + 195, + 239 + ], + "score": 0.92, + "content": "\\mathbf { S } ^ { \\prime } = \\mathbf { H } \\mathbf { V } ^ { \\prime }", + "type": "inline_equation" + }, + { + "bbox": [ + 196, + 228, + 200, + 240 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 5 + }, + { + "type": "text", + "bbox": [ + 106, + 245, + 505, + 312 + ], + "lines": [ + { + "bbox": [ + 106, + 246, + 504, + 257 + ], + "spans": [ + { + "bbox": [ + 106, + 246, + 165, + 257 + ], + "score": 1.0, + "content": "We argue that", + "type": "text" + }, + { + "bbox": [ + 166, + 246, + 176, + 256 + ], + "score": 0.49, + "content": "\\mathbf { H }", + "type": "inline_equation" + }, + { + "bbox": [ + 177, + 246, + 477, + 257 + ], + "score": 1.0, + "content": "is a horizontal step matrix through a series of assertions. First, note that", + "type": "text" + }, + { + "bbox": [ + 477, + 246, + 488, + 256 + ], + "score": 0.43, + "content": "\\mathbf { H }", + "type": "inline_equation" + }, + { + "bbox": [ + 488, + 246, + 504, + 257 + ], + "score": 1.0, + "content": "has", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 256, + 506, + 269 + ], + "spans": [ + { + "bbox": [ + 105, + 256, + 294, + 269 + ], + "score": 1.0, + "content": "exactly one non-zero entry in each of its first", + "type": "text" + }, + { + "bbox": [ + 295, + 258, + 301, + 266 + ], + "score": 0.51, + "content": "s", + "type": "inline_equation" + }, + { + "bbox": [ + 301, + 256, + 431, + 269 + ], + "score": 1.0, + "content": "columns. Also, note that since", + "type": "text" + }, + { + "bbox": [ + 432, + 257, + 438, + 266 + ], + "score": 0.78, + "content": "\\theta", + "type": "inline_equation" + }, + { + "bbox": [ + 438, + 256, + 506, + 269 + ], + "score": 1.0, + "content": "is in row-major", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 267, + 506, + 280 + ], + "spans": [ + { + "bbox": [ + 106, + 267, + 506, + 280 + ], + "score": 1.0, + "content": "order, these non-zero entries are sorted (any column to the right cannot have a non-zero entry in a", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 278, + 506, + 291 + ], + "spans": [ + { + "bbox": [ + 105, + 278, + 240, + 291 + ], + "score": 1.0, + "content": "higher row). Hence, to show that", + "type": "text" + }, + { + "bbox": [ + 240, + 279, + 251, + 289 + ], + "score": 0.36, + "content": "\\mathbf { H }", + "type": "inline_equation" + }, + { + "bbox": [ + 251, + 278, + 506, + 291 + ], + "score": 1.0, + "content": "is a horizontal step matrix, it is sufficient to argue that adjacent", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 289, + 505, + 300 + ], + "spans": [ + { + "bbox": [ + 106, + 289, + 154, + 300 + ], + "score": 1.0, + "content": "columns of", + "type": "text" + }, + { + "bbox": [ + 154, + 289, + 164, + 299 + ], + "score": 0.34, + "content": "\\mathbf { H }", + "type": "inline_equation" + }, + { + "bbox": [ + 165, + 289, + 431, + 300 + ], + "score": 1.0, + "content": "have non-zero entries at most one row apart. This is equivalent to", + "type": "text" + }, + { + "bbox": [ + 431, + 289, + 442, + 299 + ], + "score": 0.84, + "content": "\\mathbf { S } ^ { \\prime }", + "type": "inline_equation" + }, + { + "bbox": [ + 442, + 289, + 505, + 300 + ], + "score": 1.0, + "content": "having no zero", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 300, + 506, + 313 + ], + "spans": [ + { + "bbox": [ + 105, + 300, + 337, + 313 + ], + "score": 1.0, + "content": "rows between two non-zero rows, which is guaranteed by", + "type": "text" + }, + { + "bbox": [ + 338, + 300, + 351, + 312 + ], + "score": 0.88, + "content": "{ \\bf P } _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 351, + 300, + 386, + 313 + ], + "score": 1.0, + "content": ". Hence,", + "type": "text" + }, + { + "bbox": [ + 386, + 300, + 397, + 310 + ], + "score": 0.43, + "content": "\\mathbf { H }", + "type": "inline_equation" + }, + { + "bbox": [ + 397, + 300, + 506, + 313 + ], + "score": 1.0, + "content": "is a horizontal step matrix.", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 9.5 + }, + { + "type": "text", + "bbox": [ + 106, + 317, + 505, + 385 + ], + "lines": [ + { + "bbox": [ + 106, + 317, + 506, + 330 + ], + "spans": [ + { + "bbox": [ + 106, + 317, + 130, + 330 + ], + "score": 1.0, + "content": "Since", + "type": "text" + }, + { + "bbox": [ + 131, + 317, + 144, + 327 + ], + "score": 0.85, + "content": "\\mathbf { V } ^ { \\prime }", + "type": "inline_equation" + }, + { + "bbox": [ + 144, + 317, + 419, + 330 + ], + "score": 1.0, + "content": "has at most one non-zero entry per row, we may permute the rows of", + "type": "text" + }, + { + "bbox": [ + 419, + 317, + 432, + 327 + ], + "score": 0.87, + "content": "\\mathbf { V } ^ { \\prime }", + "type": "inline_equation" + }, + { + "bbox": [ + 432, + 317, + 506, + 330 + ], + "score": 1.0, + "content": "to obtain a matrix", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 327, + 505, + 340 + ], + "spans": [ + { + "bbox": [ + 106, + 328, + 116, + 338 + ], + "score": 0.53, + "content": "\\mathbf { V }", + "type": "inline_equation" + }, + { + "bbox": [ + 116, + 327, + 237, + 340 + ], + "score": 1.0, + "content": ", where the non-zero entries of", + "type": "text" + }, + { + "bbox": [ + 238, + 328, + 248, + 338 + ], + "score": 0.48, + "content": "\\mathbf { V }", + "type": "inline_equation" + }, + { + "bbox": [ + 248, + 327, + 505, + 340 + ], + "score": 1.0, + "content": "are sorted (any lower row below cannot have a non-zero entry in", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 104, + 339, + 506, + 353 + ], + "spans": [ + { + "bbox": [ + 104, + 339, + 338, + 353 + ], + "score": 1.0, + "content": "an earlier column). Hence, for some permutation matrix", + "type": "text" + }, + { + "bbox": [ + 338, + 339, + 369, + 353 + ], + "score": 0.91, + "content": "( \\mathbf { P } _ { 2 } ) ^ { - 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 369, + 339, + 373, + 353 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 373, + 339, + 441, + 353 + ], + "score": 0.92, + "content": "\\mathbf { V } = ( \\mathbf { P } _ { 2 } ) ^ { - 1 } \\mathbf { V } ^ { \\prime }", + "type": "inline_equation" + }, + { + "bbox": [ + 441, + 339, + 506, + 353 + ], + "score": 1.0, + "content": ", which implies", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 351, + 505, + 364 + ], + "spans": [ + { + "bbox": [ + 106, + 351, + 124, + 364 + ], + "score": 1.0, + "content": "that", + "type": "text" + }, + { + "bbox": [ + 124, + 352, + 173, + 363 + ], + "score": 0.92, + "content": "\\mathbf { V } ^ { \\prime } = \\mathbf { P } _ { 2 } \\mathbf { V }", + "type": "inline_equation" + }, + { + "bbox": [ + 173, + 351, + 389, + 364 + ], + "score": 1.0, + "content": ". It has exactly one non-zero entry in each of its first", + "type": "text" + }, + { + "bbox": [ + 389, + 354, + 396, + 362 + ], + "score": 0.67, + "content": "s", + "type": "inline_equation" + }, + { + "bbox": [ + 396, + 351, + 505, + 364 + ], + "score": 1.0, + "content": "columns. From the action", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 361, + 507, + 375 + ], + "spans": [ + { + "bbox": [ + 105, + 361, + 117, + 375 + ], + "score": 1.0, + "content": "of", + "type": "text" + }, + { + "bbox": [ + 117, + 363, + 131, + 374 + ], + "score": 0.89, + "content": "\\mathbf { P } _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 131, + 361, + 431, + 375 + ], + "score": 1.0, + "content": ", these non-zero entries are sorted. Therefore, by the same argument as for", + "type": "text" + }, + { + "bbox": [ + 431, + 363, + 442, + 372 + ], + "score": 0.51, + "content": "\\mathbf { H }", + "type": "inline_equation" + }, + { + "bbox": [ + 442, + 361, + 471, + 375 + ], + "score": 1.0, + "content": "above,", + "type": "text" + }, + { + "bbox": [ + 471, + 362, + 487, + 373 + ], + "score": 0.88, + "content": "\\mathbf { V } ^ { T }", + "type": "inline_equation" + }, + { + "bbox": [ + 488, + 361, + 507, + 375 + ], + "score": 1.0, + "content": "is a", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 373, + 339, + 386 + ], + "spans": [ + { + "bbox": [ + 105, + 373, + 230, + 386 + ], + "score": 1.0, + "content": "horizontal step matrix. Hence,", + "type": "text" + }, + { + "bbox": [ + 230, + 374, + 240, + 384 + ], + "score": 0.52, + "content": "\\mathbf { V }", + "type": "inline_equation" + }, + { + "bbox": [ + 240, + 373, + 339, + 386 + ], + "score": 1.0, + "content": "is a vertical step matrix.", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 15.5 + }, + { + "type": "text", + "bbox": [ + 106, + 390, + 505, + 435 + ], + "lines": [ + { + "bbox": [ + 105, + 390, + 506, + 403 + ], + "spans": [ + { + "bbox": [ + 105, + 390, + 265, + 403 + ], + "score": 1.0, + "content": "In all, we have found a decomposition", + "type": "text" + }, + { + "bbox": [ + 266, + 390, + 342, + 402 + ], + "score": 0.91, + "content": "\\mathbf { S } = \\mathbf { P } _ { 1 } \\mathbf { H } \\mathbf { P } _ { 2 } \\mathbf { V } \\mathbf { P } _ { 3 }", + "type": "inline_equation" + }, + { + "bbox": [ + 342, + 390, + 394, + 403 + ], + "score": 1.0, + "content": ", where each", + "type": "text" + }, + { + "bbox": [ + 395, + 391, + 406, + 402 + ], + "score": 0.87, + "content": "P _ { \\ell }", + "type": "inline_equation" + }, + { + "bbox": [ + 407, + 390, + 506, + 403 + ], + "score": 1.0, + "content": "is a permutation matrix", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 109, + 401, + 506, + 414 + ], + "spans": [ + { + "bbox": [ + 109, + 402, + 138, + 412 + ], + "score": 0.87, + "content": "( \\in B B ^ { * }", + "type": "inline_equation" + }, + { + "bbox": [ + 138, + 401, + 204, + 414 + ], + "score": 1.0, + "content": "by Theorem 2),", + "type": "text" + }, + { + "bbox": [ + 204, + 402, + 215, + 412 + ], + "score": 0.31, + "content": "\\mathbf { H }", + "type": "inline_equation" + }, + { + "bbox": [ + 215, + 401, + 326, + 414 + ], + "score": 1.0, + "content": "is a horizontal step matrix (", + "type": "text" + }, + { + "bbox": [ + 326, + 402, + 343, + 412 + ], + "score": 0.85, + "content": "\\in B", + "type": "inline_equation" + }, + { + "bbox": [ + 344, + 401, + 426, + 414 + ], + "score": 1.0, + "content": "by Lemma I.2), and", + "type": "text" + }, + { + "bbox": [ + 426, + 402, + 437, + 412 + ], + "score": 0.55, + "content": "\\mathbf { V }", + "type": "inline_equation" + }, + { + "bbox": [ + 437, + 401, + 506, + 414 + ], + "score": 1.0, + "content": "is a vertical step", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 412, + 505, + 425 + ], + "spans": [ + { + "bbox": [ + 105, + 412, + 137, + 425 + ], + "score": 1.0, + "content": "matrix", + "type": "text" + }, + { + "bbox": [ + 137, + 413, + 159, + 423 + ], + "score": 0.87, + "content": "( \\in B ^ { * }", + "type": "inline_equation" + }, + { + "bbox": [ + 159, + 412, + 337, + 425 + ], + "score": 1.0, + "content": "by Corollary I.3). Moreover, by Lemma G.6,", + "type": "text" + }, + { + "bbox": [ + 337, + 413, + 381, + 424 + ], + "score": 0.9, + "content": "\\mathbf { P } _ { 2 } \\in B ^ { \\ast } B", + "type": "inline_equation" + }, + { + "bbox": [ + 382, + 412, + 395, + 425 + ], + "score": 1.0, + "content": ", so", + "type": "text" + }, + { + "bbox": [ + 396, + 413, + 436, + 424 + ], + "score": 0.86, + "content": "\\mathbf { H } , \\mathbf { P } _ { 2 } , \\mathbf { V }", + "type": "inline_equation" + }, + { + "bbox": [ + 436, + 412, + 505, + 425 + ], + "score": 1.0, + "content": "can be combined", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 423, + 505, + 436 + ], + "spans": [ + { + "bbox": [ + 105, + 423, + 144, + 436 + ], + "score": 1.0, + "content": "to obtain", + "type": "text" + }, + { + "bbox": [ + 145, + 423, + 218, + 435 + ], + "score": 0.93, + "content": "\\mathbf { H } \\mathbf { P } _ { 2 } \\mathbf { \\bar { V } } \\in ( B B ^ { \\bar { * } } ) ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 218, + 423, + 289, + 436 + ], + "score": 1.0, + "content": ". By Lemma H.2,", + "type": "text" + }, + { + "bbox": [ + 289, + 423, + 339, + 435 + ], + "score": 0.92, + "content": "\\mathbf { S } \\in ( B B ^ { * } ) ^ { 4 }", + "type": "inline_equation" + }, + { + "bbox": [ + 339, + 423, + 344, + 436 + ], + "score": 1.0, + "content": ".", + "type": "text" + }, + { + "bbox": [ + 495, + 424, + 505, + 434 + ], + "score": 0.999, + "content": "□", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 20.5 + }, + { + "type": "text", + "bbox": [ + 107, + 444, + 397, + 457 + ], + "lines": [ + { + "bbox": [ + 105, + 443, + 394, + 460 + ], + "spans": [ + { + "bbox": [ + 105, + 443, + 219, + 460 + ], + "score": 1.0, + "content": "Corollary I.4. Let R be an", + "type": "text" + }, + { + "bbox": [ + 220, + 446, + 245, + 456 + ], + "score": 0.87, + "content": "n \\times n", + "type": "inline_equation" + }, + { + "bbox": [ + 245, + 443, + 306, + 460 + ], + "score": 1.0, + "content": "matrix of rank", + "type": "text" + }, + { + "bbox": [ + 306, + 447, + 311, + 455 + ], + "score": 0.26, + "content": "r", + "type": "inline_equation" + }, + { + "bbox": [ + 312, + 443, + 339, + 460 + ], + "score": 1.0, + "content": ". Then", + "type": "text" + }, + { + "bbox": [ + 339, + 444, + 394, + 457 + ], + "score": 0.92, + "content": "\\mathbf { R } \\in ( B B ^ { * } ) _ { 4 } ^ { 8 r }", + "type": "inline_equation" + } + ], + "index": 23 + } + ], + "index": 23 + }, + { + "type": "text", + "bbox": [ + 107, + 472, + 505, + 506 + ], + "lines": [ + { + "bbox": [ + 105, + 471, + 505, + 486 + ], + "spans": [ + { + "bbox": [ + 105, + 471, + 219, + 486 + ], + "score": 1.0, + "content": "Proof. We can decompose", + "type": "text" + }, + { + "bbox": [ + 219, + 473, + 230, + 483 + ], + "score": 0.43, + "content": "\\mathbf { R }", + "type": "inline_equation" + }, + { + "bbox": [ + 230, + 471, + 243, + 486 + ], + "score": 1.0, + "content": "as", + "type": "text" + }, + { + "bbox": [ + 243, + 473, + 291, + 483 + ], + "score": 0.9, + "content": "\\mathbf { R } = \\mathbf { G } \\mathbf { H } ^ { * }", + "type": "inline_equation" + }, + { + "bbox": [ + 292, + 471, + 321, + 486 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 322, + 473, + 346, + 484 + ], + "score": 0.86, + "content": "\\mathbf { G } , \\mathbf { H }", + "type": "inline_equation" + }, + { + "bbox": [ + 346, + 471, + 363, + 486 + ], + "score": 1.0, + "content": "are", + "type": "text" + }, + { + "bbox": [ + 363, + 474, + 389, + 483 + ], + "score": 0.89, + "content": "n \\times r", + "type": "inline_equation" + }, + { + "bbox": [ + 389, + 471, + 505, + 486 + ], + "score": 1.0, + "content": "matrices. With appropriate", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 483, + 505, + 496 + ], + "spans": [ + { + "bbox": [ + 106, + 483, + 284, + 496 + ], + "score": 1.0, + "content": "zero-padding, both of these can be made into", + "type": "text" + }, + { + "bbox": [ + 284, + 485, + 308, + 494 + ], + "score": 0.89, + "content": "n \\times n", + "type": "inline_equation" + }, + { + "bbox": [ + 308, + 483, + 394, + 496 + ], + "score": 1.0, + "content": "matrices with at most", + "type": "text" + }, + { + "bbox": [ + 394, + 485, + 407, + 494 + ], + "score": 0.42, + "content": "r n", + "type": "inline_equation" + }, + { + "bbox": [ + 407, + 483, + 505, + 496 + ], + "score": 1.0, + "content": "NNZ. The proof follows", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 494, + 506, + 506 + ], + "spans": [ + { + "bbox": [ + 105, + 494, + 296, + 506 + ], + "score": 1.0, + "content": "immediately from Theorem 3 and Lemma H.2.", + "type": "text" + }, + { + "bbox": [ + 494, + 495, + 506, + 506 + ], + "score": 0.999, + "content": "□", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 25 + }, + { + "type": "title", + "bbox": [ + 108, + 523, + 503, + 550 + ], + "lines": [ + { + "bbox": [ + 105, + 523, + 505, + 537 + ], + "spans": [ + { + "bbox": [ + 105, + 523, + 505, + 537 + ], + "score": 1.0, + "content": "J EXAMPLE OF K-MATRIX REPRESENTATION OF STRUCTURED MATRICES AND", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 124, + 537, + 298, + 551 + ], + "spans": [ + { + "bbox": [ + 124, + 537, + 212, + 551 + ], + "score": 1.0, + "content": "COMPARISON TO", + "type": "text" + }, + { + "bbox": [ + 213, + 537, + 232, + 549 + ], + "score": 0.69, + "content": "B \\mathcal { P }", + "type": "inline_equation" + }, + { + "bbox": [ + 232, + 537, + 298, + 551 + ], + "score": 1.0, + "content": "HIERARCHY", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 27.5 + }, + { + "type": "text", + "bbox": [ + 107, + 563, + 505, + 596 + ], + "lines": [ + { + "bbox": [ + 105, + 562, + 505, + 576 + ], + "spans": [ + { + "bbox": [ + 105, + 562, + 505, + 576 + ], + "score": 1.0, + "content": "In this appendix, we show explicitly how some common structured matrices (e.g. originating from", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 574, + 504, + 586 + ], + "spans": [ + { + "bbox": [ + 106, + 574, + 484, + 586 + ], + "score": 1.0, + "content": "fast transforms) can be represented as K-matrices. We also draw comparisons between the", + "type": "text" + }, + { + "bbox": [ + 484, + 574, + 504, + 584 + ], + "score": 0.86, + "content": "B B ^ { * }", + "type": "inline_equation" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 585, + 365, + 597 + ], + "spans": [ + { + "bbox": [ + 106, + 585, + 178, + 597 + ], + "score": 1.0, + "content": "hierarchy and the", + "type": "text" + }, + { + "bbox": [ + 178, + 585, + 194, + 595 + ], + "score": 0.81, + "content": "B \\mathcal { P }", + "type": "inline_equation" + }, + { + "bbox": [ + 194, + 585, + 365, + 597 + ], + "score": 1.0, + "content": "hierarchy introduced by Dao et al. (2019).", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 30 + }, + { + "type": "text", + "bbox": [ + 107, + 600, + 450, + 612 + ], + "lines": [ + { + "bbox": [ + 105, + 599, + 451, + 614 + ], + "spans": [ + { + "bbox": [ + 105, + 599, + 176, + 614 + ], + "score": 1.0, + "content": "Lemma J.1. Let", + "type": "text" + }, + { + "bbox": [ + 177, + 601, + 190, + 612 + ], + "score": 0.88, + "content": "{ \\bf F } _ { n }", + "type": "inline_equation" + }, + { + "bbox": [ + 191, + 599, + 358, + 614 + ], + "score": 1.0, + "content": "be the Discrete Fourier Transform of size", + "type": "text" + }, + { + "bbox": [ + 358, + 603, + 365, + 610 + ], + "score": 0.5, + "content": "n", + "type": "inline_equation" + }, + { + "bbox": [ + 365, + 599, + 392, + 614 + ], + "score": 1.0, + "content": ". Then", + "type": "text" + }, + { + "bbox": [ + 392, + 600, + 448, + 613 + ], + "score": 0.93, + "content": "\\mathbf { F } _ { n } \\in ( B B ^ { * } ) ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 448, + 599, + 451, + 614 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 32 + }, + { + "type": "text", + "bbox": [ + 106, + 627, + 505, + 652 + ], + "lines": [ + { + "bbox": [ + 106, + 627, + 505, + 640 + ], + "spans": [ + { + "bbox": [ + 106, + 627, + 280, + 640 + ], + "score": 1.0, + "content": "Proof. From Parker (1995), we can express", + "type": "text" + }, + { + "bbox": [ + 281, + 628, + 294, + 639 + ], + "score": 0.89, + "content": "{ \\bf F } _ { n }", + "type": "inline_equation" + }, + { + "bbox": [ + 295, + 627, + 306, + 640 + ], + "score": 1.0, + "content": "as", + "type": "text" + }, + { + "bbox": [ + 306, + 628, + 352, + 639 + ], + "score": 0.9, + "content": "{ \\bf F } _ { n } = { \\bf B } { \\bf P }", + "type": "inline_equation" + }, + { + "bbox": [ + 353, + 627, + 382, + 640 + ], + "score": 1.0, + "content": ", where", + "type": "text" + }, + { + "bbox": [ + 383, + 628, + 411, + 638 + ], + "score": 0.91, + "content": "\\mathbf { B } \\in B", + "type": "inline_equation" + }, + { + "bbox": [ + 411, + 627, + 428, + 640 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 429, + 628, + 438, + 638 + ], + "score": 0.77, + "content": "\\mathbf { P }", + "type": "inline_equation" + }, + { + "bbox": [ + 438, + 627, + 505, + 640 + ], + "score": 1.0, + "content": "is a permutation", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 637, + 506, + 652 + ], + "spans": [ + { + "bbox": [ + 105, + 637, + 304, + 652 + ], + "score": 1.0, + "content": "(the bit reversal permutation). From Theorem 2,", + "type": "text" + }, + { + "bbox": [ + 304, + 639, + 344, + 650 + ], + "score": 0.89, + "content": "\\mathbf { P } \\in B B ^ { * }", + "type": "inline_equation" + }, + { + "bbox": [ + 344, + 637, + 446, + 652 + ], + "score": 1.0, + "content": ". Hence, by Lemma H.2,", + "type": "text" + }, + { + "bbox": [ + 446, + 639, + 502, + 651 + ], + "score": 0.89, + "content": "\\mathbf { F } _ { n } ^ { - } \\in ( B B ^ { * } ) ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 503, + 637, + 506, + 652 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 33.5 + }, + { + "type": "text", + "bbox": [ + 106, + 671, + 419, + 683 + ], + "lines": [ + { + "bbox": [ + 105, + 671, + 420, + 685 + ], + "spans": [ + { + "bbox": [ + 105, + 671, + 176, + 685 + ], + "score": 1.0, + "content": "Lemma J.2. Let", + "type": "text" + }, + { + "bbox": [ + 177, + 672, + 192, + 682 + ], + "score": 0.87, + "content": "\\mathbf { H } _ { n }", + "type": "inline_equation" + }, + { + "bbox": [ + 192, + 671, + 336, + 685 + ], + "score": 1.0, + "content": "be the Hadamard Transform of size", + "type": "text" + }, + { + "bbox": [ + 337, + 674, + 344, + 681 + ], + "score": 0.44, + "content": "n", + "type": "inline_equation" + }, + { + "bbox": [ + 344, + 671, + 371, + 685 + ], + "score": 1.0, + "content": ". 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Let", + "type": "text" + }, + { + "bbox": [ + 177, + 721, + 189, + 732 + ], + "score": 0.88, + "content": "\\mathbf { S } _ { n }", + "type": "inline_equation" + }, + { + "bbox": [ + 190, + 719, + 344, + 734 + ], + "score": 1.0, + "content": "be the Discrete Sine Transform of size", + "type": "text" + }, + { + "bbox": [ + 345, + 723, + 352, + 730 + ], + "score": 0.49, + "content": "n", + "type": "inline_equation" + }, + { + "bbox": [ + 352, + 719, + 378, + 734 + ], + "score": 1.0, + "content": ". 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Hence,", + "type": "text" + }, + { + "bbox": [ + 169, + 218, + 231, + 229 + ], + "score": 0.92, + "content": "\\mathbf { H } \\mathbf { V } ^ { \\prime } [ i , j ] = c _ { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 231, + 217, + 261, + 230 + ], + "score": 1.0, + "content": "only if", + "type": "text" + }, + { + "bbox": [ + 261, + 218, + 325, + 230 + ], + "score": 0.93, + "content": "( i , j ) = ( i _ { k } , j _ { k } )", + "type": "inline_equation" + }, + { + "bbox": [ + 325, + 217, + 363, + 230 + ], + "score": 1.0, + "content": "for some", + "type": "text" + }, + { + "bbox": [ + 363, + 218, + 370, + 227 + ], + "score": 0.81, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 370, + 217, + 506, + 230 + ], + "score": 1.0, + "content": ", which is exactly the definition of", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 228, + 200, + 240 + ], + "spans": [ + { + "bbox": [ + 106, + 228, + 117, + 239 + ], + "score": 0.81, + "content": "\\mathbf { S } ^ { \\prime }", + "type": "inline_equation" + }, + { + "bbox": [ + 117, + 228, + 151, + 240 + ], + "score": 1.0, + "content": ". Hence,", + "type": "text" + }, + { + "bbox": [ + 152, + 228, + 195, + 239 + ], + "score": 0.92, + "content": "\\mathbf { S } ^ { \\prime } = \\mathbf { H } \\mathbf { V } ^ { \\prime }", + "type": "inline_equation" + }, + { + "bbox": [ + 196, + 228, + 200, + 240 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 5, + "bbox_fs": [ + 105, + 205, + 506, + 240 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 245, + 505, + 312 + ], + "lines": [ + { + "bbox": [ + 106, + 246, + 504, + 257 + ], + "spans": [ + { + "bbox": [ + 106, + 246, + 165, + 257 + ], + "score": 1.0, + "content": "We argue that", + "type": "text" + }, + { + "bbox": [ + 166, + 246, + 176, + 256 + ], + "score": 0.49, + "content": "\\mathbf { H }", + "type": "inline_equation" + }, + { + "bbox": [ + 177, + 246, + 477, + 257 + ], + "score": 1.0, + "content": "is a horizontal step matrix through a series of assertions. First, note that", + "type": "text" + }, + { + "bbox": [ + 477, + 246, + 488, + 256 + ], + "score": 0.43, + "content": "\\mathbf { H }", + "type": "inline_equation" + }, + { + "bbox": [ + 488, + 246, + 504, + 257 + ], + "score": 1.0, + "content": "has", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 256, + 506, + 269 + ], + "spans": [ + { + "bbox": [ + 105, + 256, + 294, + 269 + ], + "score": 1.0, + "content": "exactly one non-zero entry in each of its first", + "type": "text" + }, + { + "bbox": [ + 295, + 258, + 301, + 266 + ], + "score": 0.51, + "content": "s", + "type": "inline_equation" + }, + { + "bbox": [ + 301, + 256, + 431, + 269 + ], + "score": 1.0, + "content": "columns. Also, note that since", + "type": "text" + }, + { + "bbox": [ + 432, + 257, + 438, + 266 + ], + "score": 0.78, + "content": "\\theta", + "type": "inline_equation" + }, + { + "bbox": [ + 438, + 256, + 506, + 269 + ], + "score": 1.0, + "content": "is in row-major", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 267, + 506, + 280 + ], + "spans": [ + { + "bbox": [ + 106, + 267, + 506, + 280 + ], + "score": 1.0, + "content": "order, these non-zero entries are sorted (any column to the right cannot have a non-zero entry in a", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 278, + 506, + 291 + ], + "spans": [ + { + "bbox": [ + 105, + 278, + 240, + 291 + ], + "score": 1.0, + "content": "higher row). Hence, to show that", + "type": "text" + }, + { + "bbox": [ + 240, + 279, + 251, + 289 + ], + "score": 0.36, + "content": "\\mathbf { H }", + "type": "inline_equation" + }, + { + "bbox": [ + 251, + 278, + 506, + 291 + ], + "score": 1.0, + "content": "is a horizontal step matrix, it is sufficient to argue that adjacent", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 289, + 505, + 300 + ], + "spans": [ + { + "bbox": [ + 106, + 289, + 154, + 300 + ], + "score": 1.0, + "content": "columns of", + "type": "text" + }, + { + "bbox": [ + 154, + 289, + 164, + 299 + ], + "score": 0.34, + "content": "\\mathbf { H }", + "type": "inline_equation" + }, + { + "bbox": [ + 165, + 289, + 431, + 300 + ], + "score": 1.0, + "content": "have non-zero entries at most one row apart. This is equivalent to", + "type": "text" + }, + { + "bbox": [ + 431, + 289, + 442, + 299 + ], + "score": 0.84, + "content": "\\mathbf { S } ^ { \\prime }", + "type": "inline_equation" + }, + { + "bbox": [ + 442, + 289, + 505, + 300 + ], + "score": 1.0, + "content": "having no zero", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 300, + 506, + 313 + ], + "spans": [ + { + "bbox": [ + 105, + 300, + 337, + 313 + ], + "score": 1.0, + "content": "rows between two non-zero rows, which is guaranteed by", + "type": "text" + }, + { + "bbox": [ + 338, + 300, + 351, + 312 + ], + "score": 0.88, + "content": "{ \\bf P } _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 351, + 300, + 386, + 313 + ], + "score": 1.0, + "content": ". Hence,", + "type": "text" + }, + { + "bbox": [ + 386, + 300, + 397, + 310 + ], + "score": 0.43, + "content": "\\mathbf { H }", + "type": "inline_equation" + }, + { + "bbox": [ + 397, + 300, + 506, + 313 + ], + "score": 1.0, + "content": "is a horizontal step matrix.", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 9.5, + "bbox_fs": [ + 105, + 246, + 506, + 313 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 317, + 505, + 385 + ], + "lines": [ + { + "bbox": [ + 106, + 317, + 506, + 330 + ], + "spans": [ + { + "bbox": [ + 106, + 317, + 130, + 330 + ], + "score": 1.0, + "content": "Since", + "type": "text" + }, + { + "bbox": [ + 131, + 317, + 144, + 327 + ], + "score": 0.85, + "content": "\\mathbf { V } ^ { \\prime }", + "type": "inline_equation" + }, + { + "bbox": [ + 144, + 317, + 419, + 330 + ], + "score": 1.0, + "content": "has at most one non-zero entry per row, we may permute the rows of", + "type": "text" + }, + { + "bbox": [ + 419, + 317, + 432, + 327 + ], + "score": 0.87, + "content": "\\mathbf { V } ^ { \\prime }", + "type": "inline_equation" + }, + { + "bbox": [ + 432, + 317, + 506, + 330 + ], + "score": 1.0, + "content": "to obtain a matrix", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 327, + 505, + 340 + ], + "spans": [ + { + "bbox": [ + 106, + 328, + 116, + 338 + ], + "score": 0.53, + "content": "\\mathbf { V }", + "type": "inline_equation" + }, + { + "bbox": [ + 116, + 327, + 237, + 340 + ], + "score": 1.0, + "content": ", where the non-zero entries of", + "type": "text" + }, + { + "bbox": [ + 238, + 328, + 248, + 338 + ], + "score": 0.48, + "content": "\\mathbf { V }", + "type": "inline_equation" + }, + { + "bbox": [ + 248, + 327, + 505, + 340 + ], + "score": 1.0, + "content": "are sorted (any lower row below cannot have a non-zero entry in", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 104, + 339, + 506, + 353 + ], + "spans": [ + { + "bbox": [ + 104, + 339, + 338, + 353 + ], + "score": 1.0, + "content": "an earlier column). Hence, for some permutation matrix", + "type": "text" + }, + { + "bbox": [ + 338, + 339, + 369, + 353 + ], + "score": 0.91, + "content": "( \\mathbf { P } _ { 2 } ) ^ { - 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 369, + 339, + 373, + 353 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 373, + 339, + 441, + 353 + ], + "score": 0.92, + "content": "\\mathbf { V } = ( \\mathbf { P } _ { 2 } ) ^ { - 1 } \\mathbf { V } ^ { \\prime }", + "type": "inline_equation" + }, + { + "bbox": [ + 441, + 339, + 506, + 353 + ], + "score": 1.0, + "content": ", which implies", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 351, + 505, + 364 + ], + "spans": [ + { + "bbox": [ + 106, + 351, + 124, + 364 + ], + "score": 1.0, + "content": "that", + "type": "text" + }, + { + "bbox": [ + 124, + 352, + 173, + 363 + ], + "score": 0.92, + "content": "\\mathbf { V } ^ { \\prime } = \\mathbf { P } _ { 2 } \\mathbf { V }", + "type": "inline_equation" + }, + { + "bbox": [ + 173, + 351, + 389, + 364 + ], + "score": 1.0, + "content": ". It has exactly one non-zero entry in each of its first", + "type": "text" + }, + { + "bbox": [ + 389, + 354, + 396, + 362 + ], + "score": 0.67, + "content": "s", + "type": "inline_equation" + }, + { + "bbox": [ + 396, + 351, + 505, + 364 + ], + "score": 1.0, + "content": "columns. From the action", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 361, + 507, + 375 + ], + "spans": [ + { + "bbox": [ + 105, + 361, + 117, + 375 + ], + "score": 1.0, + "content": "of", + "type": "text" + }, + { + "bbox": [ + 117, + 363, + 131, + 374 + ], + "score": 0.89, + "content": "\\mathbf { P } _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 131, + 361, + 431, + 375 + ], + "score": 1.0, + "content": ", these non-zero entries are sorted. Therefore, by the same argument as for", + "type": "text" + }, + { + "bbox": [ + 431, + 363, + 442, + 372 + ], + "score": 0.51, + "content": "\\mathbf { H }", + "type": "inline_equation" + }, + { + "bbox": [ + 442, + 361, + 471, + 375 + ], + "score": 1.0, + "content": "above,", + "type": "text" + }, + { + "bbox": [ + 471, + 362, + 487, + 373 + ], + "score": 0.88, + "content": "\\mathbf { V } ^ { T }", + "type": "inline_equation" + }, + { + "bbox": [ + 488, + 361, + 507, + 375 + ], + "score": 1.0, + "content": "is a", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 373, + 339, + 386 + ], + "spans": [ + { + "bbox": [ + 105, + 373, + 230, + 386 + ], + "score": 1.0, + "content": "horizontal step matrix. Hence,", + "type": "text" + }, + { + "bbox": [ + 230, + 374, + 240, + 384 + ], + "score": 0.52, + "content": "\\mathbf { V }", + "type": "inline_equation" + }, + { + "bbox": [ + 240, + 373, + 339, + 386 + ], + "score": 1.0, + "content": "is a vertical step matrix.", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 15.5, + "bbox_fs": [ + 104, + 317, + 507, + 386 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 390, + 505, + 435 + ], + "lines": [ + { + "bbox": [ + 105, + 390, + 506, + 403 + ], + "spans": [ + { + "bbox": [ + 105, + 390, + 265, + 403 + ], + "score": 1.0, + "content": "In all, we have found a decomposition", + "type": "text" + }, + { + "bbox": [ + 266, + 390, + 342, + 402 + ], + "score": 0.91, + "content": "\\mathbf { S } = \\mathbf { P } _ { 1 } \\mathbf { H } \\mathbf { P } _ { 2 } \\mathbf { V } \\mathbf { P } _ { 3 }", + "type": "inline_equation" + }, + { + "bbox": [ + 342, + 390, + 394, + 403 + ], + "score": 1.0, + "content": ", where each", + "type": "text" + }, + { + "bbox": [ + 395, + 391, + 406, + 402 + ], + "score": 0.87, + "content": "P _ { \\ell }", + "type": "inline_equation" + }, + { + "bbox": [ + 407, + 390, + 506, + 403 + ], + "score": 1.0, + "content": "is a permutation matrix", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 109, + 401, + 506, + 414 + ], + "spans": [ + { + "bbox": [ + 109, + 402, + 138, + 412 + ], + "score": 0.87, + "content": "( \\in B B ^ { * }", + "type": "inline_equation" + }, + { + "bbox": [ + 138, + 401, + 204, + 414 + ], + "score": 1.0, + "content": "by Theorem 2),", + "type": "text" + }, + { + "bbox": [ + 204, + 402, + 215, + 412 + ], + "score": 0.31, + "content": "\\mathbf { H }", + "type": "inline_equation" + }, + { + "bbox": [ + 215, + 401, + 326, + 414 + ], + "score": 1.0, + "content": "is a horizontal step matrix (", + "type": "text" + }, + { + "bbox": [ + 326, + 402, + 343, + 412 + ], + "score": 0.85, + "content": "\\in B", + "type": "inline_equation" + }, + { + "bbox": [ + 344, + 401, + 426, + 414 + ], + "score": 1.0, + "content": "by Lemma I.2), and", + "type": "text" + }, + { + "bbox": [ + 426, + 402, + 437, + 412 + ], + "score": 0.55, + "content": "\\mathbf { V }", + "type": "inline_equation" + }, + { + "bbox": [ + 437, + 401, + 506, + 414 + ], + "score": 1.0, + "content": "is a vertical step", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 412, + 505, + 425 + ], + "spans": [ + { + "bbox": [ + 105, + 412, + 137, + 425 + ], + "score": 1.0, + "content": "matrix", + "type": "text" + }, + { + "bbox": [ + 137, + 413, + 159, + 423 + ], + "score": 0.87, + "content": "( \\in B ^ { * }", + "type": "inline_equation" + }, + { + "bbox": [ + 159, + 412, + 337, + 425 + ], + "score": 1.0, + "content": "by Corollary I.3). Moreover, by Lemma G.6,", + "type": "text" + }, + { + "bbox": [ + 337, + 413, + 381, + 424 + ], + "score": 0.9, + "content": "\\mathbf { P } _ { 2 } \\in B ^ { \\ast } B", + "type": "inline_equation" + }, + { + "bbox": [ + 382, + 412, + 395, + 425 + ], + "score": 1.0, + "content": ", so", + "type": "text" + }, + { + "bbox": [ + 396, + 413, + 436, + 424 + ], + "score": 0.86, + "content": "\\mathbf { H } , \\mathbf { P } _ { 2 } , \\mathbf { V }", + "type": "inline_equation" + }, + { + "bbox": [ + 436, + 412, + 505, + 425 + ], + "score": 1.0, + "content": "can be combined", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 423, + 505, + 436 + ], + "spans": [ + { + "bbox": [ + 105, + 423, + 144, + 436 + ], + "score": 1.0, + "content": "to obtain", + "type": "text" + }, + { + "bbox": [ + 145, + 423, + 218, + 435 + ], + "score": 0.93, + "content": "\\mathbf { H } \\mathbf { P } _ { 2 } \\mathbf { \\bar { V } } \\in ( B B ^ { \\bar { * } } ) ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 218, + 423, + 289, + 436 + ], + "score": 1.0, + "content": ". By Lemma H.2,", + "type": "text" + }, + { + "bbox": [ + 289, + 423, + 339, + 435 + ], + "score": 0.92, + "content": "\\mathbf { S } \\in ( B B ^ { * } ) ^ { 4 }", + "type": "inline_equation" + }, + { + "bbox": [ + 339, + 423, + 344, + 436 + ], + "score": 1.0, + "content": ".", + "type": "text" + }, + { + "bbox": [ + 495, + 424, + 505, + 434 + ], + "score": 0.999, + "content": "□", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 20.5, + "bbox_fs": [ + 105, + 390, + 506, + 436 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 444, + 397, + 457 + ], + "lines": [ + { + "bbox": [ + 105, + 443, + 394, + 460 + ], + "spans": [ + { + "bbox": [ + 105, + 443, + 219, + 460 + ], + "score": 1.0, + "content": "Corollary I.4. Let R be an", + "type": "text" + }, + { + "bbox": [ + 220, + 446, + 245, + 456 + ], + "score": 0.87, + "content": "n \\times n", + "type": "inline_equation" + }, + { + "bbox": [ + 245, + 443, + 306, + 460 + ], + "score": 1.0, + "content": "matrix of rank", + "type": "text" + }, + { + "bbox": [ + 306, + 447, + 311, + 455 + ], + "score": 0.26, + "content": "r", + "type": "inline_equation" + }, + { + "bbox": [ + 312, + 443, + 339, + 460 + ], + "score": 1.0, + "content": ". Then", + "type": "text" + }, + { + "bbox": [ + 339, + 444, + 394, + 457 + ], + "score": 0.92, + "content": "\\mathbf { R } \\in ( B B ^ { * } ) _ { 4 } ^ { 8 r }", + "type": "inline_equation" + } + ], + "index": 23 + } + ], + "index": 23, + "bbox_fs": [ + 105, + 443, + 394, + 460 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 472, + 505, + 506 + ], + "lines": [ + { + "bbox": [ + 105, + 471, + 505, + 486 + ], + "spans": [ + { + "bbox": [ + 105, + 471, + 219, + 486 + ], + "score": 1.0, + "content": "Proof. We can decompose", + "type": "text" + }, + { + "bbox": [ + 219, + 473, + 230, + 483 + ], + "score": 0.43, + "content": "\\mathbf { R }", + "type": "inline_equation" + }, + { + "bbox": [ + 230, + 471, + 243, + 486 + ], + "score": 1.0, + "content": "as", + "type": "text" + }, + { + "bbox": [ + 243, + 473, + 291, + 483 + ], + "score": 0.9, + "content": "\\mathbf { R } = \\mathbf { G } \\mathbf { H } ^ { * }", + "type": "inline_equation" + }, + { + "bbox": [ + 292, + 471, + 321, + 486 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 322, + 473, + 346, + 484 + ], + "score": 0.86, + "content": "\\mathbf { G } , \\mathbf { H }", + "type": "inline_equation" + }, + { + "bbox": [ + 346, + 471, + 363, + 486 + ], + "score": 1.0, + "content": "are", + "type": "text" + }, + { + "bbox": [ + 363, + 474, + 389, + 483 + ], + "score": 0.89, + "content": "n \\times r", + "type": "inline_equation" + }, + { + "bbox": [ + 389, + 471, + 505, + 486 + ], + "score": 1.0, + "content": "matrices. With appropriate", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 483, + 505, + 496 + ], + "spans": [ + { + "bbox": [ + 106, + 483, + 284, + 496 + ], + "score": 1.0, + "content": "zero-padding, both of these can be made into", + "type": "text" + }, + { + "bbox": [ + 284, + 485, + 308, + 494 + ], + "score": 0.89, + "content": "n \\times n", + "type": "inline_equation" + }, + { + "bbox": [ + 308, + 483, + 394, + 496 + ], + "score": 1.0, + "content": "matrices with at most", + "type": "text" + }, + { + "bbox": [ + 394, + 485, + 407, + 494 + ], + "score": 0.42, + "content": "r n", + "type": "inline_equation" + }, + { + "bbox": [ + 407, + 483, + 505, + 496 + ], + "score": 1.0, + "content": "NNZ. The proof follows", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 494, + 506, + 506 + ], + "spans": [ + { + "bbox": [ + 105, + 494, + 296, + 506 + ], + "score": 1.0, + "content": "immediately from Theorem 3 and Lemma H.2.", + "type": "text" + }, + { + "bbox": [ + 494, + 495, + 506, + 506 + ], + "score": 0.999, + "content": "□", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 25, + "bbox_fs": [ + 105, + 471, + 506, + 506 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 523, + 503, + 550 + ], + "lines": [ + { + "bbox": [ + 105, + 523, + 505, + 537 + ], + "spans": [ + { + "bbox": [ + 105, + 523, + 505, + 537 + ], + "score": 1.0, + "content": "J EXAMPLE OF K-MATRIX REPRESENTATION OF STRUCTURED MATRICES AND", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 124, + 537, + 298, + 551 + ], + "spans": [ + { + "bbox": [ + 124, + 537, + 212, + 551 + ], + "score": 1.0, + "content": "COMPARISON TO", + "type": "text" + }, + { + "bbox": [ + 213, + 537, + 232, + 549 + ], + "score": 0.69, + "content": "B \\mathcal { P }", + "type": "inline_equation" + }, + { + "bbox": [ + 232, + 537, + 298, + 551 + ], + "score": 1.0, + "content": "HIERARCHY", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 27.5 + }, + { + "type": "text", + "bbox": [ + 107, + 563, + 505, + 596 + ], + "lines": [ + { + "bbox": [ + 105, + 562, + 505, + 576 + ], + "spans": [ + { + "bbox": [ + 105, + 562, + 505, + 576 + ], + "score": 1.0, + "content": "In this appendix, we show explicitly how some common structured matrices (e.g. originating from", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 574, + 504, + 586 + ], + "spans": [ + { + "bbox": [ + 106, + 574, + 484, + 586 + ], + "score": 1.0, + "content": "fast transforms) can be represented as K-matrices. We also draw comparisons between the", + "type": "text" + }, + { + "bbox": [ + 484, + 574, + 504, + 584 + ], + "score": 0.86, + "content": "B B ^ { * }", + "type": "inline_equation" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 585, + 365, + 597 + ], + "spans": [ + { + "bbox": [ + 106, + 585, + 178, + 597 + ], + "score": 1.0, + "content": "hierarchy and the", + "type": "text" + }, + { + "bbox": [ + 178, + 585, + 194, + 595 + ], + "score": 0.81, + "content": "B \\mathcal { P }", + "type": "inline_equation" + }, + { + "bbox": [ + 194, + 585, + 365, + 597 + ], + "score": 1.0, + "content": "hierarchy introduced by Dao et al. (2019).", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 30, + "bbox_fs": [ + 105, + 562, + 505, + 597 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 600, + 450, + 612 + ], + "lines": [ + { + "bbox": [ + 105, + 599, + 451, + 614 + ], + "spans": [ + { + "bbox": [ + 105, + 599, + 176, + 614 + ], + "score": 1.0, + "content": "Lemma J.1. Let", + "type": "text" + }, + { + "bbox": [ + 177, + 601, + 190, + 612 + ], + "score": 0.88, + "content": "{ \\bf F } _ { n }", + "type": "inline_equation" + }, + { + "bbox": [ + 191, + 599, + 358, + 614 + ], + "score": 1.0, + "content": "be the Discrete Fourier Transform of size", + "type": "text" + }, + { + "bbox": [ + 358, + 603, + 365, + 610 + ], + "score": 0.5, + "content": "n", + "type": "inline_equation" + }, + { + "bbox": [ + 365, + 599, + 392, + 614 + ], + "score": 1.0, + "content": ". Then", + "type": "text" + }, + { + "bbox": [ + 392, + 600, + 448, + 613 + ], + "score": 0.93, + "content": "\\mathbf { F } _ { n } \\in ( B B ^ { * } ) ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 448, + 599, + 451, + 614 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 32, + "bbox_fs": [ + 105, + 599, + 451, + 614 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 627, + 505, + 652 + ], + "lines": [ + { + "bbox": [ + 106, + 627, + 505, + 640 + ], + "spans": [ + { + "bbox": [ + 106, + 627, + 280, + 640 + ], + "score": 1.0, + "content": "Proof. From Parker (1995), we can express", + "type": "text" + }, + { + "bbox": [ + 281, + 628, + 294, + 639 + ], + "score": 0.89, + "content": "{ \\bf F } _ { n }", + "type": "inline_equation" + }, + { + "bbox": [ + 295, + 627, + 306, + 640 + ], + "score": 1.0, + "content": "as", + "type": "text" + }, + { + "bbox": [ + 306, + 628, + 352, + 639 + ], + "score": 0.9, + "content": "{ \\bf F } _ { n } = { \\bf B } { \\bf P }", + "type": "inline_equation" + }, + { + "bbox": [ + 353, + 627, + 382, + 640 + ], + "score": 1.0, + "content": ", where", + "type": "text" + }, + { + "bbox": [ + 383, + 628, + 411, + 638 + ], + "score": 0.91, + "content": "\\mathbf { B } \\in B", + "type": "inline_equation" + }, + { + "bbox": [ + 411, + 627, + 428, + 640 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 429, + 628, + 438, + 638 + ], + "score": 0.77, + "content": "\\mathbf { P }", + "type": "inline_equation" + }, + { + "bbox": [ + 438, + 627, + 505, + 640 + ], + "score": 1.0, + "content": "is a permutation", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 637, + 506, + 652 + ], + "spans": [ + { + "bbox": [ + 105, + 637, + 304, + 652 + ], + "score": 1.0, + "content": "(the bit reversal permutation). From Theorem 2,", + "type": "text" + }, + { + "bbox": [ + 304, + 639, + 344, + 650 + ], + "score": 0.89, + "content": "\\mathbf { P } \\in B B ^ { * }", + "type": "inline_equation" + }, + { + "bbox": [ + 344, + 637, + 446, + 652 + ], + "score": 1.0, + "content": ". Hence, by Lemma H.2,", + "type": "text" + }, + { + "bbox": [ + 446, + 639, + 502, + 651 + ], + "score": 0.89, + "content": "\\mathbf { F } _ { n } ^ { - } \\in ( B B ^ { * } ) ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 503, + 637, + 506, + 652 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 33.5, + "bbox_fs": [ + 105, + 627, + 506, + 652 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 671, + 419, + 683 + ], + "lines": [ + { + "bbox": [ + 105, + 671, + 420, + 685 + ], + "spans": [ + { + "bbox": [ + 105, + 671, + 176, + 685 + ], + "score": 1.0, + "content": "Lemma J.2. Let", + "type": "text" + }, + { + "bbox": [ + 177, + 672, + 192, + 682 + ], + "score": 0.87, + "content": "\\mathbf { H } _ { n }", + "type": "inline_equation" + }, + { + "bbox": [ + 192, + 671, + 336, + 685 + ], + "score": 1.0, + "content": "be the Hadamard Transform of size", + "type": "text" + }, + { + "bbox": [ + 337, + 674, + 344, + 681 + ], + "score": 0.44, + "content": "n", + "type": "inline_equation" + }, + { + "bbox": [ + 344, + 671, + 371, + 685 + ], + "score": 1.0, + "content": ". Then", + "type": "text" + }, + { + "bbox": [ + 371, + 672, + 416, + 682 + ], + "score": 0.91, + "content": "\\mathbf { H } _ { n } \\in B B ^ { * }", + "type": "inline_equation" + }, + { + "bbox": [ + 416, + 671, + 420, + 685 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 35, + "bbox_fs": [ + 105, + 671, + 420, + 685 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 698, + 268, + 711 + ], + "lines": [ + { + "bbox": [ + 105, + 698, + 266, + 712 + ], + "spans": [ + { + "bbox": [ + 105, + 698, + 136, + 712 + ], + "score": 1.0, + "content": "Proof.", + "type": "text" + }, + { + "bbox": [ + 136, + 699, + 171, + 710 + ], + "score": 0.91, + "content": "{ \\mathbf { H } } _ { n } \\in { \\mathfrak { B } }", + "type": "inline_equation" + }, + { + "bbox": [ + 171, + 698, + 219, + 712 + ], + "score": 1.0, + "content": ", so trivially", + "type": "text" + }, + { + "bbox": [ + 220, + 699, + 266, + 710 + ], + "score": 0.92, + "content": "\\mathbf { H } _ { n } \\in B B ^ { * }", + "type": "inline_equation" + } + ], + "index": 36 + } + ], + "index": 36, + "bbox_fs": [ + 105, + 698, + 266, + 712 + ] + }, + { + "type": "text", + "bbox": [ + 105, + 720, + 436, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 719, + 437, + 734 + ], + "spans": [ + { + "bbox": [ + 106, + 719, + 176, + 734 + ], + "score": 1.0, + "content": "Lemma J.3. Let", + "type": "text" + }, + { + "bbox": [ + 177, + 721, + 189, + 732 + ], + "score": 0.88, + "content": "\\mathbf { S } _ { n }", + "type": "inline_equation" + }, + { + "bbox": [ + 190, + 719, + 344, + 734 + ], + "score": 1.0, + "content": "be the Discrete Sine Transform of size", + "type": "text" + }, + { + "bbox": [ + 345, + 723, + 352, + 730 + ], + "score": 0.49, + "content": "n", + "type": "inline_equation" + }, + { + "bbox": [ + 352, + 719, + 378, + 734 + ], + "score": 1.0, + "content": ". Then", + "type": "text" + }, + { + "bbox": [ + 379, + 720, + 434, + 732 + ], + "score": 0.93, + "content": "\\mathbf { S } _ { n } \\in ( B B ^ { * } ) ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 434, + 719, + 437, + 734 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 37 + } + ], + "index": 37, + "bbox_fs": [ + 106, + 719, + 437, + 734 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 106, + 81, + 506, + 149 + ], + "lines": [ + { + "bbox": [ + 105, + 81, + 506, + 96 + ], + "spans": [ + { + "bbox": [ + 105, + 81, + 272, + 96 + ], + "score": 1.0, + "content": "Proof. As described in Makhoul (1980),", + "type": "text" + }, + { + "bbox": [ + 272, + 83, + 286, + 94 + ], + "score": 0.88, + "content": "\\mathbf { S } _ { n }", + "type": "inline_equation" + }, + { + "bbox": [ + 286, + 81, + 506, + 96 + ], + "score": 1.0, + "content": "can be performed as a scaled permutation (separating", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 104, + 92, + 507, + 108 + ], + "spans": [ + { + "bbox": [ + 104, + 92, + 490, + 108 + ], + "score": 1.0, + "content": "the even and odd indices of the input, and reversing and negating the odd indices) composed with", + "type": "text" + }, + { + "bbox": [ + 490, + 94, + 502, + 105 + ], + "score": 0.86, + "content": "\\mathbb { F } _ { n }", + "type": "inline_equation" + }, + { + "bbox": [ + 503, + 92, + 507, + 108 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 106, + 104, + 506, + 117 + ], + "spans": [ + { + "bbox": [ + 106, + 104, + 234, + 117 + ], + "score": 1.0, + "content": "Therefore, we may decompose", + "type": "text" + }, + { + "bbox": [ + 234, + 105, + 247, + 115 + ], + "score": 0.89, + "content": "\\mathbf { S } _ { n }", + "type": "inline_equation" + }, + { + "bbox": [ + 248, + 104, + 259, + 117 + ], + "score": 1.0, + "content": "as", + "type": "text" + }, + { + "bbox": [ + 260, + 105, + 338, + 116 + ], + "score": 0.31, + "content": "\\mathbf { S } _ { n } = \\mathbf { B } ~ \\mathbf { P } _ { 2 } ~ \\mathbf { D } ~ \\mathbf { P } _ { 1 } ,", + "type": "inline_equation" + }, + { + "bbox": [ + 338, + 104, + 368, + 117 + ], + "score": 1.0, + "content": ", where", + "type": "text" + }, + { + "bbox": [ + 369, + 105, + 399, + 116 + ], + "score": 0.91, + "content": "\\mathbf { P } _ { 1 } , \\mathbf { P } _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 399, + 104, + 473, + 117 + ], + "score": 1.0, + "content": "are permutations,", + "type": "text" + }, + { + "bbox": [ + 473, + 105, + 502, + 115 + ], + "score": 0.9, + "content": "\\mathbf { B } \\in B", + "type": "inline_equation" + }, + { + "bbox": [ + 503, + 104, + 506, + 117 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 114, + 506, + 129 + ], + "spans": [ + { + "bbox": [ + 105, + 114, + 123, + 129 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 123, + 115, + 133, + 126 + ], + "score": 0.51, + "content": "\\mathbf { D }", + "type": "inline_equation" + }, + { + "bbox": [ + 134, + 114, + 218, + 129 + ], + "score": 1.0, + "content": "is a diagonal matrix.", + "type": "text" + }, + { + "bbox": [ + 218, + 116, + 232, + 126 + ], + "score": 0.72, + "content": "\\mathbf { P } _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 232, + 114, + 244, + 129 + ], + "score": 1.0, + "content": "D", + "type": "text" + }, + { + "bbox": [ + 245, + 116, + 258, + 127 + ], + "score": 0.73, + "content": "\\mathbf { P } _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 259, + 114, + 506, + 129 + ], + "score": 1.0, + "content": "is simply a permutation matrix with scaled entries, which can", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 126, + 507, + 139 + ], + "spans": [ + { + "bbox": [ + 105, + 126, + 218, + 139 + ], + "score": 1.0, + "content": "be equivalently expressed as", + "type": "text" + }, + { + "bbox": [ + 219, + 127, + 245, + 137 + ], + "score": 0.88, + "content": "\\mathbf { D } ^ { \\prime } \\mathbf { P } ^ { \\prime }", + "type": "inline_equation" + }, + { + "bbox": [ + 245, + 126, + 345, + 139 + ], + "score": 1.0, + "content": "for some diagonal matrix", + "type": "text" + }, + { + "bbox": [ + 346, + 126, + 358, + 137 + ], + "score": 0.87, + "content": "\\mathbf { D ^ { \\prime } }", + "type": "inline_equation" + }, + { + "bbox": [ + 359, + 126, + 425, + 139 + ], + "score": 1.0, + "content": "and permutation", + "type": "text" + }, + { + "bbox": [ + 425, + 126, + 437, + 136 + ], + "score": 0.84, + "content": "\\mathbf { P ^ { \\prime } }", + "type": "inline_equation" + }, + { + "bbox": [ + 437, + 126, + 507, + 139 + ], + "score": 1.0, + "content": ". By Lemma H.1,", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 109, + 135, + 506, + 151 + ], + "spans": [ + { + "bbox": [ + 109, + 137, + 160, + 148 + ], + "score": 0.9, + "content": "\\mathbf { B } \\bar { \\mathbf { D ^ { \\prime } } } \\in B B ^ { * }", + "type": "inline_equation" + }, + { + "bbox": [ + 161, + 135, + 227, + 151 + ], + "score": 1.0, + "content": ". By Theorem 2,", + "type": "text" + }, + { + "bbox": [ + 227, + 137, + 269, + 148 + ], + "score": 0.91, + "content": "\\mathbf { P } ^ { \\prime } \\in B B ^ { * }", + "type": "inline_equation" + }, + { + "bbox": [ + 270, + 135, + 369, + 151 + ], + "score": 1.0, + "content": ". Hence, by Lemma H.2,", + "type": "text" + }, + { + "bbox": [ + 370, + 137, + 425, + 149 + ], + "score": 0.93, + "content": "\\bar { \\mathbf { S } _ { n } } \\in ( B B ^ { * } ) ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 425, + 135, + 429, + 151 + ], + "score": 1.0, + "content": ".", + "type": "text" + }, + { + "bbox": [ + 494, + 137, + 506, + 149 + ], + "score": 0.999, + "content": "□", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 2.5 + }, + { + "type": "text", + "bbox": [ + 106, + 162, + 501, + 175 + ], + "lines": [ + { + "bbox": [ + 106, + 161, + 500, + 177 + ], + "spans": [ + { + "bbox": [ + 106, + 161, + 469, + 177 + ], + "score": 1.0, + "content": "Remark J.4. An analogous argument shows that the Discrete Cosine Transform is also in", + "type": "text" + }, + { + "bbox": [ + 470, + 163, + 500, + 175 + ], + "score": 0.91, + "content": "( B B ^ { * } ) ^ { 2 }", + "type": "inline_equation" + } + ], + "index": 6 + } + ], + "index": 6 + }, + { + "type": "text", + "bbox": [ + 107, + 179, + 439, + 191 + ], + "lines": [ + { + "bbox": [ + 106, + 178, + 440, + 192 + ], + "spans": [ + { + "bbox": [ + 106, + 178, + 176, + 192 + ], + "score": 1.0, + "content": "Lemma J.5. Let", + "type": "text" + }, + { + "bbox": [ + 177, + 180, + 191, + 190 + ], + "score": 0.88, + "content": "\\mathbf { C } _ { n }", + "type": "inline_equation" + }, + { + "bbox": [ + 192, + 178, + 217, + 192 + ], + "score": 1.0, + "content": "be an", + "type": "text" + }, + { + "bbox": [ + 217, + 181, + 243, + 190 + ], + "score": 0.89, + "content": "n \\times n", + "type": "inline_equation" + }, + { + "bbox": [ + 243, + 178, + 392, + 192 + ], + "score": 1.0, + "content": "circulant (convolution) matrix. Then", + "type": "text" + }, + { + "bbox": [ + 392, + 179, + 436, + 190 + ], + "score": 0.91, + "content": "\\mathbf { C } _ { n } \\in B B ^ { * }", + "type": "inline_equation" + }, + { + "bbox": [ + 437, + 178, + 440, + 192 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 7 + }, + { + "type": "text", + "bbox": [ + 106, + 211, + 505, + 250 + ], + "lines": [ + { + "bbox": [ + 105, + 210, + 504, + 226 + ], + "spans": [ + { + "bbox": [ + 105, + 210, + 350, + 226 + ], + "score": 1.0, + "content": "Proof. Using Theorem 2.6.4 of Pan (2001), we can express", + "type": "text" + }, + { + "bbox": [ + 351, + 212, + 366, + 223 + ], + "score": 0.89, + "content": "\\mathbf { C } _ { n }", + "type": "inline_equation" + }, + { + "bbox": [ + 366, + 210, + 378, + 226 + ], + "score": 1.0, + "content": "as", + "type": "text" + }, + { + "bbox": [ + 378, + 210, + 461, + 224 + ], + "score": 0.93, + "content": "\\mathbf { C } _ { n } = ( \\mathbf { F } _ { n } ) ^ { - 1 } \\mathbf { D } \\mathbf { F } _ { n }", + "type": "inline_equation" + }, + { + "bbox": [ + 461, + 210, + 489, + 226 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 490, + 212, + 504, + 224 + ], + "score": 0.88, + "content": "{ \\bf F } _ { n }", + "type": "inline_equation" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 224, + 506, + 239 + ], + "spans": [ + { + "bbox": [ + 105, + 224, + 263, + 239 + ], + "score": 1.0, + "content": "is the Discrete Fourier Transform and", + "type": "text" + }, + { + "bbox": [ + 263, + 226, + 273, + 235 + ], + "score": 0.66, + "content": "\\mathbf { D }", + "type": "inline_equation" + }, + { + "bbox": [ + 274, + 224, + 362, + 239 + ], + "score": 1.0, + "content": "is a diagonal matrix.", + "type": "text" + }, + { + "bbox": [ + 363, + 224, + 428, + 237 + ], + "score": 0.89, + "content": "\\left( \\mathbf { F } _ { n } \\right) ^ { - 1 } = \\mathbf { B } \\mathbf { \\Lambda } \\mathbf { P }", + "type": "inline_equation" + }, + { + "bbox": [ + 429, + 224, + 453, + 239 + ], + "score": 1.0, + "content": "(with", + "type": "text" + }, + { + "bbox": [ + 454, + 225, + 483, + 236 + ], + "score": 0.62, + "content": "\\mathbf { B } \\in B", + "type": "inline_equation" + }, + { + "bbox": [ + 484, + 224, + 487, + 239 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 487, + 226, + 497, + 236 + ], + "score": 0.55, + "content": "\\mathbf { P }", + "type": "inline_equation" + }, + { + "bbox": [ + 497, + 224, + 506, + 239 + ], + "score": 1.0, + "content": "a", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 104, + 236, + 368, + 252 + ], + "spans": [ + { + "bbox": [ + 104, + 237, + 239, + 252 + ], + "score": 1.0, + "content": "permutation), which implies that", + "type": "text" + }, + { + "bbox": [ + 239, + 236, + 321, + 250 + ], + "score": 0.94, + "content": "{ \\bf F } _ { n } = \\left( { \\bf P } \\right) ^ { - 1 } \\left( { \\bf B } \\right) ^ { - 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 321, + 237, + 368, + 252 + ], + "score": 1.0, + "content": ". Therefore", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 9 + }, + { + "type": "interline_equation", + "bbox": [ + 244, + 257, + 367, + 273 + ], + "lines": [ + { + "bbox": [ + 244, + 257, + 367, + 273 + ], + "spans": [ + { + "bbox": [ + 244, + 257, + 367, + 273 + ], + "score": 0.91, + "content": "{ \\bf C } _ { n } = { \\bf B } { \\bf P } { \\bf D } { \\bf \\Lambda } \\left( { \\bf P } \\right) ^ { - 1 } \\left( { \\bf B } \\right) ^ { - 1 } .", + "type": "interline_equation", + "image_path": "2093a282bede9cd9f85441018ef319fec57948e6c681841b22b1dae5d248b8fc.jpg" + } + ] + } + ], + "index": 11, + "virtual_lines": [ + { + "bbox": [ + 244, + 257, + 367, + 273 + ], + "spans": [], + "index": 11 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 280, + 506, + 314 + ], + "lines": [ + { + "bbox": [ + 106, + 281, + 505, + 293 + ], + "spans": [ + { + "bbox": [ + 106, + 281, + 505, + 293 + ], + "score": 1.0, + "content": "The middle three factors have the effect of performing a permutation, scaling each element, and", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 290, + 507, + 306 + ], + "spans": [ + { + "bbox": [ + 105, + 290, + 459, + 306 + ], + "score": 1.0, + "content": "undoing the permutation, which is equivalent to simply scaling by some diagonal matrix", + "type": "text" + }, + { + "bbox": [ + 459, + 292, + 471, + 302 + ], + "score": 0.84, + "content": "\\mathbf { D ^ { \\prime } }", + "type": "inline_equation" + }, + { + "bbox": [ + 472, + 290, + 507, + 306 + ], + "score": 1.0, + "content": ". Hence,", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 303, + 171, + 314 + ], + "spans": [ + { + "bbox": [ + 105, + 303, + 171, + 314 + ], + "score": 1.0, + "content": "we are left with", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 13 + }, + { + "type": "interline_equation", + "bbox": [ + 263, + 314, + 348, + 330 + ], + "lines": [ + { + "bbox": [ + 263, + 314, + 348, + 330 + ], + "spans": [ + { + "bbox": [ + 263, + 314, + 348, + 330 + ], + "score": 0.91, + "content": "\\mathbf { C } _ { n } = \\mathbf { B } \\mathbf { D } ^ { \\prime } \\left( \\mathbf { B } \\right) ^ { - 1 } .", + "type": "interline_equation", + "image_path": "bc6f96999e33bad0feb03e49b8f25e8891094d64091f542f506f8ec7dd10f1e3.jpg" + } + ] + } + ], + "index": 15, + "virtual_lines": [ + { + "bbox": [ + 263, + 314, + 348, + 330 + ], + "spans": [], + "index": 15 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 336, + 421, + 351 + ], + "lines": [ + { + "bbox": [ + 105, + 335, + 418, + 352 + ], + "spans": [ + { + "bbox": [ + 105, + 335, + 181, + 352 + ], + "score": 1.0, + "content": "By Lemma H.1, B", + "type": "text" + }, + { + "bbox": [ + 181, + 338, + 216, + 349 + ], + "score": 0.71, + "content": "\\mathbf { D } ^ { \\prime } \\in B", + "type": "inline_equation" + }, + { + "bbox": [ + 216, + 335, + 287, + 352 + ], + "score": 1.0, + "content": ". By Lemma H.6,", + "type": "text" + }, + { + "bbox": [ + 288, + 336, + 338, + 350 + ], + "score": 0.92, + "content": "\\left( \\mathbf { B } \\right) ^ { - 1 } \\in B ^ { * }", + "type": "inline_equation" + }, + { + "bbox": [ + 338, + 335, + 373, + 352 + ], + "score": 1.0, + "content": ". Hence,", + "type": "text" + }, + { + "bbox": [ + 373, + 338, + 418, + 349 + ], + "score": 0.9, + "content": "\\mathbf { C } _ { n } \\in B B ^ { * }", + "type": "inline_equation" + } + ], + "index": 16 + } + ], + "index": 16 + }, + { + "type": "text", + "bbox": [ + 107, + 363, + 506, + 387 + ], + "lines": [ + { + "bbox": [ + 106, + 363, + 505, + 376 + ], + "spans": [ + { + "bbox": [ + 106, + 363, + 244, + 376 + ], + "score": 1.0, + "content": "Remark J.6. We can expand any", + "type": "text" + }, + { + "bbox": [ + 245, + 365, + 270, + 374 + ], + "score": 0.89, + "content": "n \\times n", + "type": "inline_equation" + }, + { + "bbox": [ + 271, + 363, + 335, + 376 + ], + "score": 1.0, + "content": "Toeplitz matrix", + "type": "text" + }, + { + "bbox": [ + 335, + 364, + 349, + 375 + ], + "score": 0.88, + "content": "\\mathbf { T } _ { n }", + "type": "inline_equation" + }, + { + "bbox": [ + 349, + 363, + 376, + 376 + ], + "score": 1.0, + "content": "into a", + "type": "text" + }, + { + "bbox": [ + 377, + 364, + 412, + 374 + ], + "score": 0.9, + "content": "2 n \\times 2 n", + "type": "inline_equation" + }, + { + "bbox": [ + 413, + 363, + 505, + 376 + ], + "score": 1.0, + "content": "circulant matrix (with", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 375, + 423, + 387 + ], + "spans": [ + { + "bbox": [ + 106, + 375, + 147, + 387 + ], + "score": 1.0, + "content": "upper left", + "type": "text" + }, + { + "bbox": [ + 147, + 376, + 173, + 385 + ], + "score": 0.88, + "content": "n \\times n", + "type": "inline_equation" + }, + { + "bbox": [ + 173, + 375, + 251, + 387 + ], + "score": 1.0, + "content": "submatrix equal to", + "type": "text" + }, + { + "bbox": [ + 251, + 375, + 266, + 386 + ], + "score": 0.86, + "content": "\\mathbf { T } _ { n }", + "type": "inline_equation" + }, + { + "bbox": [ + 266, + 375, + 304, + 387 + ], + "score": 1.0, + "content": "). Hence,", + "type": "text" + }, + { + "bbox": [ + 304, + 375, + 361, + 387 + ], + "score": 0.93, + "content": "\\mathbf { T } _ { n } \\in ( B B ^ { * } ) _ { 2 } ^ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 362, + 375, + 423, + 387 + ], + "score": 1.0, + "content": "by Lemma J.5.", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 17.5 + }, + { + "type": "text", + "bbox": [ + 106, + 396, + 459, + 408 + ], + "lines": [ + { + "bbox": [ + 105, + 396, + 460, + 410 + ], + "spans": [ + { + "bbox": [ + 105, + 396, + 396, + 410 + ], + "score": 1.0, + "content": "The Fastfood matrix class (Le et al., 2013) can be tightly captured in the", + "type": "text" + }, + { + "bbox": [ + 397, + 397, + 416, + 407 + ], + "score": 0.88, + "content": "B B ^ { * }", + "type": "inline_equation" + }, + { + "bbox": [ + 416, + 396, + 460, + 410 + ], + "score": 1.0, + "content": "hierarchy:", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 19 + }, + { + "type": "text", + "bbox": [ + 106, + 412, + 506, + 436 + ], + "lines": [ + { + "bbox": [ + 106, + 412, + 506, + 425 + ], + "spans": [ + { + "bbox": [ + 106, + 412, + 292, + 425 + ], + "score": 1.0, + "content": "Lemma J.7. The product SHDPHB where", + "type": "text" + }, + { + "bbox": [ + 293, + 413, + 326, + 424 + ], + "score": 0.51, + "content": "\\mathbf { S } , \\mathbf { D } , \\mathbf { B }", + "type": "inline_equation" + }, + { + "bbox": [ + 326, + 412, + 422, + 425 + ], + "score": 1.0, + "content": "are diagonal matrices,", + "type": "text" + }, + { + "bbox": [ + 423, + 413, + 433, + 423 + ], + "score": 0.43, + "content": "\\mathbf { H }", + "type": "inline_equation" + }, + { + "bbox": [ + 433, + 412, + 506, + 425 + ], + "score": 1.0, + "content": "is the Hadamard", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 423, + 331, + 437 + ], + "spans": [ + { + "bbox": [ + 105, + 423, + 168, + 437 + ], + "score": 1.0, + "content": "transform, and", + "type": "text" + }, + { + "bbox": [ + 168, + 425, + 177, + 434 + ], + "score": 0.45, + "content": "\\mathbf { P }", + "type": "inline_equation" + }, + { + "bbox": [ + 178, + 423, + 297, + 437 + ], + "score": 1.0, + "content": "is a permutation matrix, is in", + "type": "text" + }, + { + "bbox": [ + 297, + 424, + 327, + 436 + ], + "score": 0.91, + "content": "( B B ^ { * } ) ^ { 3 }", + "type": "inline_equation" + }, + { + "bbox": [ + 328, + 423, + 331, + 437 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 20.5 + }, + { + "type": "text", + "bbox": [ + 106, + 455, + 505, + 478 + ], + "lines": [ + { + "bbox": [ + 106, + 455, + 505, + 468 + ], + "spans": [ + { + "bbox": [ + 106, + 455, + 273, + 468 + ], + "score": 1.0, + "content": "Proof. We have shown in Lemma J.2 that", + "type": "text" + }, + { + "bbox": [ + 274, + 456, + 314, + 466 + ], + "score": 0.91, + "content": "\\mathbf { H } \\in B B ^ { * }", + "type": "inline_equation" + }, + { + "bbox": [ + 314, + 455, + 407, + 468 + ], + "score": 1.0, + "content": ", and in Theorem 2 that", + "type": "text" + }, + { + "bbox": [ + 407, + 456, + 446, + 466 + ], + "score": 0.91, + "content": "\\mathbf { P } \\in B B ^ { * }", + "type": "inline_equation" + }, + { + "bbox": [ + 447, + 455, + 475, + 468 + ], + "score": 1.0, + "content": ". Since", + "type": "text" + }, + { + "bbox": [ + 475, + 456, + 495, + 466 + ], + "score": 0.89, + "content": "B B ^ { * }", + "type": "inline_equation" + }, + { + "bbox": [ + 495, + 455, + 505, + 468 + ], + "score": 1.0, + "content": "is", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 465, + 502, + 479 + ], + "spans": [ + { + "bbox": [ + 105, + 465, + 384, + 479 + ], + "score": 1.0, + "content": "closed under diagonal multiplication (Lemma H.1), we conclude that", + "type": "text" + }, + { + "bbox": [ + 384, + 467, + 477, + 479 + ], + "score": 0.56, + "content": "\\mathbf { S H D P H B } \\in ( B B ^ { * } ) ^ { 3 }", + "type": "inline_equation" + }, + { + "bbox": [ + 477, + 465, + 488, + 479 + ], + "score": 1.0, + "content": ".", + "type": "text" + }, + { + "bbox": [ + 496, + 469, + 502, + 475 + ], + "score": 0.0, + "content": "", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 22.5 + }, + { + "type": "text", + "bbox": [ + 108, + 497, + 505, + 521 + ], + "lines": [ + { + "bbox": [ + 106, + 497, + 505, + 510 + ], + "spans": [ + { + "bbox": [ + 106, + 497, + 505, + 510 + ], + "score": 1.0, + "content": "The two classes of matrices introduced in Moczulski et al. (2016), called AFDF and ACDC, are also", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 509, + 261, + 522 + ], + "spans": [ + { + "bbox": [ + 106, + 509, + 196, + 522 + ], + "score": 1.0, + "content": "tightly captured in the", + "type": "text" + }, + { + "bbox": [ + 197, + 509, + 216, + 520 + ], + "score": 0.88, + "content": "B B ^ { * }", + "type": "inline_equation" + }, + { + "bbox": [ + 216, + 509, + 261, + 522 + ], + "score": 1.0, + "content": "hierarchy:", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 24.5 + }, + { + "type": "text", + "bbox": [ + 107, + 525, + 504, + 548 + ], + "lines": [ + { + "bbox": [ + 105, + 524, + 502, + 538 + ], + "spans": [ + { + "bbox": [ + 105, + 524, + 176, + 538 + ], + "score": 1.0, + "content": "Lemma J.8. Let", + "type": "text" + }, + { + "bbox": [ + 176, + 525, + 219, + 536 + ], + "score": 0.75, + "content": "\\mathbf { A } \\mathbf { F } ^ { - 1 } \\mathbf { D } \\mathbf { F }", + "type": "inline_equation" + }, + { + "bbox": [ + 220, + 524, + 483, + 538 + ], + "score": 1.0, + "content": "be a product of a diagonal matrix A, the inverse Fourier transform", + "type": "text" + }, + { + "bbox": [ + 484, + 525, + 502, + 536 + ], + "score": 0.89, + "content": "\\mathbf { F } ^ { - 1 }", + "type": "inline_equation" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 536, + 439, + 549 + ], + "spans": [ + { + "bbox": [ + 105, + 536, + 362, + 549 + ], + "score": 1.0, + "content": "another diagonal matrix D, and the Fourier transform F. Then", + "type": "text" + }, + { + "bbox": [ + 362, + 536, + 435, + 547 + ], + "score": 0.89, + "content": "\\mathbf { A } \\mathbf { F } ^ { - 1 } \\mathbf { D } \\mathbf { F } \\in B B ^ { * }", + "type": "inline_equation" + }, + { + "bbox": [ + 435, + 536, + 439, + 549 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 26.5 + }, + { + "type": "text", + "bbox": [ + 106, + 552, + 505, + 576 + ], + "lines": [ + { + "bbox": [ + 105, + 552, + 506, + 566 + ], + "spans": [ + { + "bbox": [ + 105, + 552, + 122, + 566 + ], + "score": 1.0, + "content": "Let", + "type": "text" + }, + { + "bbox": [ + 122, + 552, + 168, + 564 + ], + "score": 0.83, + "content": "\\mathbf { A C } ^ { - 1 } \\mathbf { D C }", + "type": "inline_equation" + }, + { + "bbox": [ + 169, + 552, + 446, + 566 + ], + "score": 1.0, + "content": "be a product of a diagonal matrix A, the inverse cosine transform", + "type": "text" + }, + { + "bbox": [ + 446, + 553, + 466, + 564 + ], + "score": 0.89, + "content": "\\mathbf { C } ^ { - 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 466, + 552, + 506, + 566 + ], + "score": 1.0, + "content": ", another", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 561, + 416, + 577 + ], + "spans": [ + { + "bbox": [ + 105, + 561, + 325, + 577 + ], + "score": 1.0, + "content": "diagonal matrix D, and the cosine transform C. Then", + "type": "text" + }, + { + "bbox": [ + 325, + 564, + 412, + 577 + ], + "score": 0.9, + "content": "\\mathbf { A } \\mathbf { C } ^ { - 1 } \\mathbf { D } \\mathbf { C } \\in ( B B ^ { * } ) ^ { 4 }", + "type": "inline_equation" + }, + { + "bbox": [ + 412, + 561, + 416, + 577 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 28.5 + }, + { + "type": "text", + "bbox": [ + 107, + 595, + 504, + 618 + ], + "lines": [ + { + "bbox": [ + 105, + 594, + 505, + 609 + ], + "spans": [ + { + "bbox": [ + 105, + 594, + 282, + 609 + ], + "score": 1.0, + "content": "Proof. We have argued in Lemma J.5 that", + "type": "text" + }, + { + "bbox": [ + 283, + 596, + 350, + 606 + ], + "score": 0.9, + "content": "\\mathbf { F } ^ { - 1 } \\mathbf { D } \\mathbf { F } \\in B B ^ { * }", + "type": "inline_equation" + }, + { + "bbox": [ + 350, + 594, + 381, + 609 + ], + "score": 1.0, + "content": ". Since", + "type": "text" + }, + { + "bbox": [ + 381, + 596, + 401, + 606 + ], + "score": 0.89, + "content": "B B ^ { * }", + "type": "inline_equation" + }, + { + "bbox": [ + 401, + 594, + 505, + 609 + ], + "score": 1.0, + "content": "is closed under diagonal", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 606, + 372, + 619 + ], + "spans": [ + { + "bbox": [ + 105, + 606, + 295, + 619 + ], + "score": 1.0, + "content": "multiplication (Lemma H.1), we conclude that", + "type": "text" + }, + { + "bbox": [ + 295, + 607, + 368, + 618 + ], + "score": 0.9, + "content": "\\mathbf { A } \\mathbf { F } ^ { - 1 } \\mathbf { D } \\mathbf { F } \\in B B ^ { * }", + "type": "inline_equation" + }, + { + "bbox": [ + 368, + 606, + 372, + 619 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 30.5 + }, + { + "type": "text", + "bbox": [ + 106, + 622, + 502, + 647 + ], + "lines": [ + { + "bbox": [ + 105, + 622, + 504, + 636 + ], + "spans": [ + { + "bbox": [ + 105, + 622, + 190, + 636 + ], + "score": 1.0, + "content": "We have shown that", + "type": "text" + }, + { + "bbox": [ + 191, + 623, + 243, + 636 + ], + "score": 0.92, + "content": "\\mathbf { C } \\in ( B B ^ { * } ) ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 243, + 622, + 258, + 636 + ], + "score": 1.0, + "content": ", so", + "type": "text" + }, + { + "bbox": [ + 259, + 623, + 322, + 636 + ], + "score": 0.92, + "content": "\\mathbf { C } ^ { - 1 } \\in ( B B ^ { * } ) ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 322, + 622, + 383, + 636 + ], + "score": 1.0, + "content": "as well. Since", + "type": "text" + }, + { + "bbox": [ + 383, + 624, + 402, + 634 + ], + "score": 0.88, + "content": "B B ^ { * }", + "type": "inline_equation" + }, + { + "bbox": [ + 403, + 622, + 504, + 636 + ], + "score": 1.0, + "content": "is closed under diagonal", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 633, + 501, + 648 + ], + "spans": [ + { + "bbox": [ + 105, + 633, + 295, + 648 + ], + "score": 1.0, + "content": "multiplication (Lemma H.1), we conclude that", + "type": "text" + }, + { + "bbox": [ + 295, + 635, + 382, + 647 + ], + "score": 0.89, + "content": "\\mathbf { A C } ^ { - 1 } \\mathbf { D C } \\in ( B B ^ { * } ) ^ { 4 }", + "type": "inline_equation" + }, + { + "bbox": [ + 382, + 633, + 386, + 648 + ], + "score": 1.0, + "content": ".", + "type": "text" + }, + { + "bbox": [ + 497, + 640, + 501, + 643 + ], + "score": 0.0, + "content": "", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 32.5 + }, + { + "type": "text", + "bbox": [ + 106, + 660, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 660, + 506, + 673 + ], + "spans": [ + { + "bbox": [ + 106, + 660, + 506, + 673 + ], + "score": 1.0, + "content": "Remark J.9. Within each butterfly factor matrix of the DFT (excluding the bit reversal permutation)", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 671, + 506, + 685 + ], + "spans": [ + { + "bbox": [ + 105, + 671, + 506, + 685 + ], + "score": 1.0, + "content": "and the Hadamard transform, the columns are pairwise orthogonal and have norm 2. Hence, we can√", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 682, + 506, + 696 + ], + "spans": [ + { + "bbox": [ + 106, + 682, + 187, + 696 + ], + "score": 1.0, + "content": "divide all factors by", + "type": "text" + }, + { + "bbox": [ + 187, + 682, + 201, + 695 + ], + "score": 0.86, + "content": "\\sqrt { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 201, + 682, + 506, + 696 + ], + "score": 1.0, + "content": "to make orthogonal factor matrices. To counteract this scaling, we can add a", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 104, + 694, + 506, + 713 + ], + "spans": [ + { + "bbox": [ + 104, + 694, + 236, + 712 + ], + "score": 1.0, + "content": "diagonal matrix with 2log2(n)", + "type": "text" + }, + { + "bbox": [ + 193, + 695, + 261, + 711 + ], + "score": 0.9, + "content": "{ \\sqrt { 2 } } ^ { \\log _ { 2 } ( n ) } = { \\sqrt { n } }", + "type": "inline_equation" + }, + { + "bbox": [ + 231, + 696, + 506, + 713 + ], + "score": 1.0, + "content": "= n in all entries to the factorization. By doing this we can place", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 709, + 506, + 722 + ], + "spans": [ + { + "bbox": [ + 105, + 709, + 392, + 722 + ], + "score": 1.0, + "content": "all of the above transforms in the OBB hierarchy (defined in Appendix", + "type": "text" + }, + { + "bbox": [ + 392, + 711, + 401, + 720 + ], + "score": 0.41, + "content": "K _ { \\cdot }", + "type": "inline_equation" + }, + { + "bbox": [ + 401, + 709, + 506, + 722 + ], + "score": 1.0, + "content": ") with the same width and", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 106, + 721, + 176, + 734 + ], + "spans": [ + { + "bbox": [ + 106, + 721, + 176, + 734 + ], + "score": 1.0, + "content": "expansion factor.", + "type": "text" + } + ], + "index": 39 + } + ], + "index": 36.5 + } + ], + "page_idx": 33, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 107, + 27, + 293, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 26, + 294, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 26, + 294, + 38 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2020", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 300, + 751, + 311, + 760 + ], + "lines": [ + { + "bbox": [ + 298, + 750, + 313, + 764 + ], + "spans": [ + { + "bbox": [ + 298, + 750, + 313, + 764 + ], + "score": 1.0, + "content": "", + "type": "text", + "height": 14, + "width": 15 + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 495, + 338, + 504, + 348 + ], + "lines": [ + { + "bbox": [ + 496, + 340, + 504, + 348 + ], + "spans": [ + { + "bbox": [ + 496, + 340, + 504, + 348 + ], + "score": 0.997, + "content": "□", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 106, + 81, + 506, + 149 + ], + "lines": [ + { + "bbox": [ + 105, + 81, + 506, + 96 + ], + "spans": [ + { + "bbox": [ + 105, + 81, + 272, + 96 + ], + "score": 1.0, + "content": "Proof. As described in Makhoul (1980),", + "type": "text" + }, + { + "bbox": [ + 272, + 83, + 286, + 94 + ], + "score": 0.88, + "content": "\\mathbf { S } _ { n }", + "type": "inline_equation" + }, + { + "bbox": [ + 286, + 81, + 506, + 96 + ], + "score": 1.0, + "content": "can be performed as a scaled permutation (separating", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 104, + 92, + 507, + 108 + ], + "spans": [ + { + "bbox": [ + 104, + 92, + 490, + 108 + ], + "score": 1.0, + "content": "the even and odd indices of the input, and reversing and negating the odd indices) composed with", + "type": "text" + }, + { + "bbox": [ + 490, + 94, + 502, + 105 + ], + "score": 0.86, + "content": "\\mathbb { F } _ { n }", + "type": "inline_equation" + }, + { + "bbox": [ + 503, + 92, + 507, + 108 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 106, + 104, + 506, + 117 + ], + "spans": [ + { + "bbox": [ + 106, + 104, + 234, + 117 + ], + "score": 1.0, + "content": "Therefore, we may decompose", + "type": "text" + }, + { + "bbox": [ + 234, + 105, + 247, + 115 + ], + "score": 0.89, + "content": "\\mathbf { S } _ { n }", + "type": "inline_equation" + }, + { + "bbox": [ + 248, + 104, + 259, + 117 + ], + "score": 1.0, + "content": "as", + "type": "text" + }, + { + "bbox": [ + 260, + 105, + 338, + 116 + ], + "score": 0.31, + "content": "\\mathbf { S } _ { n } = \\mathbf { B } ~ \\mathbf { P } _ { 2 } ~ \\mathbf { D } ~ \\mathbf { P } _ { 1 } ,", + "type": "inline_equation" + }, + { + "bbox": [ + 338, + 104, + 368, + 117 + ], + "score": 1.0, + "content": ", where", + "type": "text" + }, + { + "bbox": [ + 369, + 105, + 399, + 116 + ], + "score": 0.91, + "content": "\\mathbf { P } _ { 1 } , \\mathbf { P } _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 399, + 104, + 473, + 117 + ], + "score": 1.0, + "content": "are permutations,", + "type": "text" + }, + { + "bbox": [ + 473, + 105, + 502, + 115 + ], + "score": 0.9, + "content": "\\mathbf { B } \\in B", + "type": "inline_equation" + }, + { + "bbox": [ + 503, + 104, + 506, + 117 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 114, + 506, + 129 + ], + "spans": [ + { + "bbox": [ + 105, + 114, + 123, + 129 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 123, + 115, + 133, + 126 + ], + "score": 0.51, + "content": "\\mathbf { D }", + "type": "inline_equation" + }, + { + "bbox": [ + 134, + 114, + 218, + 129 + ], + "score": 1.0, + "content": "is a diagonal matrix.", + "type": "text" + }, + { + "bbox": [ + 218, + 116, + 232, + 126 + ], + "score": 0.72, + "content": "\\mathbf { P } _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 232, + 114, + 244, + 129 + ], + "score": 1.0, + "content": "D", + "type": "text" + }, + { + "bbox": [ + 245, + 116, + 258, + 127 + ], + "score": 0.73, + "content": "\\mathbf { P } _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 259, + 114, + 506, + 129 + ], + "score": 1.0, + "content": "is simply a permutation matrix with scaled entries, which can", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 126, + 507, + 139 + ], + "spans": [ + { + "bbox": [ + 105, + 126, + 218, + 139 + ], + "score": 1.0, + "content": "be equivalently expressed as", + "type": "text" + }, + { + "bbox": [ + 219, + 127, + 245, + 137 + ], + "score": 0.88, + "content": "\\mathbf { D } ^ { \\prime } \\mathbf { P } ^ { \\prime }", + "type": "inline_equation" + }, + { + "bbox": [ + 245, + 126, + 345, + 139 + ], + "score": 1.0, + "content": "for some diagonal matrix", + "type": "text" + }, + { + "bbox": [ + 346, + 126, + 358, + 137 + ], + "score": 0.87, + "content": "\\mathbf { D ^ { \\prime } }", + "type": "inline_equation" + }, + { + "bbox": [ + 359, + 126, + 425, + 139 + ], + "score": 1.0, + "content": "and permutation", + "type": "text" + }, + { + "bbox": [ + 425, + 126, + 437, + 136 + ], + "score": 0.84, + "content": "\\mathbf { P ^ { \\prime } }", + "type": "inline_equation" + }, + { + "bbox": [ + 437, + 126, + 507, + 139 + ], + "score": 1.0, + "content": ". By Lemma H.1,", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 109, + 135, + 506, + 151 + ], + "spans": [ + { + "bbox": [ + 109, + 137, + 160, + 148 + ], + "score": 0.9, + "content": "\\mathbf { B } \\bar { \\mathbf { D ^ { \\prime } } } \\in B B ^ { * }", + "type": "inline_equation" + }, + { + "bbox": [ + 161, + 135, + 227, + 151 + ], + "score": 1.0, + "content": ". By Theorem 2,", + "type": "text" + }, + { + "bbox": [ + 227, + 137, + 269, + 148 + ], + "score": 0.91, + "content": "\\mathbf { P } ^ { \\prime } \\in B B ^ { * }", + "type": "inline_equation" + }, + { + "bbox": [ + 270, + 135, + 369, + 151 + ], + "score": 1.0, + "content": ". Hence, by Lemma H.2,", + "type": "text" + }, + { + "bbox": [ + 370, + 137, + 425, + 149 + ], + "score": 0.93, + "content": "\\bar { \\mathbf { S } _ { n } } \\in ( B B ^ { * } ) ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 425, + 135, + 429, + 151 + ], + "score": 1.0, + "content": ".", + "type": "text" + }, + { + "bbox": [ + 494, + 137, + 506, + 149 + ], + "score": 0.999, + "content": "□", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 2.5, + "bbox_fs": [ + 104, + 81, + 507, + 151 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 162, + 501, + 175 + ], + "lines": [ + { + "bbox": [ + 106, + 161, + 500, + 177 + ], + "spans": [ + { + "bbox": [ + 106, + 161, + 469, + 177 + ], + "score": 1.0, + "content": "Remark J.4. An analogous argument shows that the Discrete Cosine Transform is also in", + "type": "text" + }, + { + "bbox": [ + 470, + 163, + 500, + 175 + ], + "score": 0.91, + "content": "( B B ^ { * } ) ^ { 2 }", + "type": "inline_equation" + } + ], + "index": 6 + } + ], + "index": 6, + "bbox_fs": [ + 106, + 161, + 500, + 177 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 179, + 439, + 191 + ], + "lines": [ + { + "bbox": [ + 106, + 178, + 440, + 192 + ], + "spans": [ + { + "bbox": [ + 106, + 178, + 176, + 192 + ], + "score": 1.0, + "content": "Lemma J.5. Let", + "type": "text" + }, + { + "bbox": [ + 177, + 180, + 191, + 190 + ], + "score": 0.88, + "content": "\\mathbf { C } _ { n }", + "type": "inline_equation" + }, + { + "bbox": [ + 192, + 178, + 217, + 192 + ], + "score": 1.0, + "content": "be an", + "type": "text" + }, + { + "bbox": [ + 217, + 181, + 243, + 190 + ], + "score": 0.89, + "content": "n \\times n", + "type": "inline_equation" + }, + { + "bbox": [ + 243, + 178, + 392, + 192 + ], + "score": 1.0, + "content": "circulant (convolution) matrix. Then", + "type": "text" + }, + { + "bbox": [ + 392, + 179, + 436, + 190 + ], + "score": 0.91, + "content": "\\mathbf { C } _ { n } \\in B B ^ { * }", + "type": "inline_equation" + }, + { + "bbox": [ + 437, + 178, + 440, + 192 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 7, + "bbox_fs": [ + 106, + 178, + 440, + 192 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 211, + 505, + 250 + ], + "lines": [ + { + "bbox": [ + 105, + 210, + 504, + 226 + ], + "spans": [ + { + "bbox": [ + 105, + 210, + 350, + 226 + ], + "score": 1.0, + "content": "Proof. Using Theorem 2.6.4 of Pan (2001), we can express", + "type": "text" + }, + { + "bbox": [ + 351, + 212, + 366, + 223 + ], + "score": 0.89, + "content": "\\mathbf { C } _ { n }", + "type": "inline_equation" + }, + { + "bbox": [ + 366, + 210, + 378, + 226 + ], + "score": 1.0, + "content": "as", + "type": "text" + }, + { + "bbox": [ + 378, + 210, + 461, + 224 + ], + "score": 0.93, + "content": "\\mathbf { C } _ { n } = ( \\mathbf { F } _ { n } ) ^ { - 1 } \\mathbf { D } \\mathbf { F } _ { n }", + "type": "inline_equation" + }, + { + "bbox": [ + 461, + 210, + 489, + 226 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 490, + 212, + 504, + 224 + ], + "score": 0.88, + "content": "{ \\bf F } _ { n }", + "type": "inline_equation" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 224, + 506, + 239 + ], + "spans": [ + { + "bbox": [ + 105, + 224, + 263, + 239 + ], + "score": 1.0, + "content": "is the Discrete Fourier Transform and", + "type": "text" + }, + { + "bbox": [ + 263, + 226, + 273, + 235 + ], + "score": 0.66, + "content": "\\mathbf { D }", + "type": "inline_equation" + }, + { + "bbox": [ + 274, + 224, + 362, + 239 + ], + "score": 1.0, + "content": "is a diagonal matrix.", + "type": "text" + }, + { + "bbox": [ + 363, + 224, + 428, + 237 + ], + "score": 0.89, + "content": "\\left( \\mathbf { F } _ { n } \\right) ^ { - 1 } = \\mathbf { B } \\mathbf { \\Lambda } \\mathbf { P }", + "type": "inline_equation" + }, + { + "bbox": [ + 429, + 224, + 453, + 239 + ], + "score": 1.0, + "content": "(with", + "type": "text" + }, + { + "bbox": [ + 454, + 225, + 483, + 236 + ], + "score": 0.62, + "content": "\\mathbf { B } \\in B", + "type": "inline_equation" + }, + { + "bbox": [ + 484, + 224, + 487, + 239 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 487, + 226, + 497, + 236 + ], + "score": 0.55, + "content": "\\mathbf { P }", + "type": "inline_equation" + }, + { + "bbox": [ + 497, + 224, + 506, + 239 + ], + "score": 1.0, + "content": "a", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 104, + 236, + 368, + 252 + ], + "spans": [ + { + "bbox": [ + 104, + 237, + 239, + 252 + ], + "score": 1.0, + "content": "permutation), which implies that", + "type": "text" + }, + { + "bbox": [ + 239, + 236, + 321, + 250 + ], + "score": 0.94, + "content": "{ \\bf F } _ { n } = \\left( { \\bf P } \\right) ^ { - 1 } \\left( { \\bf B } \\right) ^ { - 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 321, + 237, + 368, + 252 + ], + "score": 1.0, + "content": ". Therefore", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 9, + "bbox_fs": [ + 104, + 210, + 506, + 252 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 244, + 257, + 367, + 273 + ], + "lines": [ + { + "bbox": [ + 244, + 257, + 367, + 273 + ], + "spans": [ + { + "bbox": [ + 244, + 257, + 367, + 273 + ], + "score": 0.91, + "content": "{ \\bf C } _ { n } = { \\bf B } { \\bf P } { \\bf D } { \\bf \\Lambda } \\left( { \\bf P } \\right) ^ { - 1 } \\left( { \\bf B } \\right) ^ { - 1 } .", + "type": "interline_equation", + "image_path": "2093a282bede9cd9f85441018ef319fec57948e6c681841b22b1dae5d248b8fc.jpg" + } + ] + } + ], + "index": 11, + "virtual_lines": [ + { + "bbox": [ + 244, + 257, + 367, + 273 + ], + "spans": [], + "index": 11 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 280, + 506, + 314 + ], + "lines": [ + { + "bbox": [ + 106, + 281, + 505, + 293 + ], + "spans": [ + { + "bbox": [ + 106, + 281, + 505, + 293 + ], + "score": 1.0, + "content": "The middle three factors have the effect of performing a permutation, scaling each element, and", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 290, + 507, + 306 + ], + "spans": [ + { + "bbox": [ + 105, + 290, + 459, + 306 + ], + "score": 1.0, + "content": "undoing the permutation, which is equivalent to simply scaling by some diagonal matrix", + "type": "text" + }, + { + "bbox": [ + 459, + 292, + 471, + 302 + ], + "score": 0.84, + "content": "\\mathbf { D ^ { \\prime } }", + "type": "inline_equation" + }, + { + "bbox": [ + 472, + 290, + 507, + 306 + ], + "score": 1.0, + "content": ". Hence,", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 303, + 171, + 314 + ], + "spans": [ + { + "bbox": [ + 105, + 303, + 171, + 314 + ], + "score": 1.0, + "content": "we are left with", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 13, + "bbox_fs": [ + 105, + 281, + 507, + 314 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 263, + 314, + 348, + 330 + ], + "lines": [ + { + "bbox": [ + 263, + 314, + 348, + 330 + ], + "spans": [ + { + "bbox": [ + 263, + 314, + 348, + 330 + ], + "score": 0.91, + "content": "\\mathbf { C } _ { n } = \\mathbf { B } \\mathbf { D } ^ { \\prime } \\left( \\mathbf { B } \\right) ^ { - 1 } .", + "type": "interline_equation", + "image_path": "bc6f96999e33bad0feb03e49b8f25e8891094d64091f542f506f8ec7dd10f1e3.jpg" + } + ] + } + ], + "index": 15, + "virtual_lines": [ + { + "bbox": [ + 263, + 314, + 348, + 330 + ], + "spans": [], + "index": 15 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 336, + 421, + 351 + ], + "lines": [ + { + "bbox": [ + 105, + 335, + 418, + 352 + ], + "spans": [ + { + "bbox": [ + 105, + 335, + 181, + 352 + ], + "score": 1.0, + "content": "By Lemma H.1, B", + "type": "text" + }, + { + "bbox": [ + 181, + 338, + 216, + 349 + ], + "score": 0.71, + "content": "\\mathbf { D } ^ { \\prime } \\in B", + "type": "inline_equation" + }, + { + "bbox": [ + 216, + 335, + 287, + 352 + ], + "score": 1.0, + "content": ". By Lemma H.6,", + "type": "text" + }, + { + "bbox": [ + 288, + 336, + 338, + 350 + ], + "score": 0.92, + "content": "\\left( \\mathbf { B } \\right) ^ { - 1 } \\in B ^ { * }", + "type": "inline_equation" + }, + { + "bbox": [ + 338, + 335, + 373, + 352 + ], + "score": 1.0, + "content": ". Hence,", + "type": "text" + }, + { + "bbox": [ + 373, + 338, + 418, + 349 + ], + "score": 0.9, + "content": "\\mathbf { C } _ { n } \\in B B ^ { * }", + "type": "inline_equation" + } + ], + "index": 16 + } + ], + "index": 16, + "bbox_fs": [ + 105, + 335, + 418, + 352 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 363, + 506, + 387 + ], + "lines": [ + { + "bbox": [ + 106, + 363, + 505, + 376 + ], + "spans": [ + { + "bbox": [ + 106, + 363, + 244, + 376 + ], + "score": 1.0, + "content": "Remark J.6. We can expand any", + "type": "text" + }, + { + "bbox": [ + 245, + 365, + 270, + 374 + ], + "score": 0.89, + "content": "n \\times n", + "type": "inline_equation" + }, + { + "bbox": [ + 271, + 363, + 335, + 376 + ], + "score": 1.0, + "content": "Toeplitz matrix", + "type": "text" + }, + { + "bbox": [ + 335, + 364, + 349, + 375 + ], + "score": 0.88, + "content": "\\mathbf { T } _ { n }", + "type": "inline_equation" + }, + { + "bbox": [ + 349, + 363, + 376, + 376 + ], + "score": 1.0, + "content": "into a", + "type": "text" + }, + { + "bbox": [ + 377, + 364, + 412, + 374 + ], + "score": 0.9, + "content": "2 n \\times 2 n", + "type": "inline_equation" + }, + { + "bbox": [ + 413, + 363, + 505, + 376 + ], + "score": 1.0, + "content": "circulant matrix (with", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 375, + 423, + 387 + ], + "spans": [ + { + "bbox": [ + 106, + 375, + 147, + 387 + ], + "score": 1.0, + "content": "upper left", + "type": "text" + }, + { + "bbox": [ + 147, + 376, + 173, + 385 + ], + "score": 0.88, + "content": "n \\times n", + "type": "inline_equation" + }, + { + "bbox": [ + 173, + 375, + 251, + 387 + ], + "score": 1.0, + "content": "submatrix equal to", + "type": "text" + }, + { + "bbox": [ + 251, + 375, + 266, + 386 + ], + "score": 0.86, + "content": "\\mathbf { T } _ { n }", + "type": "inline_equation" + }, + { + "bbox": [ + 266, + 375, + 304, + 387 + ], + "score": 1.0, + "content": "). Hence,", + "type": "text" + }, + { + "bbox": [ + 304, + 375, + 361, + 387 + ], + "score": 0.93, + "content": "\\mathbf { T } _ { n } \\in ( B B ^ { * } ) _ { 2 } ^ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 362, + 375, + 423, + 387 + ], + "score": 1.0, + "content": "by Lemma J.5.", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 17.5, + "bbox_fs": [ + 106, + 363, + 505, + 387 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 396, + 459, + 408 + ], + "lines": [ + { + "bbox": [ + 105, + 396, + 460, + 410 + ], + "spans": [ + { + "bbox": [ + 105, + 396, + 396, + 410 + ], + "score": 1.0, + "content": "The Fastfood matrix class (Le et al., 2013) can be tightly captured in the", + "type": "text" + }, + { + "bbox": [ + 397, + 397, + 416, + 407 + ], + "score": 0.88, + "content": "B B ^ { * }", + "type": "inline_equation" + }, + { + "bbox": [ + 416, + 396, + 460, + 410 + ], + "score": 1.0, + "content": "hierarchy:", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 19, + "bbox_fs": [ + 105, + 396, + 460, + 410 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 412, + 506, + 436 + ], + "lines": [ + { + "bbox": [ + 106, + 412, + 506, + 425 + ], + "spans": [ + { + "bbox": [ + 106, + 412, + 292, + 425 + ], + "score": 1.0, + "content": "Lemma J.7. The product SHDPHB where", + "type": "text" + }, + { + "bbox": [ + 293, + 413, + 326, + 424 + ], + "score": 0.51, + "content": "\\mathbf { S } , \\mathbf { D } , \\mathbf { B }", + "type": "inline_equation" + }, + { + "bbox": [ + 326, + 412, + 422, + 425 + ], + "score": 1.0, + "content": "are diagonal matrices,", + "type": "text" + }, + { + "bbox": [ + 423, + 413, + 433, + 423 + ], + "score": 0.43, + "content": "\\mathbf { H }", + "type": "inline_equation" + }, + { + "bbox": [ + 433, + 412, + 506, + 425 + ], + "score": 1.0, + "content": "is the Hadamard", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 423, + 331, + 437 + ], + "spans": [ + { + "bbox": [ + 105, + 423, + 168, + 437 + ], + "score": 1.0, + "content": "transform, and", + "type": "text" + }, + { + "bbox": [ + 168, + 425, + 177, + 434 + ], + "score": 0.45, + "content": "\\mathbf { P }", + "type": "inline_equation" + }, + { + "bbox": [ + 178, + 423, + 297, + 437 + ], + "score": 1.0, + "content": "is a permutation matrix, is in", + "type": "text" + }, + { + "bbox": [ + 297, + 424, + 327, + 436 + ], + "score": 0.91, + "content": "( B B ^ { * } ) ^ { 3 }", + "type": "inline_equation" + }, + { + "bbox": [ + 328, + 423, + 331, + 437 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 20.5, + "bbox_fs": [ + 105, + 412, + 506, + 437 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 455, + 505, + 478 + ], + "lines": [ + { + "bbox": [ + 106, + 455, + 505, + 468 + ], + "spans": [ + { + "bbox": [ + 106, + 455, + 273, + 468 + ], + "score": 1.0, + "content": "Proof. We have shown in Lemma J.2 that", + "type": "text" + }, + { + "bbox": [ + 274, + 456, + 314, + 466 + ], + "score": 0.91, + "content": "\\mathbf { H } \\in B B ^ { * }", + "type": "inline_equation" + }, + { + "bbox": [ + 314, + 455, + 407, + 468 + ], + "score": 1.0, + "content": ", and in Theorem 2 that", + "type": "text" + }, + { + "bbox": [ + 407, + 456, + 446, + 466 + ], + "score": 0.91, + "content": "\\mathbf { P } \\in B B ^ { * }", + "type": "inline_equation" + }, + { + "bbox": [ + 447, + 455, + 475, + 468 + ], + "score": 1.0, + "content": ". Since", + "type": "text" + }, + { + "bbox": [ + 475, + 456, + 495, + 466 + ], + "score": 0.89, + "content": "B B ^ { * }", + "type": "inline_equation" + }, + { + "bbox": [ + 495, + 455, + 505, + 468 + ], + "score": 1.0, + "content": "is", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 465, + 502, + 479 + ], + "spans": [ + { + "bbox": [ + 105, + 465, + 384, + 479 + ], + "score": 1.0, + "content": "closed under diagonal multiplication (Lemma H.1), we conclude that", + "type": "text" + }, + { + "bbox": [ + 384, + 467, + 477, + 479 + ], + "score": 0.56, + "content": "\\mathbf { S H D P H B } \\in ( B B ^ { * } ) ^ { 3 }", + "type": "inline_equation" + }, + { + "bbox": [ + 477, + 465, + 488, + 479 + ], + "score": 1.0, + "content": ".", + "type": "text" + }, + { + "bbox": [ + 496, + 469, + 502, + 475 + ], + "score": 0.0, + "content": "", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 22.5, + "bbox_fs": [ + 105, + 455, + 505, + 479 + ] + }, + { + "type": "text", + "bbox": [ + 108, + 497, + 505, + 521 + ], + "lines": [ + { + "bbox": [ + 106, + 497, + 505, + 510 + ], + "spans": [ + { + "bbox": [ + 106, + 497, + 505, + 510 + ], + "score": 1.0, + "content": "The two classes of matrices introduced in Moczulski et al. (2016), called AFDF and ACDC, are also", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 509, + 261, + 522 + ], + "spans": [ + { + "bbox": [ + 106, + 509, + 196, + 522 + ], + "score": 1.0, + "content": "tightly captured in the", + "type": "text" + }, + { + "bbox": [ + 197, + 509, + 216, + 520 + ], + "score": 0.88, + "content": "B B ^ { * }", + "type": "inline_equation" + }, + { + "bbox": [ + 216, + 509, + 261, + 522 + ], + "score": 1.0, + "content": "hierarchy:", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 24.5, + "bbox_fs": [ + 106, + 497, + 505, + 522 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 525, + 504, + 548 + ], + "lines": [ + { + "bbox": [ + 105, + 524, + 502, + 538 + ], + "spans": [ + { + "bbox": [ + 105, + 524, + 176, + 538 + ], + "score": 1.0, + "content": "Lemma J.8. Let", + "type": "text" + }, + { + "bbox": [ + 176, + 525, + 219, + 536 + ], + "score": 0.75, + "content": "\\mathbf { A } \\mathbf { F } ^ { - 1 } \\mathbf { D } \\mathbf { F }", + "type": "inline_equation" + }, + { + "bbox": [ + 220, + 524, + 483, + 538 + ], + "score": 1.0, + "content": "be a product of a diagonal matrix A, the inverse Fourier transform", + "type": "text" + }, + { + "bbox": [ + 484, + 525, + 502, + 536 + ], + "score": 0.89, + "content": "\\mathbf { F } ^ { - 1 }", + "type": "inline_equation" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 536, + 439, + 549 + ], + "spans": [ + { + "bbox": [ + 105, + 536, + 362, + 549 + ], + "score": 1.0, + "content": "another diagonal matrix D, and the Fourier transform F. Then", + "type": "text" + }, + { + "bbox": [ + 362, + 536, + 435, + 547 + ], + "score": 0.89, + "content": "\\mathbf { A } \\mathbf { F } ^ { - 1 } \\mathbf { D } \\mathbf { F } \\in B B ^ { * }", + "type": "inline_equation" + }, + { + "bbox": [ + 435, + 536, + 439, + 549 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 26.5, + "bbox_fs": [ + 105, + 524, + 502, + 549 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 552, + 505, + 576 + ], + "lines": [ + { + "bbox": [ + 105, + 552, + 506, + 566 + ], + "spans": [ + { + "bbox": [ + 105, + 552, + 122, + 566 + ], + "score": 1.0, + "content": "Let", + "type": "text" + }, + { + "bbox": [ + 122, + 552, + 168, + 564 + ], + "score": 0.83, + "content": "\\mathbf { A C } ^ { - 1 } \\mathbf { D C }", + "type": "inline_equation" + }, + { + "bbox": [ + 169, + 552, + 446, + 566 + ], + "score": 1.0, + "content": "be a product of a diagonal matrix A, the inverse cosine transform", + "type": "text" + }, + { + "bbox": [ + 446, + 553, + 466, + 564 + ], + "score": 0.89, + "content": "\\mathbf { C } ^ { - 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 466, + 552, + 506, + 566 + ], + "score": 1.0, + "content": ", another", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 561, + 416, + 577 + ], + "spans": [ + { + "bbox": [ + 105, + 561, + 325, + 577 + ], + "score": 1.0, + "content": "diagonal matrix D, and the cosine transform C. Then", + "type": "text" + }, + { + "bbox": [ + 325, + 564, + 412, + 577 + ], + "score": 0.9, + "content": "\\mathbf { A } \\mathbf { C } ^ { - 1 } \\mathbf { D } \\mathbf { C } \\in ( B B ^ { * } ) ^ { 4 }", + "type": "inline_equation" + }, + { + "bbox": [ + 412, + 561, + 416, + 577 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 28.5, + "bbox_fs": [ + 105, + 552, + 506, + 577 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 595, + 504, + 618 + ], + "lines": [ + { + "bbox": [ + 105, + 594, + 505, + 609 + ], + "spans": [ + { + "bbox": [ + 105, + 594, + 282, + 609 + ], + "score": 1.0, + "content": "Proof. We have argued in Lemma J.5 that", + "type": "text" + }, + { + "bbox": [ + 283, + 596, + 350, + 606 + ], + "score": 0.9, + "content": "\\mathbf { F } ^ { - 1 } \\mathbf { D } \\mathbf { F } \\in B B ^ { * }", + "type": "inline_equation" + }, + { + "bbox": [ + 350, + 594, + 381, + 609 + ], + "score": 1.0, + "content": ". Since", + "type": "text" + }, + { + "bbox": [ + 381, + 596, + 401, + 606 + ], + "score": 0.89, + "content": "B B ^ { * }", + "type": "inline_equation" + }, + { + "bbox": [ + 401, + 594, + 505, + 609 + ], + "score": 1.0, + "content": "is closed under diagonal", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 606, + 372, + 619 + ], + "spans": [ + { + "bbox": [ + 105, + 606, + 295, + 619 + ], + "score": 1.0, + "content": "multiplication (Lemma H.1), we conclude that", + "type": "text" + }, + { + "bbox": [ + 295, + 607, + 368, + 618 + ], + "score": 0.9, + "content": "\\mathbf { A } \\mathbf { F } ^ { - 1 } \\mathbf { D } \\mathbf { F } \\in B B ^ { * }", + "type": "inline_equation" + }, + { + "bbox": [ + 368, + 606, + 372, + 619 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 30.5, + "bbox_fs": [ + 105, + 594, + 505, + 619 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 622, + 502, + 647 + ], + "lines": [ + { + "bbox": [ + 105, + 622, + 504, + 636 + ], + "spans": [ + { + "bbox": [ + 105, + 622, + 190, + 636 + ], + "score": 1.0, + "content": "We have shown that", + "type": "text" + }, + { + "bbox": [ + 191, + 623, + 243, + 636 + ], + "score": 0.92, + "content": "\\mathbf { C } \\in ( B B ^ { * } ) ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 243, + 622, + 258, + 636 + ], + "score": 1.0, + "content": ", so", + "type": "text" + }, + { + "bbox": [ + 259, + 623, + 322, + 636 + ], + "score": 0.92, + "content": "\\mathbf { C } ^ { - 1 } \\in ( B B ^ { * } ) ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 322, + 622, + 383, + 636 + ], + "score": 1.0, + "content": "as well. Since", + "type": "text" + }, + { + "bbox": [ + 383, + 624, + 402, + 634 + ], + "score": 0.88, + "content": "B B ^ { * }", + "type": "inline_equation" + }, + { + "bbox": [ + 403, + 622, + 504, + 636 + ], + "score": 1.0, + "content": "is closed under diagonal", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 633, + 501, + 648 + ], + "spans": [ + { + "bbox": [ + 105, + 633, + 295, + 648 + ], + "score": 1.0, + "content": "multiplication (Lemma H.1), we conclude that", + "type": "text" + }, + { + "bbox": [ + 295, + 635, + 382, + 647 + ], + "score": 0.89, + "content": "\\mathbf { A C } ^ { - 1 } \\mathbf { D C } \\in ( B B ^ { * } ) ^ { 4 }", + "type": "inline_equation" + }, + { + "bbox": [ + 382, + 633, + 386, + 648 + ], + "score": 1.0, + "content": ".", + "type": "text" + }, + { + "bbox": [ + 497, + 640, + 501, + 643 + ], + "score": 0.0, + "content": "", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 32.5, + "bbox_fs": [ + 105, + 622, + 504, + 648 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 660, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 660, + 506, + 673 + ], + "spans": [ + { + "bbox": [ + 106, + 660, + 506, + 673 + ], + "score": 1.0, + "content": "Remark J.9. Within each butterfly factor matrix of the DFT (excluding the bit reversal permutation)", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 671, + 506, + 685 + ], + "spans": [ + { + "bbox": [ + 105, + 671, + 506, + 685 + ], + "score": 1.0, + "content": "and the Hadamard transform, the columns are pairwise orthogonal and have norm 2. Hence, we can√", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 682, + 506, + 696 + ], + "spans": [ + { + "bbox": [ + 106, + 682, + 187, + 696 + ], + "score": 1.0, + "content": "divide all factors by", + "type": "text" + }, + { + "bbox": [ + 187, + 682, + 201, + 695 + ], + "score": 0.86, + "content": "\\sqrt { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 201, + 682, + 506, + 696 + ], + "score": 1.0, + "content": "to make orthogonal factor matrices. To counteract this scaling, we can add a", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 104, + 694, + 506, + 713 + ], + "spans": [ + { + "bbox": [ + 104, + 694, + 236, + 712 + ], + "score": 1.0, + "content": "diagonal matrix with 2log2(n)", + "type": "text" + }, + { + "bbox": [ + 193, + 695, + 261, + 711 + ], + "score": 0.9, + "content": "{ \\sqrt { 2 } } ^ { \\log _ { 2 } ( n ) } = { \\sqrt { n } }", + "type": "inline_equation" + }, + { + "bbox": [ + 231, + 696, + 506, + 713 + ], + "score": 1.0, + "content": "= n in all entries to the factorization. By doing this we can place", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 709, + 506, + 722 + ], + "spans": [ + { + "bbox": [ + 105, + 709, + 392, + 722 + ], + "score": 1.0, + "content": "all of the above transforms in the OBB hierarchy (defined in Appendix", + "type": "text" + }, + { + "bbox": [ + 392, + 711, + 401, + 720 + ], + "score": 0.41, + "content": "K _ { \\cdot }", + "type": "inline_equation" + }, + { + "bbox": [ + 401, + 709, + 506, + 722 + ], + "score": 1.0, + "content": ") with the same width and", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 106, + 721, + 176, + 734 + ], + "spans": [ + { + "bbox": [ + 106, + 721, + 176, + 734 + ], + "score": 1.0, + "content": "expansion factor.", + "type": "text" + } + ], + "index": 39 + } + ], + "index": 36.5, + "bbox_fs": [ + 104, + 660, + 506, + 734 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "title", + "bbox": [ + 108, + 82, + 284, + 93 + ], + "lines": [ + { + "bbox": [ + 105, + 82, + 286, + 95 + ], + "spans": [ + { + "bbox": [ + 105, + 82, + 286, + 95 + ], + "score": 1.0, + "content": "J.1 MULTI-DIMENSIONAL TRANSFORMS", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "text", + "bbox": [ + 105, + 102, + 504, + 126 + ], + "lines": [ + { + "bbox": [ + 105, + 102, + 505, + 115 + ], + "spans": [ + { + "bbox": [ + 105, + 102, + 505, + 115 + ], + "score": 1.0, + "content": "Here, we show that, using larger matrices, we are able to similarly capture multi-dimensional versions", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 106, + 114, + 205, + 127 + ], + "spans": [ + { + "bbox": [ + 106, + 114, + 205, + 127 + ], + "score": 1.0, + "content": "of the above transforms.", + "type": "text" + } + ], + "index": 2 + } + ], + "index": 1.5 + }, + { + "type": "text", + "bbox": [ + 107, + 128, + 504, + 151 + ], + "lines": [ + { + "bbox": [ + 105, + 127, + 504, + 141 + ], + "spans": [ + { + "bbox": [ + 105, + 127, + 181, + 141 + ], + "score": 1.0, + "content": "Lemma J.10. Let", + "type": "text" + }, + { + "bbox": [ + 182, + 128, + 195, + 140 + ], + "score": 0.86, + "content": "{ \\bf F } _ { n } ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 195, + 127, + 469, + 141 + ], + "score": 1.0, + "content": "be the 2-dimensional Discrete Fourier Transform (represented as an", + "type": "text" + }, + { + "bbox": [ + 470, + 128, + 504, + 139 + ], + "score": 0.9, + "content": "n ^ { 2 } \\times n ^ { 2 }", + "type": "inline_equation" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 137, + 223, + 152 + ], + "spans": [ + { + "bbox": [ + 105, + 137, + 164, + 152 + ], + "score": 1.0, + "content": "matrix). Then", + "type": "text" + }, + { + "bbox": [ + 164, + 139, + 220, + 151 + ], + "score": 0.92, + "content": "\\mathbf { F } _ { n } ^ { 2 } \\in \\tilde { ( B B ^ { \\ast } ) } ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 220, + 137, + 223, + 152 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 3.5 + }, + { + "type": "text", + "bbox": [ + 107, + 163, + 505, + 230 + ], + "lines": [ + { + "bbox": [ + 106, + 163, + 505, + 175 + ], + "spans": [ + { + "bbox": [ + 106, + 163, + 439, + 175 + ], + "score": 1.0, + "content": "Proof. The separation property of the 2-D DFT allows us to express its action on an", + "type": "text" + }, + { + "bbox": [ + 440, + 165, + 465, + 174 + ], + "score": 0.89, + "content": "n \\times n", + "type": "inline_equation" + }, + { + "bbox": [ + 465, + 163, + 505, + 175 + ], + "score": 1.0, + "content": "matrix as", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 174, + 505, + 186 + ], + "spans": [ + { + "bbox": [ + 106, + 174, + 505, + 186 + ], + "score": 1.0, + "content": "the composition of a 1-D DFT on each of its rows and a 1-D DFT on each of its columns. If we view", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 183, + 506, + 198 + ], + "spans": [ + { + "bbox": [ + 105, + 183, + 183, + 198 + ], + "score": 1.0, + "content": "the 2-D DFT as an", + "type": "text" + }, + { + "bbox": [ + 183, + 185, + 217, + 196 + ], + "score": 0.91, + "content": "n ^ { 2 } \\times n ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 218, + 183, + 475, + 198 + ], + "score": 1.0, + "content": "matrix, its input and outputs will both be column vectors of size", + "type": "text" + }, + { + "bbox": [ + 475, + 185, + 487, + 195 + ], + "score": 0.86, + "content": "n ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 487, + 183, + 506, + 198 + ], + "score": 1.0, + "content": ". As", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 196, + 505, + 209 + ], + "spans": [ + { + "bbox": [ + 105, + 196, + 505, + 209 + ], + "score": 1.0, + "content": "our convention, we list the entries of the input vector in the row-major order corresponding to the", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 107, + 207, + 505, + 219 + ], + "spans": [ + { + "bbox": [ + 107, + 208, + 132, + 218 + ], + "score": 0.88, + "content": "n \\times n", + "type": "inline_equation" + }, + { + "bbox": [ + 132, + 207, + 505, + 219 + ], + "score": 1.0, + "content": "input matrix. Then, we consider the 2-D DFT in four steps, where the first two steps perform", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 217, + 449, + 231 + ], + "spans": [ + { + "bbox": [ + 105, + 217, + 449, + 231 + ], + "score": 1.0, + "content": "the 1-D DFT row-wise, and the second two steps perform the 1-D DFT column-wise:", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 7.5 + }, + { + "type": "text", + "bbox": [ + 107, + 235, + 225, + 246 + ], + "lines": [ + { + "bbox": [ + 106, + 235, + 227, + 248 + ], + "spans": [ + { + "bbox": [ + 106, + 235, + 227, + 248 + ], + "score": 1.0, + "content": "Step 1: Permute the columns:", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 11 + }, + { + "type": "text", + "bbox": [ + 107, + 252, + 505, + 296 + ], + "lines": [ + { + "bbox": [ + 106, + 252, + 505, + 264 + ], + "spans": [ + { + "bbox": [ + 106, + 252, + 505, + 264 + ], + "score": 1.0, + "content": "We permute the columns (with a bit reversal permutation), which performs a bit reversal permutation", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 263, + 505, + 275 + ], + "spans": [ + { + "bbox": [ + 106, + 263, + 505, + 275 + ], + "score": 1.0, + "content": "on each row. Viewing the input as a vector, this step corresponds to left multiplication by a permutation", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 274, + 505, + 286 + ], + "spans": [ + { + "bbox": [ + 106, + 275, + 134, + 286 + ], + "score": 1.0, + "content": "matrix", + "type": "text" + }, + { + "bbox": [ + 134, + 274, + 148, + 285 + ], + "score": 0.88, + "content": "\\mathbf { P } _ { c }", + "type": "inline_equation" + }, + { + "bbox": [ + 148, + 275, + 332, + 286 + ], + "score": 1.0, + "content": "that permutes the entries of each chunk of size", + "type": "text" + }, + { + "bbox": [ + 333, + 276, + 340, + 284 + ], + "score": 0.72, + "content": "n", + "type": "inline_equation" + }, + { + "bbox": [ + 340, + 275, + 505, + 286 + ], + "score": 1.0, + "content": "of the input vector. Step 2: Multiply each", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 285, + 208, + 297 + ], + "spans": [ + { + "bbox": [ + 105, + 285, + 208, + 297 + ], + "score": 1.0, + "content": "row by a butterfly matrix", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 13.5 + }, + { + "type": "text", + "bbox": [ + 107, + 301, + 505, + 341 + ], + "lines": [ + { + "bbox": [ + 106, + 302, + 505, + 313 + ], + "spans": [ + { + "bbox": [ + 106, + 302, + 505, + 313 + ], + "score": 1.0, + "content": "Since the entries of the input were listed in row major order, this step is achieved through multiplication", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 312, + 506, + 325 + ], + "spans": [ + { + "bbox": [ + 105, + 312, + 230, + 325 + ], + "score": 1.0, + "content": "by a block diagonal matrix of", + "type": "text" + }, + { + "bbox": [ + 230, + 315, + 237, + 323 + ], + "score": 0.78, + "content": "n", + "type": "inline_equation" + }, + { + "bbox": [ + 237, + 312, + 342, + 325 + ], + "score": 1.0, + "content": "butterfly matrices of size", + "type": "text" + }, + { + "bbox": [ + 342, + 315, + 349, + 323 + ], + "score": 0.75, + "content": "n", + "type": "inline_equation" + }, + { + "bbox": [ + 349, + 312, + 506, + 325 + ], + "score": 1.0, + "content": ", which can be viewed as a product of", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 104, + 320, + 296, + 343 + ], + "spans": [ + { + "bbox": [ + 104, + 321, + 204, + 343 + ], + "score": 1.0, + "content": "butterfly factor matrices", + "type": "text" + }, + { + "bbox": [ + 205, + 323, + 291, + 343 + ], + "score": 0.92, + "content": "\\mathbf { B } _ { n } ^ { ( n ^ { 2 } ) } \\dots \\mathbf { B } _ { \\frac { n } { 2 } } ^ { ( n ^ { 2 } ) } \\mathbf { \\dot { B } } _ { 2 } ^ { ( n ^ { 2 } ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 292, + 320, + 296, + 343 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 17 + }, + { + "type": "text", + "bbox": [ + 107, + 346, + 211, + 358 + ], + "lines": [ + { + "bbox": [ + 106, + 346, + 212, + 359 + ], + "spans": [ + { + "bbox": [ + 106, + 346, + 212, + 359 + ], + "score": 1.0, + "content": "Step 3: Permute the rows:", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 19 + }, + { + "type": "text", + "bbox": [ + 107, + 363, + 505, + 419 + ], + "lines": [ + { + "bbox": [ + 106, + 363, + 505, + 375 + ], + "spans": [ + { + "bbox": [ + 106, + 363, + 505, + 375 + ], + "score": 1.0, + "content": "We permute the rows (with a bit reversal permutation), which performs a bit reversal permutation", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 374, + 505, + 387 + ], + "spans": [ + { + "bbox": [ + 105, + 374, + 433, + 387 + ], + "score": 1.0, + "content": "on each column. This corresponds to left multiplication by a permutation matrix", + "type": "text" + }, + { + "bbox": [ + 433, + 375, + 446, + 385 + ], + "score": 0.87, + "content": "\\mathbf { P } _ { r }", + "type": "inline_equation" + }, + { + "bbox": [ + 447, + 374, + 505, + 387 + ], + "score": 1.0, + "content": ". Since we are", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 385, + 505, + 398 + ], + "spans": [ + { + "bbox": [ + 105, + 385, + 187, + 398 + ], + "score": 1.0, + "content": "permuting the rows,", + "type": "text" + }, + { + "bbox": [ + 188, + 385, + 201, + 396 + ], + "score": 0.88, + "content": "\\mathbf { P } _ { r }", + "type": "inline_equation" + }, + { + "bbox": [ + 202, + 385, + 386, + 398 + ], + "score": 1.0, + "content": "permutes the entries at the granularity of each", + "type": "text" + }, + { + "bbox": [ + 386, + 387, + 393, + 395 + ], + "score": 0.75, + "content": "n", + "type": "inline_equation" + }, + { + "bbox": [ + 394, + 385, + 505, + 398 + ], + "score": 1.0, + "content": "-chunk. Since Steps 1 and 2", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 395, + 505, + 409 + ], + "spans": [ + { + "bbox": [ + 105, + 395, + 302, + 409 + ], + "score": 1.0, + "content": "each performed an identical computation to each", + "type": "text" + }, + { + "bbox": [ + 302, + 398, + 309, + 406 + ], + "score": 0.76, + "content": "n", + "type": "inline_equation" + }, + { + "bbox": [ + 309, + 395, + 505, + 409 + ], + "score": 1.0, + "content": "-chunk we can move this row permutation before", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 407, + 343, + 419 + ], + "spans": [ + { + "bbox": [ + 105, + 407, + 181, + 419 + ], + "score": 1.0, + "content": "Step 2, combining", + "type": "text" + }, + { + "bbox": [ + 182, + 407, + 195, + 419 + ], + "score": 0.88, + "content": "\\mathbf { P } _ { c }", + "type": "inline_equation" + }, + { + "bbox": [ + 195, + 407, + 213, + 419 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 213, + 407, + 226, + 418 + ], + "score": 0.88, + "content": "\\mathbf { P } _ { r }", + "type": "inline_equation" + }, + { + "bbox": [ + 227, + 407, + 329, + 419 + ], + "score": 1.0, + "content": "into a single permutation", + "type": "text" + }, + { + "bbox": [ + 330, + 407, + 339, + 417 + ], + "score": 0.63, + "content": "\\mathbf { P }", + "type": "inline_equation" + }, + { + "bbox": [ + 339, + 407, + 343, + 419 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 22 + }, + { + "type": "text", + "bbox": [ + 107, + 424, + 312, + 435 + ], + "lines": [ + { + "bbox": [ + 106, + 423, + 312, + 438 + ], + "spans": [ + { + "bbox": [ + 106, + 423, + 312, + 438 + ], + "score": 1.0, + "content": "Step 4: Multiply each column by a butterfly matrix", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 25 + }, + { + "type": "text", + "bbox": [ + 106, + 441, + 505, + 520 + ], + "lines": [ + { + "bbox": [ + 106, + 441, + 505, + 452 + ], + "spans": [ + { + "bbox": [ + 106, + 441, + 505, + 452 + ], + "score": 1.0, + "content": "Consider multiplication by the first factor matrix. In each row, this matrix is taking linear combinations", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 451, + 506, + 464 + ], + "spans": [ + { + "bbox": [ + 106, + 451, + 279, + 464 + ], + "score": 1.0, + "content": "of adjacent column entries. In our length-", + "type": "text" + }, + { + "bbox": [ + 279, + 451, + 291, + 462 + ], + "score": 0.87, + "content": "\\cdot n ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 291, + 451, + 441, + 464 + ], + "score": 1.0, + "content": "vector, these entries will be exactly", + "type": "text" + }, + { + "bbox": [ + 441, + 454, + 448, + 462 + ], + "score": 0.69, + "content": "n", + "type": "inline_equation" + }, + { + "bbox": [ + 448, + 451, + 506, + 464 + ], + "score": 1.0, + "content": "indices apart.", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 103, + 458, + 510, + 484 + ], + "spans": [ + { + "bbox": [ + 103, + 458, + 400, + 484 + ], + "score": 1.0, + "content": "Therefore this multiplication can be handled by a butterfly factor matrix", + "type": "text" + }, + { + "bbox": [ + 401, + 462, + 425, + 479 + ], + "score": 0.92, + "content": "\\mathbf { B } _ { 2 n } ^ { ( n ^ { 2 } ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 425, + 458, + 510, + 484 + ], + "score": 1.0, + "content": ". Similarly, we find", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 477, + 506, + 490 + ], + "spans": [ + { + "bbox": [ + 106, + 477, + 506, + 490 + ], + "score": 1.0, + "content": "that this butterfly multiplication can be expressed as multiplication by a product of butterfly factor", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 487, + 505, + 513 + ], + "spans": [ + { + "bbox": [ + 106, + 488, + 142, + 506 + ], + "score": 1.0, + "content": "matrices", + "type": "text" + }, + { + "bbox": [ + 142, + 488, + 229, + 510 + ], + "score": 0.91, + "content": "\\mathbf { B } _ { n ^ { 2 } } ^ { ( n ^ { 2 } ) } \\cdot \\cdot . . \\mathbf { B } _ { \\frac { n ^ { 2 } } { 2 } } ^ { ( n ^ { 2 } ) } \\mathbf { B } _ { 2 n } ^ { ( n ^ { 2 } ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 229, + 487, + 231, + 513 + ], + "score": 1.0, + "content": ".", + "type": "text" + }, + { + "bbox": [ + 231, + 491, + 505, + 505 + ], + "score": 1.0, + "content": "Combined with the factor matrices from Step 2, these form a butterfly", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 509, + 190, + 520 + ], + "spans": [ + { + "bbox": [ + 105, + 509, + 135, + 520 + ], + "score": 1.0, + "content": "matrix", + "type": "text" + }, + { + "bbox": [ + 135, + 509, + 145, + 519 + ], + "score": 0.36, + "content": "\\mathbf { B }", + "type": "inline_equation" + }, + { + "bbox": [ + 145, + 509, + 174, + 520 + ], + "score": 1.0, + "content": "of size", + "type": "text" + }, + { + "bbox": [ + 174, + 509, + 186, + 519 + ], + "score": 0.84, + "content": "n ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 186, + 509, + 190, + 520 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 28.5 + }, + { + "type": "text", + "bbox": [ + 107, + 525, + 505, + 560 + ], + "lines": [ + { + "bbox": [ + 105, + 525, + 504, + 538 + ], + "spans": [ + { + "bbox": [ + 105, + 525, + 494, + 538 + ], + "score": 1.0, + "content": "In all, we see that the 2-D DFT may be realized as multiplication by a permutation matrix", + "type": "text" + }, + { + "bbox": [ + 495, + 526, + 504, + 536 + ], + "score": 0.53, + "content": "\\mathbf { P }", + "type": "inline_equation" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 535, + 506, + 550 + ], + "spans": [ + { + "bbox": [ + 105, + 535, + 506, + 550 + ], + "score": 1.0, + "content": "followed by multiplication by a butterfly matrix B. The same argument as Lemma J.1 shows that", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 107, + 546, + 505, + 561 + ], + "spans": [ + { + "bbox": [ + 107, + 547, + 163, + 560 + ], + "score": 0.92, + "content": "\\mathbf { F } _ { n } ^ { 2 } \\in ( B B ^ { * } ) ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 163, + 546, + 168, + 561 + ], + "score": 1.0, + "content": ".", + "type": "text" + }, + { + "bbox": [ + 495, + 549, + 505, + 558 + ], + "score": 0.999, + "content": "□", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 33 + }, + { + "type": "text", + "bbox": [ + 108, + 565, + 505, + 599 + ], + "lines": [ + { + "bbox": [ + 105, + 565, + 506, + 579 + ], + "spans": [ + { + "bbox": [ + 105, + 565, + 506, + 579 + ], + "score": 1.0, + "content": "Remark J.11. An analogous argument (using the separation property of the respective transforms)", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 576, + 506, + 590 + ], + "spans": [ + { + "bbox": [ + 105, + 576, + 435, + 590 + ], + "score": 1.0, + "content": "can be used to argue that 2-D Discrete Sine and Discrete Cosine transforms are in", + "type": "text" + }, + { + "bbox": [ + 435, + 576, + 466, + 588 + ], + "score": 0.91, + "content": "( B B ^ { * } ) ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 466, + 576, + 506, + 590 + ], + "score": 1.0, + "content": ", and that", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 588, + 266, + 600 + ], + "spans": [ + { + "bbox": [ + 105, + 588, + 243, + 600 + ], + "score": 1.0, + "content": "2-D Hadamard Transforms are in", + "type": "text" + }, + { + "bbox": [ + 243, + 588, + 262, + 598 + ], + "score": 0.88, + "content": "B B ^ { * }", + "type": "inline_equation" + }, + { + "bbox": [ + 262, + 588, + 266, + 600 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 37 + } + ], + "index": 36 + }, + { + "type": "text", + "bbox": [ + 107, + 601, + 427, + 614 + ], + "lines": [ + { + "bbox": [ + 105, + 600, + 424, + 615 + ], + "spans": [ + { + "bbox": [ + 105, + 600, + 181, + 615 + ], + "score": 1.0, + "content": "Lemma J.12. Let", + "type": "text" + }, + { + "bbox": [ + 182, + 601, + 196, + 614 + ], + "score": 0.9, + "content": "{ \\bf C } _ { n } ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 196, + 600, + 380, + 615 + ], + "score": 1.0, + "content": "be a 2-dimensional convolution matrix. Then", + "type": "text" + }, + { + "bbox": [ + 380, + 602, + 424, + 614 + ], + "score": 0.94, + "content": "\\mathbf { C } _ { n } ^ { 2 } \\in B B ^ { * }", + "type": "inline_equation" + } + ], + "index": 38 + } + ], + "index": 38 + }, + { + "type": "text", + "bbox": [ + 106, + 625, + 505, + 681 + ], + "lines": [ + { + "bbox": [ + 104, + 624, + 504, + 650 + ], + "spans": [ + { + "bbox": [ + 104, + 624, + 246, + 650 + ], + "score": 1.0, + "content": "Proof. We can express a 2-D convis the 2-D Fourier transform and", + "type": "text" + }, + { + "bbox": [ + 279, + 624, + 317, + 650 + ], + "score": 1.0, + "content": "matrix as is the in", + "type": "text" + }, + { + "bbox": [ + 318, + 625, + 398, + 638 + ], + "score": 0.93, + "content": "{ \\bf C } _ { n } ^ { 2 } = ( { \\bf F } _ { n } ^ { 2 } ) ^ { - 1 } { \\bf D } { \\bf F } _ { n } ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 398, + 624, + 429, + 650 + ], + "score": 1.0, + "content": ", where ansform", + "type": "text" + }, + { + "bbox": [ + 429, + 626, + 439, + 636 + ], + "score": 0.65, + "content": "\\mathbf { D }", + "type": "inline_equation" + }, + { + "bbox": [ + 440, + 624, + 489, + 650 + ], + "score": 1.0, + "content": "is diagonal, From the pr", + "type": "text" + }, + { + "bbox": [ + 490, + 625, + 504, + 638 + ], + "score": 0.88, + "content": "{ \\bf F } _ { n } ^ { 2 }", + "type": "inline_equation" + } + ], + "index": 39 + }, + { + "bbox": [ + 246, + 636, + 278, + 649 + ], + "spans": [ + { + "bbox": [ + 246, + 636, + 278, + 649 + ], + "score": 0.93, + "content": "( \\mathbf { F } _ { n } ^ { 2 } ) ^ { - 1 }", + "type": "inline_equation" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 646, + 507, + 661 + ], + "spans": [ + { + "bbox": [ + 105, + 646, + 309, + 661 + ], + "score": 1.0, + "content": "of Lemma J.10, we see that that we can express", + "type": "text" + }, + { + "bbox": [ + 310, + 648, + 324, + 660 + ], + "score": 0.9, + "content": "{ \\bf F } _ { n } ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 324, + 646, + 387, + 661 + ], + "score": 1.0, + "content": "(and similarly", + "type": "text" + }, + { + "bbox": [ + 387, + 648, + 421, + 660 + ], + "score": 0.91, + "content": "( \\mathbf { F } _ { n } ^ { 2 } ) ^ { - 1 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 421, + 646, + 507, + 661 + ], + "score": 1.0, + "content": "as the product of a", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 658, + 506, + 672 + ], + "spans": [ + { + "bbox": [ + 105, + 658, + 506, + 672 + ], + "score": 1.0, + "content": "butterfly matrix and a permutation matrix. The rest of the argument is analogous to the proof of", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 106, + 669, + 505, + 682 + ], + "spans": [ + { + "bbox": [ + 106, + 669, + 156, + 682 + ], + "score": 1.0, + "content": "Lemma J.5.", + "type": "text" + }, + { + "bbox": [ + 495, + 671, + 505, + 681 + ], + "score": 0.999, + "content": "□", + "type": "text" + } + ], + "index": 43 + } + ], + "index": 41 + }, + { + "type": "text", + "bbox": [ + 107, + 687, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 688, + 506, + 699 + ], + "spans": [ + { + "bbox": [ + 106, + 688, + 369, + 699 + ], + "score": 1.0, + "content": "Remark J.13. Using an inductive argument, we can show that all", + "type": "text" + }, + { + "bbox": [ + 369, + 689, + 375, + 698 + ], + "score": 0.62, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 375, + 688, + 428, + 699 + ], + "score": 1.0, + "content": "-dimensional", + "type": "text" + }, + { + "bbox": [ + 429, + 688, + 459, + 698 + ], + "score": 0.83, + "content": "( k \\in \\mathbb { Z } )", + "type": "inline_equation" + }, + { + "bbox": [ + 459, + 688, + 506, + 699 + ], + "score": 1.0, + "content": "variants of", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 698, + 506, + 711 + ], + "spans": [ + { + "bbox": [ + 105, + 698, + 251, + 711 + ], + "score": 1.0, + "content": "the above transforms, expressed as", + "type": "text" + }, + { + "bbox": [ + 251, + 699, + 285, + 709 + ], + "score": 0.91, + "content": "n ^ { k } \\times n ^ { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 286, + 698, + 393, + 711 + ], + "score": 1.0, + "content": "matrices are contained in", + "type": "text" + }, + { + "bbox": [ + 393, + 699, + 413, + 709 + ], + "score": 0.86, + "content": "B B ^ { * }", + "type": "inline_equation" + }, + { + "bbox": [ + 413, + 698, + 426, + 711 + ], + "score": 1.0, + "content": "or", + "type": "text" + }, + { + "bbox": [ + 426, + 699, + 457, + 711 + ], + "score": 0.91, + "content": "( B B ^ { * } ) ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 457, + 698, + 506, + 711 + ], + "score": 1.0, + "content": ". To do this,", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 710, + 504, + 722 + ], + "spans": [ + { + "bbox": [ + 105, + 710, + 385, + 722 + ], + "score": 1.0, + "content": "we use the separation property of the transforms to break them into a", + "type": "text" + }, + { + "bbox": [ + 386, + 710, + 409, + 720 + ], + "score": 0.87, + "content": "k - 1", + "type": "inline_equation" + }, + { + "bbox": [ + 410, + 710, + 504, + 722 + ], + "score": 1.0, + "content": "-dimensional transform", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 720, + 370, + 733 + ], + "spans": [ + { + "bbox": [ + 105, + 720, + 267, + 733 + ], + "score": 1.0, + "content": "(the inductive hypothesis) followed by a", + "type": "text" + }, + { + "bbox": [ + 268, + 722, + 273, + 730 + ], + "score": 0.57, + "content": "^ { l }", + "type": "inline_equation" + }, + { + "bbox": [ + 273, + 720, + 370, + 733 + ], + "score": 1.0, + "content": "-dimensional transform.", + "type": "text" + } + ], + "index": 47 + } + ], + "index": 45.5 + } + ], + "page_idx": 34, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 107, + 27, + 293, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 26, + 294, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 26, + 294, + 38 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2020", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 300, + 751, + 311, + 760 + ], + "lines": [ + { + "bbox": [ + 298, + 750, + 313, + 763 + ], + "spans": [ + { + "bbox": [ + 298, + 750, + 313, + 763 + ], + "score": 1.0, + "content": "35", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "title", + "bbox": [ + 108, + 82, + 284, + 93 + ], + "lines": [ + { + "bbox": [ + 105, + 82, + 286, + 95 + ], + "spans": [ + { + "bbox": [ + 105, + 82, + 286, + 95 + ], + "score": 1.0, + "content": "J.1 MULTI-DIMENSIONAL TRANSFORMS", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "text", + "bbox": [ + 105, + 102, + 504, + 126 + ], + "lines": [ + { + "bbox": [ + 105, + 102, + 505, + 115 + ], + "spans": [ + { + "bbox": [ + 105, + 102, + 505, + 115 + ], + "score": 1.0, + "content": "Here, we show that, using larger matrices, we are able to similarly capture multi-dimensional versions", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 106, + 114, + 205, + 127 + ], + "spans": [ + { + "bbox": [ + 106, + 114, + 205, + 127 + ], + "score": 1.0, + "content": "of the above transforms.", + "type": "text" + } + ], + "index": 2 + } + ], + "index": 1.5, + "bbox_fs": [ + 105, + 102, + 505, + 127 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 128, + 504, + 151 + ], + "lines": [ + { + "bbox": [ + 105, + 127, + 504, + 141 + ], + "spans": [ + { + "bbox": [ + 105, + 127, + 181, + 141 + ], + "score": 1.0, + "content": "Lemma J.10. Let", + "type": "text" + }, + { + "bbox": [ + 182, + 128, + 195, + 140 + ], + "score": 0.86, + "content": "{ \\bf F } _ { n } ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 195, + 127, + 469, + 141 + ], + "score": 1.0, + "content": "be the 2-dimensional Discrete Fourier Transform (represented as an", + "type": "text" + }, + { + "bbox": [ + 470, + 128, + 504, + 139 + ], + "score": 0.9, + "content": "n ^ { 2 } \\times n ^ { 2 }", + "type": "inline_equation" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 137, + 223, + 152 + ], + "spans": [ + { + "bbox": [ + 105, + 137, + 164, + 152 + ], + "score": 1.0, + "content": "matrix). Then", + "type": "text" + }, + { + "bbox": [ + 164, + 139, + 220, + 151 + ], + "score": 0.92, + "content": "\\mathbf { F } _ { n } ^ { 2 } \\in \\tilde { ( B B ^ { \\ast } ) } ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 220, + 137, + 223, + 152 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 3.5, + "bbox_fs": [ + 105, + 127, + 504, + 152 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 163, + 505, + 230 + ], + "lines": [ + { + "bbox": [ + 106, + 163, + 505, + 175 + ], + "spans": [ + { + "bbox": [ + 106, + 163, + 439, + 175 + ], + "score": 1.0, + "content": "Proof. The separation property of the 2-D DFT allows us to express its action on an", + "type": "text" + }, + { + "bbox": [ + 440, + 165, + 465, + 174 + ], + "score": 0.89, + "content": "n \\times n", + "type": "inline_equation" + }, + { + "bbox": [ + 465, + 163, + 505, + 175 + ], + "score": 1.0, + "content": "matrix as", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 174, + 505, + 186 + ], + "spans": [ + { + "bbox": [ + 106, + 174, + 505, + 186 + ], + "score": 1.0, + "content": "the composition of a 1-D DFT on each of its rows and a 1-D DFT on each of its columns. If we view", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 183, + 506, + 198 + ], + "spans": [ + { + "bbox": [ + 105, + 183, + 183, + 198 + ], + "score": 1.0, + "content": "the 2-D DFT as an", + "type": "text" + }, + { + "bbox": [ + 183, + 185, + 217, + 196 + ], + "score": 0.91, + "content": "n ^ { 2 } \\times n ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 218, + 183, + 475, + 198 + ], + "score": 1.0, + "content": "matrix, its input and outputs will both be column vectors of size", + "type": "text" + }, + { + "bbox": [ + 475, + 185, + 487, + 195 + ], + "score": 0.86, + "content": "n ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 487, + 183, + 506, + 198 + ], + "score": 1.0, + "content": ". As", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 196, + 505, + 209 + ], + "spans": [ + { + "bbox": [ + 105, + 196, + 505, + 209 + ], + "score": 1.0, + "content": "our convention, we list the entries of the input vector in the row-major order corresponding to the", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 107, + 207, + 505, + 219 + ], + "spans": [ + { + "bbox": [ + 107, + 208, + 132, + 218 + ], + "score": 0.88, + "content": "n \\times n", + "type": "inline_equation" + }, + { + "bbox": [ + 132, + 207, + 505, + 219 + ], + "score": 1.0, + "content": "input matrix. Then, we consider the 2-D DFT in four steps, where the first two steps perform", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 217, + 449, + 231 + ], + "spans": [ + { + "bbox": [ + 105, + 217, + 449, + 231 + ], + "score": 1.0, + "content": "the 1-D DFT row-wise, and the second two steps perform the 1-D DFT column-wise:", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 7.5, + "bbox_fs": [ + 105, + 163, + 506, + 231 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 235, + 225, + 246 + ], + "lines": [ + { + "bbox": [ + 106, + 235, + 227, + 248 + ], + "spans": [ + { + "bbox": [ + 106, + 235, + 227, + 248 + ], + "score": 1.0, + "content": "Step 1: Permute the columns:", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 11, + "bbox_fs": [ + 106, + 235, + 227, + 248 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 252, + 505, + 296 + ], + "lines": [ + { + "bbox": [ + 106, + 252, + 505, + 264 + ], + "spans": [ + { + "bbox": [ + 106, + 252, + 505, + 264 + ], + "score": 1.0, + "content": "We permute the columns (with a bit reversal permutation), which performs a bit reversal permutation", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 263, + 505, + 275 + ], + "spans": [ + { + "bbox": [ + 106, + 263, + 505, + 275 + ], + "score": 1.0, + "content": "on each row. Viewing the input as a vector, this step corresponds to left multiplication by a permutation", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 274, + 505, + 286 + ], + "spans": [ + { + "bbox": [ + 106, + 275, + 134, + 286 + ], + "score": 1.0, + "content": "matrix", + "type": "text" + }, + { + "bbox": [ + 134, + 274, + 148, + 285 + ], + "score": 0.88, + "content": "\\mathbf { P } _ { c }", + "type": "inline_equation" + }, + { + "bbox": [ + 148, + 275, + 332, + 286 + ], + "score": 1.0, + "content": "that permutes the entries of each chunk of size", + "type": "text" + }, + { + "bbox": [ + 333, + 276, + 340, + 284 + ], + "score": 0.72, + "content": "n", + "type": "inline_equation" + }, + { + "bbox": [ + 340, + 275, + 505, + 286 + ], + "score": 1.0, + "content": "of the input vector. Step 2: Multiply each", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 285, + 208, + 297 + ], + "spans": [ + { + "bbox": [ + 105, + 285, + 208, + 297 + ], + "score": 1.0, + "content": "row by a butterfly matrix", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 13.5, + "bbox_fs": [ + 105, + 252, + 505, + 297 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 301, + 505, + 341 + ], + "lines": [ + { + "bbox": [ + 106, + 302, + 505, + 313 + ], + "spans": [ + { + "bbox": [ + 106, + 302, + 505, + 313 + ], + "score": 1.0, + "content": "Since the entries of the input were listed in row major order, this step is achieved through multiplication", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 312, + 506, + 325 + ], + "spans": [ + { + "bbox": [ + 105, + 312, + 230, + 325 + ], + "score": 1.0, + "content": "by a block diagonal matrix of", + "type": "text" + }, + { + "bbox": [ + 230, + 315, + 237, + 323 + ], + "score": 0.78, + "content": "n", + "type": "inline_equation" + }, + { + "bbox": [ + 237, + 312, + 342, + 325 + ], + "score": 1.0, + "content": "butterfly matrices of size", + "type": "text" + }, + { + "bbox": [ + 342, + 315, + 349, + 323 + ], + "score": 0.75, + "content": "n", + "type": "inline_equation" + }, + { + "bbox": [ + 349, + 312, + 506, + 325 + ], + "score": 1.0, + "content": ", which can be viewed as a product of", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 104, + 320, + 296, + 343 + ], + "spans": [ + { + "bbox": [ + 104, + 321, + 204, + 343 + ], + "score": 1.0, + "content": "butterfly factor matrices", + "type": "text" + }, + { + "bbox": [ + 205, + 323, + 291, + 343 + ], + "score": 0.92, + "content": "\\mathbf { B } _ { n } ^ { ( n ^ { 2 } ) } \\dots \\mathbf { B } _ { \\frac { n } { 2 } } ^ { ( n ^ { 2 } ) } \\mathbf { \\dot { B } } _ { 2 } ^ { ( n ^ { 2 } ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 292, + 320, + 296, + 343 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 17, + "bbox_fs": [ + 104, + 302, + 506, + 343 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 346, + 211, + 358 + ], + "lines": [ + { + "bbox": [ + 106, + 346, + 212, + 359 + ], + "spans": [ + { + "bbox": [ + 106, + 346, + 212, + 359 + ], + "score": 1.0, + "content": "Step 3: Permute the rows:", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 19, + "bbox_fs": [ + 106, + 346, + 212, + 359 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 363, + 505, + 419 + ], + "lines": [ + { + "bbox": [ + 106, + 363, + 505, + 375 + ], + "spans": [ + { + "bbox": [ + 106, + 363, + 505, + 375 + ], + "score": 1.0, + "content": "We permute the rows (with a bit reversal permutation), which performs a bit reversal permutation", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 374, + 505, + 387 + ], + "spans": [ + { + "bbox": [ + 105, + 374, + 433, + 387 + ], + "score": 1.0, + "content": "on each column. This corresponds to left multiplication by a permutation matrix", + "type": "text" + }, + { + "bbox": [ + 433, + 375, + 446, + 385 + ], + "score": 0.87, + "content": "\\mathbf { P } _ { r }", + "type": "inline_equation" + }, + { + "bbox": [ + 447, + 374, + 505, + 387 + ], + "score": 1.0, + "content": ". Since we are", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 385, + 505, + 398 + ], + "spans": [ + { + "bbox": [ + 105, + 385, + 187, + 398 + ], + "score": 1.0, + "content": "permuting the rows,", + "type": "text" + }, + { + "bbox": [ + 188, + 385, + 201, + 396 + ], + "score": 0.88, + "content": "\\mathbf { P } _ { r }", + "type": "inline_equation" + }, + { + "bbox": [ + 202, + 385, + 386, + 398 + ], + "score": 1.0, + "content": "permutes the entries at the granularity of each", + "type": "text" + }, + { + "bbox": [ + 386, + 387, + 393, + 395 + ], + "score": 0.75, + "content": "n", + "type": "inline_equation" + }, + { + "bbox": [ + 394, + 385, + 505, + 398 + ], + "score": 1.0, + "content": "-chunk. Since Steps 1 and 2", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 395, + 505, + 409 + ], + "spans": [ + { + "bbox": [ + 105, + 395, + 302, + 409 + ], + "score": 1.0, + "content": "each performed an identical computation to each", + "type": "text" + }, + { + "bbox": [ + 302, + 398, + 309, + 406 + ], + "score": 0.76, + "content": "n", + "type": "inline_equation" + }, + { + "bbox": [ + 309, + 395, + 505, + 409 + ], + "score": 1.0, + "content": "-chunk we can move this row permutation before", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 407, + 343, + 419 + ], + "spans": [ + { + "bbox": [ + 105, + 407, + 181, + 419 + ], + "score": 1.0, + "content": "Step 2, combining", + "type": "text" + }, + { + "bbox": [ + 182, + 407, + 195, + 419 + ], + "score": 0.88, + "content": "\\mathbf { P } _ { c }", + "type": "inline_equation" + }, + { + "bbox": [ + 195, + 407, + 213, + 419 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 213, + 407, + 226, + 418 + ], + "score": 0.88, + "content": "\\mathbf { P } _ { r }", + "type": "inline_equation" + }, + { + "bbox": [ + 227, + 407, + 329, + 419 + ], + "score": 1.0, + "content": "into a single permutation", + "type": "text" + }, + { + "bbox": [ + 330, + 407, + 339, + 417 + ], + "score": 0.63, + "content": "\\mathbf { P }", + "type": "inline_equation" + }, + { + "bbox": [ + 339, + 407, + 343, + 419 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 22, + "bbox_fs": [ + 105, + 363, + 505, + 419 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 424, + 312, + 435 + ], + "lines": [ + { + "bbox": [ + 106, + 423, + 312, + 438 + ], + "spans": [ + { + "bbox": [ + 106, + 423, + 312, + 438 + ], + "score": 1.0, + "content": "Step 4: Multiply each column by a butterfly matrix", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 25, + "bbox_fs": [ + 106, + 423, + 312, + 438 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 441, + 505, + 520 + ], + "lines": [ + { + "bbox": [ + 106, + 441, + 505, + 452 + ], + "spans": [ + { + "bbox": [ + 106, + 441, + 505, + 452 + ], + "score": 1.0, + "content": "Consider multiplication by the first factor matrix. In each row, this matrix is taking linear combinations", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 451, + 506, + 464 + ], + "spans": [ + { + "bbox": [ + 106, + 451, + 279, + 464 + ], + "score": 1.0, + "content": "of adjacent column entries. In our length-", + "type": "text" + }, + { + "bbox": [ + 279, + 451, + 291, + 462 + ], + "score": 0.87, + "content": "\\cdot n ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 291, + 451, + 441, + 464 + ], + "score": 1.0, + "content": "vector, these entries will be exactly", + "type": "text" + }, + { + "bbox": [ + 441, + 454, + 448, + 462 + ], + "score": 0.69, + "content": "n", + "type": "inline_equation" + }, + { + "bbox": [ + 448, + 451, + 506, + 464 + ], + "score": 1.0, + "content": "indices apart.", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 103, + 458, + 510, + 484 + ], + "spans": [ + { + "bbox": [ + 103, + 458, + 400, + 484 + ], + "score": 1.0, + "content": "Therefore this multiplication can be handled by a butterfly factor matrix", + "type": "text" + }, + { + "bbox": [ + 401, + 462, + 425, + 479 + ], + "score": 0.92, + "content": "\\mathbf { B } _ { 2 n } ^ { ( n ^ { 2 } ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 425, + 458, + 510, + 484 + ], + "score": 1.0, + "content": ". Similarly, we find", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 477, + 506, + 490 + ], + "spans": [ + { + "bbox": [ + 106, + 477, + 506, + 490 + ], + "score": 1.0, + "content": "that this butterfly multiplication can be expressed as multiplication by a product of butterfly factor", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 487, + 505, + 513 + ], + "spans": [ + { + "bbox": [ + 106, + 488, + 142, + 506 + ], + "score": 1.0, + "content": "matrices", + "type": "text" + }, + { + "bbox": [ + 142, + 488, + 229, + 510 + ], + "score": 0.91, + "content": "\\mathbf { B } _ { n ^ { 2 } } ^ { ( n ^ { 2 } ) } \\cdot \\cdot . . \\mathbf { B } _ { \\frac { n ^ { 2 } } { 2 } } ^ { ( n ^ { 2 } ) } \\mathbf { B } _ { 2 n } ^ { ( n ^ { 2 } ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 229, + 487, + 231, + 513 + ], + "score": 1.0, + "content": ".", + "type": "text" + }, + { + "bbox": [ + 231, + 491, + 505, + 505 + ], + "score": 1.0, + "content": "Combined with the factor matrices from Step 2, these form a butterfly", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 509, + 190, + 520 + ], + "spans": [ + { + "bbox": [ + 105, + 509, + 135, + 520 + ], + "score": 1.0, + "content": "matrix", + "type": "text" + }, + { + "bbox": [ + 135, + 509, + 145, + 519 + ], + "score": 0.36, + "content": "\\mathbf { B }", + "type": "inline_equation" + }, + { + "bbox": [ + 145, + 509, + 174, + 520 + ], + "score": 1.0, + "content": "of size", + "type": "text" + }, + { + "bbox": [ + 174, + 509, + 186, + 519 + ], + "score": 0.84, + "content": "n ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 186, + 509, + 190, + 520 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 28.5, + "bbox_fs": [ + 103, + 441, + 510, + 520 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 525, + 505, + 560 + ], + "lines": [ + { + "bbox": [ + 105, + 525, + 504, + 538 + ], + "spans": [ + { + "bbox": [ + 105, + 525, + 494, + 538 + ], + "score": 1.0, + "content": "In all, we see that the 2-D DFT may be realized as multiplication by a permutation matrix", + "type": "text" + }, + { + "bbox": [ + 495, + 526, + 504, + 536 + ], + "score": 0.53, + "content": "\\mathbf { P }", + "type": "inline_equation" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 535, + 506, + 550 + ], + "spans": [ + { + "bbox": [ + 105, + 535, + 506, + 550 + ], + "score": 1.0, + "content": "followed by multiplication by a butterfly matrix B. The same argument as Lemma J.1 shows that", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 107, + 546, + 505, + 561 + ], + "spans": [ + { + "bbox": [ + 107, + 547, + 163, + 560 + ], + "score": 0.92, + "content": "\\mathbf { F } _ { n } ^ { 2 } \\in ( B B ^ { * } ) ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 163, + 546, + 168, + 561 + ], + "score": 1.0, + "content": ".", + "type": "text" + }, + { + "bbox": [ + 495, + 549, + 505, + 558 + ], + "score": 0.999, + "content": "□", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 33, + "bbox_fs": [ + 105, + 525, + 506, + 561 + ] + }, + { + "type": "text", + "bbox": [ + 108, + 565, + 505, + 599 + ], + "lines": [ + { + "bbox": [ + 105, + 565, + 506, + 579 + ], + "spans": [ + { + "bbox": [ + 105, + 565, + 506, + 579 + ], + "score": 1.0, + "content": "Remark J.11. An analogous argument (using the separation property of the respective transforms)", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 576, + 506, + 590 + ], + "spans": [ + { + "bbox": [ + 105, + 576, + 435, + 590 + ], + "score": 1.0, + "content": "can be used to argue that 2-D Discrete Sine and Discrete Cosine transforms are in", + "type": "text" + }, + { + "bbox": [ + 435, + 576, + 466, + 588 + ], + "score": 0.91, + "content": "( B B ^ { * } ) ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 466, + 576, + 506, + 590 + ], + "score": 1.0, + "content": ", and that", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 588, + 266, + 600 + ], + "spans": [ + { + "bbox": [ + 105, + 588, + 243, + 600 + ], + "score": 1.0, + "content": "2-D Hadamard Transforms are in", + "type": "text" + }, + { + "bbox": [ + 243, + 588, + 262, + 598 + ], + "score": 0.88, + "content": "B B ^ { * }", + "type": "inline_equation" + }, + { + "bbox": [ + 262, + 588, + 266, + 600 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 37 + } + ], + "index": 36, + "bbox_fs": [ + 105, + 565, + 506, + 600 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 601, + 427, + 614 + ], + "lines": [ + { + "bbox": [ + 105, + 600, + 424, + 615 + ], + "spans": [ + { + "bbox": [ + 105, + 600, + 181, + 615 + ], + "score": 1.0, + "content": "Lemma J.12. Let", + "type": "text" + }, + { + "bbox": [ + 182, + 601, + 196, + 614 + ], + "score": 0.9, + "content": "{ \\bf C } _ { n } ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 196, + 600, + 380, + 615 + ], + "score": 1.0, + "content": "be a 2-dimensional convolution matrix. Then", + "type": "text" + }, + { + "bbox": [ + 380, + 602, + 424, + 614 + ], + "score": 0.94, + "content": "\\mathbf { C } _ { n } ^ { 2 } \\in B B ^ { * }", + "type": "inline_equation" + } + ], + "index": 38 + } + ], + "index": 38, + "bbox_fs": [ + 105, + 600, + 424, + 615 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 625, + 505, + 681 + ], + "lines": [ + { + "bbox": [ + 104, + 624, + 504, + 650 + ], + "spans": [ + { + "bbox": [ + 104, + 624, + 246, + 650 + ], + "score": 1.0, + "content": "Proof. We can express a 2-D convis the 2-D Fourier transform and", + "type": "text" + }, + { + "bbox": [ + 279, + 624, + 317, + 650 + ], + "score": 1.0, + "content": "matrix as is the in", + "type": "text" + }, + { + "bbox": [ + 318, + 625, + 398, + 638 + ], + "score": 0.93, + "content": "{ \\bf C } _ { n } ^ { 2 } = ( { \\bf F } _ { n } ^ { 2 } ) ^ { - 1 } { \\bf D } { \\bf F } _ { n } ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 398, + 624, + 429, + 650 + ], + "score": 1.0, + "content": ", where ansform", + "type": "text" + }, + { + "bbox": [ + 429, + 626, + 439, + 636 + ], + "score": 0.65, + "content": "\\mathbf { D }", + "type": "inline_equation" + }, + { + "bbox": [ + 440, + 624, + 489, + 650 + ], + "score": 1.0, + "content": "is diagonal, From the pr", + "type": "text" + }, + { + "bbox": [ + 490, + 625, + 504, + 638 + ], + "score": 0.88, + "content": "{ \\bf F } _ { n } ^ { 2 }", + "type": "inline_equation" + } + ], + "index": 39 + }, + { + "bbox": [ + 246, + 636, + 278, + 649 + ], + "spans": [ + { + "bbox": [ + 246, + 636, + 278, + 649 + ], + "score": 0.93, + "content": "( \\mathbf { F } _ { n } ^ { 2 } ) ^ { - 1 }", + "type": "inline_equation" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 646, + 507, + 661 + ], + "spans": [ + { + "bbox": [ + 105, + 646, + 309, + 661 + ], + "score": 1.0, + "content": "of Lemma J.10, we see that that we can express", + "type": "text" + }, + { + "bbox": [ + 310, + 648, + 324, + 660 + ], + "score": 0.9, + "content": "{ \\bf F } _ { n } ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 324, + 646, + 387, + 661 + ], + "score": 1.0, + "content": "(and similarly", + "type": "text" + }, + { + "bbox": [ + 387, + 648, + 421, + 660 + ], + "score": 0.91, + "content": "( \\mathbf { F } _ { n } ^ { 2 } ) ^ { - 1 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 421, + 646, + 507, + 661 + ], + "score": 1.0, + "content": "as the product of a", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 658, + 506, + 672 + ], + "spans": [ + { + "bbox": [ + 105, + 658, + 506, + 672 + ], + "score": 1.0, + "content": "butterfly matrix and a permutation matrix. The rest of the argument is analogous to the proof of", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 106, + 669, + 505, + 682 + ], + "spans": [ + { + "bbox": [ + 106, + 669, + 156, + 682 + ], + "score": 1.0, + "content": "Lemma J.5.", + "type": "text" + }, + { + "bbox": [ + 495, + 671, + 505, + 681 + ], + "score": 0.999, + "content": "□", + "type": "text" + } + ], + "index": 43 + } + ], + "index": 41, + "bbox_fs": [ + 104, + 624, + 507, + 682 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 687, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 688, + 506, + 699 + ], + "spans": [ + { + "bbox": [ + 106, + 688, + 369, + 699 + ], + "score": 1.0, + "content": "Remark J.13. Using an inductive argument, we can show that all", + "type": "text" + }, + { + "bbox": [ + 369, + 689, + 375, + 698 + ], + "score": 0.62, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 375, + 688, + 428, + 699 + ], + "score": 1.0, + "content": "-dimensional", + "type": "text" + }, + { + "bbox": [ + 429, + 688, + 459, + 698 + ], + "score": 0.83, + "content": "( k \\in \\mathbb { Z } )", + "type": "inline_equation" + }, + { + "bbox": [ + 459, + 688, + 506, + 699 + ], + "score": 1.0, + "content": "variants of", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 698, + 506, + 711 + ], + "spans": [ + { + "bbox": [ + 105, + 698, + 251, + 711 + ], + "score": 1.0, + "content": "the above transforms, expressed as", + "type": "text" + }, + { + "bbox": [ + 251, + 699, + 285, + 709 + ], + "score": 0.91, + "content": "n ^ { k } \\times n ^ { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 286, + 698, + 393, + 711 + ], + "score": 1.0, + "content": "matrices are contained in", + "type": "text" + }, + { + "bbox": [ + 393, + 699, + 413, + 709 + ], + "score": 0.86, + "content": "B B ^ { * }", + "type": "inline_equation" + }, + { + "bbox": [ + 413, + 698, + 426, + 711 + ], + "score": 1.0, + "content": "or", + "type": "text" + }, + { + "bbox": [ + 426, + 699, + 457, + 711 + ], + "score": 0.91, + "content": "( B B ^ { * } ) ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 457, + 698, + 506, + 711 + ], + "score": 1.0, + "content": ". To do this,", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 710, + 504, + 722 + ], + "spans": [ + { + "bbox": [ + 105, + 710, + 385, + 722 + ], + "score": 1.0, + "content": "we use the separation property of the transforms to break them into a", + "type": "text" + }, + { + "bbox": [ + 386, + 710, + 409, + 720 + ], + "score": 0.87, + "content": "k - 1", + "type": "inline_equation" + }, + { + "bbox": [ + 410, + 710, + 504, + 722 + ], + "score": 1.0, + "content": "-dimensional transform", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 720, + 370, + 733 + ], + "spans": [ + { + "bbox": [ + 105, + 720, + 267, + 733 + ], + "score": 1.0, + "content": "(the inductive hypothesis) followed by a", + "type": "text" + }, + { + "bbox": [ + 268, + 722, + 273, + 730 + ], + "score": 0.57, + "content": "^ { l }", + "type": "inline_equation" + }, + { + "bbox": [ + 273, + 720, + 370, + 733 + ], + "score": 1.0, + "content": "-dimensional transform.", + "type": "text" + } + ], + "index": 47 + } + ], + "index": 45.5, + "bbox_fs": [ + 105, + 688, + 506, + 733 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "title", + "bbox": [ + 107, + 81, + 376, + 94 + ], + "lines": [ + { + "bbox": [ + 106, + 81, + 378, + 95 + ], + "spans": [ + { + "bbox": [ + 106, + 81, + 378, + 95 + ], + "score": 1.0, + "content": "K THE ORTHOGONAL KALEIDOSCOPE HIERARCHY", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "text", + "bbox": [ + 107, + 105, + 505, + 183 + ], + "lines": [ + { + "bbox": [ + 105, + 105, + 506, + 119 + ], + "spans": [ + { + "bbox": [ + 105, + 105, + 506, + 119 + ], + "score": 1.0, + "content": "Through practical application of the butterfly matrices, it has been found useful to constrain them", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 117, + 505, + 130 + ], + "spans": [ + { + "bbox": [ + 105, + 117, + 505, + 130 + ], + "score": 1.0, + "content": "in orthogonality. In Section K.1 we will modify the existing kaleidoscope hierarchy to create the", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 128, + 505, + 141 + ], + "spans": [ + { + "bbox": [ + 105, + 128, + 505, + 141 + ], + "score": 1.0, + "content": "orthogonal kaleidoscope hierarchy OBB. Then, in Section K.2, we will argue that all orthogonal", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 139, + 505, + 152 + ], + "spans": [ + { + "bbox": [ + 105, + 139, + 415, + 152 + ], + "score": 1.0, + "content": "matrices, and as a result all matrices, can also be expressed in this hierarchy in", + "type": "text" + }, + { + "bbox": [ + 415, + 139, + 437, + 151 + ], + "score": 0.92, + "content": "O ( n )", + "type": "inline_equation" + }, + { + "bbox": [ + 438, + 139, + 505, + 152 + ], + "score": 1.0, + "content": "width. Lastly, in", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 106, + 150, + 504, + 162 + ], + "spans": [ + { + "bbox": [ + 106, + 150, + 504, + 162 + ], + "score": 1.0, + "content": "Section K.3, we will argue that permutation matrices and sparse matrices also exist in this hierarchy", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 160, + 505, + 174 + ], + "spans": [ + { + "bbox": [ + 105, + 160, + 117, + 174 + ], + "score": 1.0, + "content": "in", + "type": "text" + }, + { + "bbox": [ + 117, + 161, + 138, + 173 + ], + "score": 0.87, + "content": "O ( 1 )", + "type": "inline_equation" + }, + { + "bbox": [ + 139, + 160, + 505, + 174 + ], + "score": 1.0, + "content": "width, which in turn implies a corresponding result for matrices with low-depth arithmetic", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 172, + 141, + 184 + ], + "spans": [ + { + "bbox": [ + 105, + 172, + 141, + 184 + ], + "score": 1.0, + "content": "circuits.", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 4 + }, + { + "type": "title", + "bbox": [ + 107, + 196, + 185, + 208 + ], + "lines": [ + { + "bbox": [ + 106, + 196, + 186, + 209 + ], + "spans": [ + { + "bbox": [ + 106, + 196, + 186, + 209 + ], + "score": 1.0, + "content": "K.1 DEFINITION", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 8 + }, + { + "type": "text", + "bbox": [ + 107, + 217, + 504, + 240 + ], + "lines": [ + { + "bbox": [ + 106, + 217, + 505, + 230 + ], + "spans": [ + { + "bbox": [ + 106, + 217, + 505, + 230 + ], + "score": 1.0, + "content": "The definition of the orthogonal butterfly is identical to the original butterfly, with the constraint that", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 106, + 228, + 382, + 241 + ], + "spans": [ + { + "bbox": [ + 106, + 228, + 382, + 241 + ], + "score": 1.0, + "content": "all butterfly factors are orthogonal. We specify this definition below:", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 9.5 + }, + { + "type": "text", + "bbox": [ + 108, + 243, + 502, + 267 + ], + "lines": [ + { + "bbox": [ + 106, + 241, + 505, + 256 + ], + "spans": [ + { + "bbox": [ + 106, + 241, + 430, + 256 + ], + "score": 1.0, + "content": "Definition K.1 (Analog of Definition 2.1). An orthogonal butterfly factor of size", + "type": "text" + }, + { + "bbox": [ + 431, + 243, + 456, + 254 + ], + "score": 0.91, + "content": "k \\geq 2", + "type": "inline_equation" + }, + { + "bbox": [ + 457, + 241, + 505, + 256 + ], + "score": 1.0, + "content": "(denoted as", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 253, + 298, + 268 + ], + "spans": [ + { + "bbox": [ + 106, + 253, + 121, + 267 + ], + "score": 0.86, + "content": "\\widetilde { \\mathbf B } _ { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 121, + 254, + 298, + 268 + ], + "score": 1.0, + "content": ") is a butterfly factor that is also orthogonal.", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 11.5 + }, + { + "type": "text", + "bbox": [ + 108, + 270, + 504, + 295 + ], + "lines": [ + { + "bbox": [ + 105, + 268, + 506, + 284 + ], + "spans": [ + { + "bbox": [ + 105, + 268, + 446, + 284 + ], + "score": 1.0, + "content": "Definition K.2 (Analog of Definition 2.3). An orthogonal butterfly matrix of size", + "type": "text" + }, + { + "bbox": [ + 446, + 273, + 453, + 280 + ], + "score": 0.45, + "content": "n", + "type": "inline_equation" + }, + { + "bbox": [ + 453, + 268, + 506, + 284 + ], + "score": 1.0, + "content": "(denoted as", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 281, + 418, + 297 + ], + "spans": [ + { + "bbox": [ + 106, + 281, + 127, + 294 + ], + "score": 0.68, + "content": "\\widetilde { \\mathbf { B } } ^ { ( n ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 128, + 281, + 418, + 297 + ], + "score": 1.0, + "content": ") is a butterfly matrix with all butterfly factor matrices being orthogonal.", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 13.5 + }, + { + "type": "text", + "bbox": [ + 108, + 303, + 504, + 326 + ], + "lines": [ + { + "bbox": [ + 105, + 302, + 505, + 317 + ], + "spans": [ + { + "bbox": [ + 105, + 302, + 505, + 317 + ], + "score": 1.0, + "content": "Note that the above definition implies that an orthogonal butterfly matrix, as well as its conjugate", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 315, + 207, + 327 + ], + "spans": [ + { + "bbox": [ + 105, + 315, + 207, + 327 + ], + "score": 1.0, + "content": "transpose, is orthogonal.", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 15.5 + }, + { + "type": "text", + "bbox": [ + 108, + 331, + 502, + 344 + ], + "lines": [ + { + "bbox": [ + 107, + 331, + 504, + 345 + ], + "spans": [ + { + "bbox": [ + 107, + 331, + 504, + 345 + ], + "score": 1.0, + "content": "The orthogonal hierarchy definition nearly mimics the original hierarchy Definition 2.4, as follows:", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 17 + }, + { + "type": "title", + "bbox": [ + 107, + 346, + 170, + 358 + ], + "lines": [ + { + "bbox": [ + 106, + 347, + 171, + 360 + ], + "spans": [ + { + "bbox": [ + 106, + 347, + 171, + 360 + ], + "score": 1.0, + "content": "Definition K.3.", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 18 + }, + { + "type": "text", + "bbox": [ + 130, + 365, + 506, + 468 + ], + "lines": [ + { + "bbox": [ + 130, + 364, + 409, + 381 + ], + "spans": [ + { + "bbox": [ + 130, + 364, + 203, + 381 + ], + "score": 1.0, + "content": "• We say that an", + "type": "text" + }, + { + "bbox": [ + 203, + 368, + 228, + 378 + ], + "score": 0.86, + "content": "n \\times n", + "type": "inline_equation" + }, + { + "bbox": [ + 228, + 364, + 258, + 381 + ], + "score": 1.0, + "content": "matrix", + "type": "text" + }, + { + "bbox": [ + 258, + 365, + 290, + 379 + ], + "score": 0.9, + "content": "\\mathbf { M } \\in \\widetilde { B }", + "type": "inline_equation" + }, + { + "bbox": [ + 290, + 364, + 360, + 381 + ], + "score": 1.0, + "content": "if we can express", + "type": "text" + }, + { + "bbox": [ + 361, + 365, + 405, + 378 + ], + "score": 0.9, + "content": "\\mathbf { M } = \\widetilde { \\mathbf { B } } ^ { ( n ) }", + "type": "inline_equation" + }, + { 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"inline_equation" + }, + { + "bbox": [ + 424, + 383, + 428, + 408 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 131, + 410, + 505, + 426 + ], + "spans": [ + { + "bbox": [ + 131, + 410, + 208, + 426 + ], + "score": 1.0, + "content": "• We say that an", + "type": "text" + }, + { + "bbox": [ + 208, + 414, + 235, + 423 + ], + "score": 0.88, + "content": "n \\times n", + "type": "inline_equation" + }, + { + "bbox": [ + 235, + 410, + 267, + 426 + ], + "score": 1.0, + "content": "matrix", + "type": "text" + }, + { + "bbox": [ + 267, + 412, + 318, + 424 + ], + "score": 0.78, + "content": "\\mathbf { M } \\in \\mathcal { O B B }", + "type": "inline_equation" + }, + { + "bbox": [ + 318, + 410, + 395, + 426 + ], + "score": 1.0, + "content": "if we can express", + "type": "text" + }, + { + "bbox": [ + 395, + 412, + 464, + 424 + ], + "score": 0.9, + "content": "\\mathbf { M } = \\mathbf { M } _ { 1 } \\mathbf { D } \\mathbf { M } _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 464, + 410, + 505, + 426 + ], + "score": 1.0, + "content": "for some", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 142, + 424, + 461, + 438 + ], + "spans": [ + { + "bbox": [ + 142, + 424, + 221, + 437 + ], + "score": 0.92, + "content": "\\mathbf { M } _ { 1 } \\in \\widetilde { B } , \\mathbf { M } _ { 2 } \\in \\widetilde { B } ^ { * }", + "type": "inline_equation" + }, + { + "bbox": [ + 221, + 424, + 362, + 438 + ], + "score": 1.0, + "content": ", and diagonal matrix D. Note that", + "type": "text" + }, + { + "bbox": [ + 363, + 426, + 372, + 435 + ], + "score": 0.45, + "content": "\\mathbf { D }", + "type": "inline_equation" + }, + { + "bbox": [ + 373, + 424, + 461, + 438 + ], + "score": 1.0, + "content": "need not be full rank.", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 132, + 443, + 506, + 459 + ], + "spans": [ + { + "bbox": [ + 132, + 443, + 254, + 459 + ], + "score": 1.0, + "content": "• Width w and expansion e in", + "type": "text" + }, + { + "bbox": [ + 255, + 444, + 291, + 456 + ], + "score": 0.91, + "content": "( O B B ) _ { e } ^ { w }", + "type": "inline_equation" + }, + { + "bbox": [ + 291, + 443, + 506, + 459 + ], + "score": 1.0, + "content": "mimic the same definition as in the original hierarchy,", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 141, + 454, + 354, + 469 + ], + "spans": [ + { + "bbox": [ + 141, + 454, + 232, + 469 + ], + "score": 1.0, + "content": "using OBB instead of", + "type": "text" + }, + { + "bbox": [ + 232, + 456, + 251, + 466 + ], + "score": 0.88, + "content": "B B ^ { * }", + "type": "inline_equation" + }, + { + "bbox": [ + 252, + 454, + 294, + 469 + ], + "score": 1.0, + "content": ", such that", + "type": "text" + }, + { + "bbox": [ + 295, + 455, + 351, + 468 + ], + "score": 0.93, + "content": "\\mathbf { E } \\in ( \\mathcal { O } B B ) ^ { w }", + "type": "inline_equation" + }, + { + "bbox": [ + 351, + 454, + 354, + 469 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 21.5 + }, + { + "type": "text", + "bbox": [ + 106, + 476, + 360, + 487 + ], + "lines": [ + { + "bbox": [ + 105, + 474, + 361, + 489 + ], + "spans": [ + { + "bbox": [ + 105, + 474, + 288, + 489 + ], + "score": 1.0, + "content": "By padding if necessary, we will assume that", + "type": "text" + }, + { + "bbox": [ + 288, + 478, + 295, + 486 + ], + "score": 0.77, + "content": "n", + "type": "inline_equation" + }, + { + "bbox": [ + 296, + 474, + 361, + 489 + ], + "score": 1.0, + "content": "is a power of 2.", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 25 + }, + { + "type": "title", + "bbox": [ + 107, + 500, + 197, + 512 + ], + "lines": [ + { + "bbox": [ + 106, + 500, + 197, + 513 + ], + "spans": [ + { + "bbox": [ + 106, + 500, + 197, + 513 + ], + "score": 1.0, + "content": "K.2 EXPRESSIVITY", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 26 + }, + { + "type": "text", + "bbox": [ + 107, + 521, + 505, + 554 + ], + "lines": [ + { + "bbox": [ + 105, + 520, + 505, + 534 + ], + "spans": [ + { + "bbox": [ + 105, + 520, + 459, + 534 + ], + "score": 1.0, + "content": "In this subsection we prove that all orthogonal (resp. unitary) matrices are contained in", + "type": "text" + }, + { + "bbox": [ + 459, + 521, + 487, + 532 + ], + "score": 0.88, + "content": "O B B ^ { n }", + "type": "inline_equation" + }, + { + "bbox": [ + 488, + 520, + 505, + 534 + ], + "score": 1.0, + "content": ". To", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 531, + 506, + 546 + ], + "spans": [ + { + "bbox": [ + 105, + 531, + 377, + 546 + ], + "score": 1.0, + "content": "do this, we consider the class of Householder reflections, given by", + "type": "text" + }, + { + "bbox": [ + 377, + 533, + 417, + 543 + ], + "score": 0.85, + "content": "\\mathbf { I } - 2 \\mathbf { u } \\mathbf { u } ^ { * }", + "type": "inline_equation" + }, + { + "bbox": [ + 417, + 531, + 506, + 546 + ], + "score": 1.0, + "content": "for any unit vector u", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 543, + 194, + 555 + ], + "spans": [ + { + "bbox": [ + 106, + 543, + 194, + 555 + ], + "score": 1.0, + "content": "(Householder, 1958):", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 28 + }, + { + "type": "text", + "bbox": [ + 106, + 557, + 444, + 570 + ], + "lines": [ + { + "bbox": [ + 104, + 555, + 446, + 572 + ], + "spans": [ + { + "bbox": [ + 104, + 555, + 446, + 572 + ], + "score": 1.0, + "content": "Lemma K.1. All Householder reflections are in OBB with inner diagonal matrix I.", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 30 + }, + { + "type": "text", + "bbox": [ + 106, + 578, + 503, + 601 + ], + "lines": [ + { + "bbox": [ + 106, + 578, + 505, + 591 + ], + "spans": [ + { + "bbox": [ + 106, + 578, + 505, + 591 + ], + "score": 1.0, + "content": "We will prove this lemma shortly. First, we use this lemma to present a decomposition for all", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 590, + 250, + 603 + ], + "spans": [ + { + "bbox": [ + 106, + 590, + 250, + 603 + ], + "score": 1.0, + "content": "orthogonal (resp. unitary) matrices.", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 31.5 + }, + { + "type": "text", + "bbox": [ + 109, + 604, + 450, + 616 + ], + "lines": [ + { + "bbox": [ + 106, + 601, + 450, + 619 + ], + "spans": [ + { + "bbox": [ + 106, + 601, + 217, + 619 + ], + "score": 1.0, + "content": "Lemma K.2. Let M be an", + "type": "text" + }, + { + "bbox": [ + 217, + 605, + 243, + 615 + ], + "score": 0.87, + "content": "n \\times n", + "type": "inline_equation" + }, + { + "bbox": [ + 243, + 601, + 376, + 619 + ], + "score": 1.0, + "content": "orthogonal/unitary matrix. Then", + "type": "text" + }, + { + "bbox": [ + 376, + 603, + 445, + 616 + ], + "score": 0.91, + "content": "\\mathbf { M } \\in ( O B B ) ^ { n - 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 446, + 601, + 450, + 619 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 33 + }, + { + "type": "text", + "bbox": [ + 107, + 627, + 505, + 662 + ], + "lines": [ + { + "bbox": [ + 105, + 627, + 505, + 641 + ], + "spans": [ + { + "bbox": [ + 105, + 627, + 435, + 641 + ], + "score": 1.0, + "content": "Proof. We consider the QR decomposition of M. It is known that we can compose", + "type": "text" + }, + { + "bbox": [ + 435, + 628, + 448, + 639 + ], + "score": 0.3, + "content": "\\mathbf { M }", + "type": "inline_equation" + }, + { + "bbox": [ + 448, + 627, + 505, + 641 + ], + "score": 1.0, + "content": "into a product", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 638, + 505, + 651 + ], + "spans": [ + { + "bbox": [ + 105, + 638, + 117, + 651 + ], + "score": 1.0, + "content": "of", + "type": "text" + }, + { + "bbox": [ + 117, + 640, + 142, + 650 + ], + "score": 0.87, + "content": "n - 1", + "type": "inline_equation" + }, + { + "bbox": [ + 142, + 638, + 505, + 651 + ], + "score": 1.0, + "content": "Householder reflections and an orthogonal/unitary diagonal matrix (Householder, 1958).10", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 650, + 344, + 662 + ], + "spans": [ + { + "bbox": [ + 105, + 650, + 317, + 662 + ], + "score": 1.0, + "content": "From Lemma K.1, each Householder reflection is in", + "type": "text" + }, + { + "bbox": [ + 317, + 650, + 340, + 660 + ], + "score": 0.3, + "content": "o s B", + "type": "inline_equation" + }, + { + "bbox": [ + 340, + 650, + 344, + 662 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 35 + }, + { + "type": "text", + "bbox": [ + 107, + 667, + 505, + 704 + ], + "lines": [ + { + "bbox": [ + 106, + 667, + 505, + 680 + ], + "spans": [ + { + "bbox": [ + 106, + 667, + 254, + 680 + ], + "score": 1.0, + "content": "To complete the proof, we argue that", + "type": "text" + }, + { + "bbox": [ + 255, + 667, + 265, + 677 + ], + "score": 0.55, + "content": "\\mathbf { R }", + "type": "inline_equation" + }, + { + "bbox": [ + 265, + 667, + 477, + 680 + ], + "score": 1.0, + "content": "can be folded into the rightmost butterfly matrix. Let", + "type": "text" + }, + { + "bbox": [ + 478, + 667, + 492, + 679 + ], + "score": 0.89, + "content": "\\mathbf { Q } _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 492, + 667, + 505, + 680 + ], + "score": 1.0, + "content": "be", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 103, + 675, + 507, + 696 + ], + "spans": [ + { + "bbox": [ + 103, + 675, + 267, + 696 + ], + "score": 1.0, + "content": "the rightmost butterfly factor matrix in", + "type": "text" + }, + { + "bbox": [ + 267, + 678, + 315, + 692 + ], + "score": 0.85, + "content": "\\mathbf { Q } ~ ( \\in \\widetilde { \\mathbf { B } } _ { n } ^ { ( n ) } )", + "type": "inline_equation" + }, + { + "bbox": [ + 315, + 675, + 416, + 696 + ], + "score": 1.0, + "content": ". Right multiplication of", + "type": "text" + }, + { + "bbox": [ + 417, + 680, + 431, + 692 + ], + "score": 0.92, + "content": "\\mathbf { Q } _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 432, + 675, + 445, + 696 + ], + "score": 1.0, + "content": "by", + "type": "text" + }, + { + "bbox": [ + 446, + 681, + 456, + 691 + ], + "score": 0.48, + "content": "\\mathbf { R }", + "type": "inline_equation" + }, + { + "bbox": [ + 456, + 675, + 507, + 696 + ], + "score": 1.0, + "content": "scales each", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 106, + 691, + 505, + 705 + ], + "spans": [ + { + "bbox": [ + 106, + 691, + 152, + 705 + ], + "score": 1.0, + "content": "columns of", + "type": "text" + }, + { + "bbox": [ + 153, + 692, + 167, + 703 + ], + "score": 0.88, + "content": "\\mathbf { Q } _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 167, + 691, + 203, + 705 + ], + "score": 1.0, + "content": "by some", + "type": "text" + }, + { + "bbox": [ + 203, + 692, + 228, + 702 + ], + "score": 0.9, + "content": "c \\in \\mathbb { C }", + "type": "inline_equation" + }, + { + "bbox": [ + 228, + 691, + 249, + 705 + ], + "score": 1.0, + "content": "with", + "type": "text" + }, + { + "bbox": [ + 249, + 692, + 284, + 704 + ], + "score": 0.92, + "content": "| | c | | = 1", + "type": "inline_equation" + }, + { + "bbox": [ + 284, + 691, + 288, + 705 + ], + "score": 1.0, + "content": "(", + "type": "text" + }, + { + "bbox": [ + 288, + 692, + 298, + 702 + ], + "score": 0.67, + "content": "\\mathbf { R }", + "type": "inline_equation" + }, + { + "bbox": [ + 298, + 691, + 505, + 705 + ], + "score": 1.0, + "content": "is unitary diagonal). This preserves both the sparsity", + "type": "text" + } + ], + "index": 39 + } + ], + "index": 38 + } + ], + "page_idx": 35, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 106, + 711, + 506, + 732 + ], + "lines": [ + { + "bbox": [ + 117, + 708, + 506, + 725 + ], + "spans": [ + { + "bbox": [ + 117, + 711, + 132, + 722 + ], + "score": 0.76, + "content": "{ } ^ { 1 0 } \\mathbf { Q }", + "type": "inline_equation" + }, + { + "bbox": [ + 132, + 708, + 268, + 725 + ], + "score": 1.0, + "content": "is the (orthogonal/unitary) product of", + "type": "text" + }, + { + "bbox": [ + 268, + 712, + 290, + 721 + ], + "score": 0.86, + "content": "n - 1", + "type": "inline_equation" + }, + { + "bbox": [ + 291, + 708, + 381, + 725 + ], + "score": 1.0, + "content": "Householder reflections.", + "type": "text" + }, + { + "bbox": [ + 382, + 712, + 390, + 721 + ], + "score": 0.35, + "content": "\\mathbf { R }", + "type": "inline_equation" + }, + { + "bbox": [ + 391, + 708, + 506, + 725 + ], + "score": 1.0, + "content": ", the remaining upper triangular", + "type": "text" + } + ] + }, + { + "bbox": [ + 105, + 721, + 440, + 733 + ], + "spans": [ + { + "bbox": [ + 105, + 721, + 440, + 733 + ], + "score": 1.0, + "content": "matrix after performing these reflections, is itself orthogonal/unitary, and therefore diagonal.", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 107, + 27, + 294, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 26, + 294, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 26, + 294, + 38 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2020", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 300, + 751, + 311, + 760 + ], + "lines": [ + { + "bbox": [ + 298, + 750, + 313, + 763 + ], + "spans": [ + { + "bbox": [ + 298, + 750, + 313, + 763 + ], + "score": 1.0, + "content": "36", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "title", + "bbox": [ + 107, + 81, + 376, + 94 + ], + "lines": [ + { + "bbox": [ + 106, + 81, + 378, + 95 + ], + "spans": [ + { + "bbox": [ + 106, + 81, + 378, + 95 + ], + "score": 1.0, + "content": "K THE ORTHOGONAL KALEIDOSCOPE HIERARCHY", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "text", + "bbox": [ + 107, + 105, + 505, + 183 + ], + "lines": [ + { + "bbox": [ + 105, + 105, + 506, + 119 + ], + "spans": [ + { + "bbox": [ + 105, + 105, + 506, + 119 + ], + "score": 1.0, + "content": "Through practical application of the butterfly matrices, it has been found useful to constrain them", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 117, + 505, + 130 + ], + "spans": [ + { + "bbox": [ + 105, + 117, + 505, + 130 + ], + "score": 1.0, + "content": "in orthogonality. In Section K.1 we will modify the existing kaleidoscope hierarchy to create the", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 128, + 505, + 141 + ], + "spans": [ + { + "bbox": [ + 105, + 128, + 505, + 141 + ], + "score": 1.0, + "content": "orthogonal kaleidoscope hierarchy OBB. Then, in Section K.2, we will argue that all orthogonal", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 139, + 505, + 152 + ], + "spans": [ + { + "bbox": [ + 105, + 139, + 415, + 152 + ], + "score": 1.0, + "content": "matrices, and as a result all matrices, can also be expressed in this hierarchy in", + "type": "text" + }, + { + "bbox": [ + 415, + 139, + 437, + 151 + ], + "score": 0.92, + "content": "O ( n )", + "type": "inline_equation" + }, + { + "bbox": [ + 438, + 139, + 505, + 152 + ], + "score": 1.0, + "content": "width. Lastly, in", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 106, + 150, + 504, + 162 + ], + "spans": [ + { + "bbox": [ + 106, + 150, + 504, + 162 + ], + "score": 1.0, + "content": "Section K.3, we will argue that permutation matrices and sparse matrices also exist in this hierarchy", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 160, + 505, + 174 + ], + "spans": [ + { + "bbox": [ + 105, + 160, + 117, + 174 + ], + "score": 1.0, + "content": "in", + "type": "text" + }, + { + "bbox": [ + 117, + 161, + 138, + 173 + ], + "score": 0.87, + "content": "O ( 1 )", + "type": "inline_equation" + }, + { + "bbox": [ + 139, + 160, + 505, + 174 + ], + "score": 1.0, + "content": "width, which in turn implies a corresponding result for matrices with low-depth arithmetic", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 172, + 141, + 184 + ], + "spans": [ + { + "bbox": [ + 105, + 172, + 141, + 184 + ], + "score": 1.0, + "content": "circuits.", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 4, + "bbox_fs": [ + 105, + 105, + 506, + 184 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 196, + 185, + 208 + ], + "lines": [ + { + "bbox": [ + 106, + 196, + 186, + 209 + ], + "spans": [ + { + "bbox": [ + 106, + 196, + 186, + 209 + ], + "score": 1.0, + "content": "K.1 DEFINITION", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 8 + }, + { + "type": "text", + "bbox": [ + 107, + 217, + 504, + 240 + ], + "lines": [ + { + "bbox": [ + 106, + 217, + 505, + 230 + ], + "spans": [ + { + "bbox": [ + 106, + 217, + 505, + 230 + ], + "score": 1.0, + "content": "The definition of the orthogonal butterfly is identical to the original butterfly, with the constraint that", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 106, + 228, + 382, + 241 + ], + "spans": [ + { + "bbox": [ + 106, + 228, + 382, + 241 + ], + "score": 1.0, + "content": "all butterfly factors are orthogonal. We specify this definition below:", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 9.5, + "bbox_fs": [ + 106, + 217, + 505, + 241 + ] + }, + { + "type": "text", + "bbox": [ + 108, + 243, + 502, + 267 + ], + "lines": [ + { + "bbox": [ + 106, + 241, + 505, + 256 + ], + "spans": [ + { + "bbox": [ + 106, + 241, + 430, + 256 + ], + "score": 1.0, + "content": "Definition K.1 (Analog of Definition 2.1). An orthogonal butterfly factor of size", + "type": "text" + }, + { + "bbox": [ + 431, + 243, + 456, + 254 + ], + "score": 0.91, + "content": "k \\geq 2", + "type": "inline_equation" + }, + { + "bbox": [ + 457, + 241, + 505, + 256 + ], + "score": 1.0, + "content": "(denoted as", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 253, + 298, + 268 + ], + "spans": [ + { + "bbox": [ + 106, + 253, + 121, + 267 + ], + "score": 0.86, + "content": "\\widetilde { \\mathbf B } _ { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 121, + 254, + 298, + 268 + ], + "score": 1.0, + "content": ") is a butterfly factor that is also orthogonal.", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 11.5, + "bbox_fs": [ + 106, + 241, + 505, + 268 + ] + }, + { + "type": "text", + "bbox": [ + 108, + 270, + 504, + 295 + ], + "lines": [ + { + "bbox": [ + 105, + 268, + 506, + 284 + ], + "spans": [ + { + "bbox": [ + 105, + 268, + 446, + 284 + ], + "score": 1.0, + "content": "Definition K.2 (Analog of Definition 2.3). An orthogonal butterfly matrix of size", + "type": "text" + }, + { + "bbox": [ + 446, + 273, + 453, + 280 + ], + "score": 0.45, + "content": "n", + "type": "inline_equation" + }, + { + "bbox": [ + 453, + 268, + 506, + 284 + ], + "score": 1.0, + "content": "(denoted as", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 281, + 418, + 297 + ], + "spans": [ + { + "bbox": [ + 106, + 281, + 127, + 294 + ], + "score": 0.68, + "content": "\\widetilde { \\mathbf { B } } ^ { ( n ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 128, + 281, + 418, + 297 + ], + "score": 1.0, + "content": ") is a butterfly matrix with all butterfly factor matrices being orthogonal.", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 13.5, + "bbox_fs": [ + 105, + 268, + 506, + 297 + ] + }, + { + "type": "text", + "bbox": [ + 108, + 303, + 504, + 326 + ], + "lines": [ + { + "bbox": [ + 105, + 302, + 505, + 317 + ], + "spans": [ + { + "bbox": [ + 105, + 302, + 505, + 317 + ], + "score": 1.0, + "content": "Note that the above definition implies that an orthogonal butterfly matrix, as well as its conjugate", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 315, + 207, + 327 + ], + "spans": [ + { + "bbox": [ + 105, + 315, + 207, + 327 + ], + "score": 1.0, + "content": "transpose, is orthogonal.", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 15.5, + "bbox_fs": [ + 105, + 302, + 505, + 327 + ] + }, + { + "type": "text", + "bbox": [ + 108, + 331, + 502, + 344 + ], + "lines": [ + { + "bbox": [ + 107, + 331, + 504, + 345 + ], + "spans": [ + { + "bbox": [ + 107, + 331, + 504, + 345 + ], + "score": 1.0, + "content": "The orthogonal hierarchy definition nearly mimics the original hierarchy Definition 2.4, as follows:", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 17, + "bbox_fs": [ + 107, + 331, + 504, + 345 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 346, + 170, + 358 + ], + "lines": [ + { + "bbox": [ + 106, + 347, + 171, + 360 + ], + "spans": [ + { + "bbox": [ + 106, + 347, + 171, + 360 + ], + "score": 1.0, + "content": "Definition K.3.", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 18 + }, + { + "type": "text", + "bbox": [ + 130, + 365, + 506, + 468 + ], + "lines": [ + { + "bbox": [ + 130, + 364, + 409, + 381 + ], + "spans": [ + { + "bbox": [ + 130, + 364, + 203, + 381 + ], + "score": 1.0, + "content": "• We say that an", + "type": "text" + }, + { + "bbox": [ + 203, + 368, + 228, + 378 + ], + "score": 0.86, + "content": "n \\times n", + "type": "inline_equation" + }, + { + "bbox": [ + 228, + 364, + 258, + 381 + ], + "score": 1.0, + "content": "matrix", + "type": "text" + }, + { + "bbox": [ + 258, + 365, + 290, + 379 + ], + "score": 0.9, + "content": "\\mathbf { M } \\in \\widetilde { B }", + "type": "inline_equation" + }, + { + "bbox": [ + 290, + 364, + 360, + 381 + ], + "score": 1.0, + "content": "if we can express", + "type": "text" + }, + { + "bbox": [ + 361, + 365, + 405, + 378 + ], + "score": 0.9, + "content": "\\mathbf { M } = \\widetilde { \\mathbf { B } } ^ { ( n ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 406, + 364, + 409, + 381 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 128, + 383, + 428, + 408 + ], + "spans": [ + { + "bbox": [ + 128, + 383, + 203, + 408 + ], + "score": 1.0, + "content": "• We say that an", + "type": "text" + }, + { + "bbox": [ + 203, + 392, + 228, + 401 + ], + "score": 0.86, + "content": "n \\times n", + "type": "inline_equation" + }, + { + "bbox": [ + 228, + 383, + 258, + 408 + ], + "score": 1.0, + "content": "matrix", + "type": "text" + }, + { + "bbox": [ + 258, + 388, + 293, + 402 + ], + "score": 0.9, + "content": "\\mathbf { M } \\in \\widetilde { B } ^ { \\ast }", + "type": "inline_equation" + }, + { + "bbox": [ + 294, + 383, + 365, + 408 + ], + "score": 1.0, + "content": "if we can express", + "type": "text" + }, + { + "bbox": [ + 365, + 385, + 424, + 407 + ], + "score": 0.88, + "content": "\\mathbf { M } = \\left[ \\widetilde { \\mathbf { B } } ^ { ( n ) } \\right] ^ { * }", + "type": "inline_equation" + }, + { + "bbox": [ + 424, + 383, + 428, + 408 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 131, + 410, + 505, + 426 + ], + "spans": [ + { + "bbox": [ + 131, + 410, + 208, + 426 + ], + "score": 1.0, + "content": "• We say that an", + "type": "text" + }, + { + "bbox": [ + 208, + 414, + 235, + 423 + ], + "score": 0.88, + "content": "n \\times n", + "type": "inline_equation" + }, + { + "bbox": [ + 235, + 410, + 267, + 426 + ], + "score": 1.0, + "content": "matrix", + "type": "text" + }, + { + "bbox": [ + 267, + 412, + 318, + 424 + ], + "score": 0.78, + "content": "\\mathbf { M } \\in \\mathcal { O B B }", + "type": "inline_equation" + }, + { + "bbox": [ + 318, + 410, + 395, + 426 + ], + "score": 1.0, + "content": "if we can express", + "type": "text" + }, + { + "bbox": [ + 395, + 412, + 464, + 424 + ], + "score": 0.9, + "content": "\\mathbf { M } = \\mathbf { M } _ { 1 } \\mathbf { D } \\mathbf { M } _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 464, + 410, + 505, + 426 + ], + "score": 1.0, + "content": "for some", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 142, + 424, + 461, + 438 + ], + "spans": [ + { + "bbox": [ + 142, + 424, + 221, + 437 + ], + "score": 0.92, + "content": "\\mathbf { M } _ { 1 } \\in \\widetilde { B } , \\mathbf { M } _ { 2 } \\in \\widetilde { B } ^ { * }", + "type": "inline_equation" + }, + { + "bbox": [ + 221, + 424, + 362, + 438 + ], + "score": 1.0, + "content": ", and diagonal matrix D. Note that", + "type": "text" + }, + { + "bbox": [ + 363, + 426, + 372, + 435 + ], + "score": 0.45, + "content": "\\mathbf { D }", + "type": "inline_equation" + }, + { + "bbox": [ + 373, + 424, + 461, + 438 + ], + "score": 1.0, + "content": "need not be full rank.", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 132, + 443, + 506, + 459 + ], + "spans": [ + { + "bbox": [ + 132, + 443, + 254, + 459 + ], + "score": 1.0, + "content": "• Width w and expansion e in", + "type": "text" + }, + { + "bbox": [ + 255, + 444, + 291, + 456 + ], + "score": 0.91, + "content": "( O B B ) _ { e } ^ { w }", + "type": "inline_equation" + }, + { + "bbox": [ + 291, + 443, + 506, + 459 + ], + "score": 1.0, + "content": "mimic the same definition as in the original hierarchy,", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 141, + 454, + 354, + 469 + ], + "spans": [ + { + "bbox": [ + 141, + 454, + 232, + 469 + ], + "score": 1.0, + "content": "using OBB instead of", + "type": "text" + }, + { + "bbox": [ + 232, + 456, + 251, + 466 + ], + "score": 0.88, + "content": "B B ^ { * }", + "type": "inline_equation" + }, + { + "bbox": [ + 252, + 454, + 294, + 469 + ], + "score": 1.0, + "content": ", such that", + "type": "text" + }, + { + "bbox": [ + 295, + 455, + 351, + 468 + ], + "score": 0.93, + "content": "\\mathbf { E } \\in ( \\mathcal { O } B B ) ^ { w }", + "type": "inline_equation" + }, + { + "bbox": [ + 351, + 454, + 354, + 469 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 21.5, + "bbox_fs": [ + 128, + 364, + 506, + 469 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 476, + 360, + 487 + ], + "lines": [ + { + "bbox": [ + 105, + 474, + 361, + 489 + ], + "spans": [ + { + "bbox": [ + 105, + 474, + 288, + 489 + ], + "score": 1.0, + "content": "By padding if necessary, we will assume that", + "type": "text" + }, + { + "bbox": [ + 288, + 478, + 295, + 486 + ], + "score": 0.77, + "content": "n", + "type": "inline_equation" + }, + { + "bbox": [ + 296, + 474, + 361, + 489 + ], + "score": 1.0, + "content": "is a power of 2.", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 25, + "bbox_fs": [ + 105, + 474, + 361, + 489 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 500, + 197, + 512 + ], + "lines": [ + { + "bbox": [ + 106, + 500, + 197, + 513 + ], + "spans": [ + { + "bbox": [ + 106, + 500, + 197, + 513 + ], + "score": 1.0, + "content": "K.2 EXPRESSIVITY", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 26 + }, + { + "type": "text", + "bbox": [ + 107, + 521, + 505, + 554 + ], + "lines": [ + { + "bbox": [ + 105, + 520, + 505, + 534 + ], + "spans": [ + { + "bbox": [ + 105, + 520, + 459, + 534 + ], + "score": 1.0, + "content": "In this subsection we prove that all orthogonal (resp. unitary) matrices are contained in", + "type": "text" + }, + { + "bbox": [ + 459, + 521, + 487, + 532 + ], + "score": 0.88, + "content": "O B B ^ { n }", + "type": "inline_equation" + }, + { + "bbox": [ + 488, + 520, + 505, + 534 + ], + "score": 1.0, + "content": ". To", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 531, + 506, + 546 + ], + "spans": [ + { + "bbox": [ + 105, + 531, + 377, + 546 + ], + "score": 1.0, + "content": "do this, we consider the class of Householder reflections, given by", + "type": "text" + }, + { + "bbox": [ + 377, + 533, + 417, + 543 + ], + "score": 0.85, + "content": "\\mathbf { I } - 2 \\mathbf { u } \\mathbf { u } ^ { * }", + "type": "inline_equation" + }, + { + "bbox": [ + 417, + 531, + 506, + 546 + ], + "score": 1.0, + "content": "for any unit vector u", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 543, + 194, + 555 + ], + "spans": [ + { + "bbox": [ + 106, + 543, + 194, + 555 + ], + "score": 1.0, + "content": "(Householder, 1958):", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 28, + "bbox_fs": [ + 105, + 520, + 506, + 555 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 557, + 444, + 570 + ], + "lines": [ + { + "bbox": [ + 104, + 555, + 446, + 572 + ], + "spans": [ + { + "bbox": [ + 104, + 555, + 446, + 572 + ], + "score": 1.0, + "content": "Lemma K.1. All Householder reflections are in OBB with inner diagonal matrix I.", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 30, + "bbox_fs": [ + 104, + 555, + 446, + 572 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 578, + 503, + 601 + ], + "lines": [ + { + "bbox": [ + 106, + 578, + 505, + 591 + ], + "spans": [ + { + "bbox": [ + 106, + 578, + 505, + 591 + ], + "score": 1.0, + "content": "We will prove this lemma shortly. First, we use this lemma to present a decomposition for all", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 590, + 250, + 603 + ], + "spans": [ + { + "bbox": [ + 106, + 590, + 250, + 603 + ], + "score": 1.0, + "content": "orthogonal (resp. unitary) matrices.", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 31.5, + "bbox_fs": [ + 106, + 578, + 505, + 603 + ] + }, + { + "type": "text", + "bbox": [ + 109, + 604, + 450, + 616 + ], + "lines": [ + { + "bbox": [ + 106, + 601, + 450, + 619 + ], + "spans": [ + { + "bbox": [ + 106, + 601, + 217, + 619 + ], + "score": 1.0, + "content": "Lemma K.2. Let M be an", + "type": "text" + }, + { + "bbox": [ + 217, + 605, + 243, + 615 + ], + "score": 0.87, + "content": "n \\times n", + "type": "inline_equation" + }, + { + "bbox": [ + 243, + 601, + 376, + 619 + ], + "score": 1.0, + "content": "orthogonal/unitary matrix. Then", + "type": "text" + }, + { + "bbox": [ + 376, + 603, + 445, + 616 + ], + "score": 0.91, + "content": "\\mathbf { M } \\in ( O B B ) ^ { n - 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 446, + 601, + 450, + 619 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 33, + "bbox_fs": [ + 106, + 601, + 450, + 619 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 627, + 505, + 662 + ], + "lines": [ + { + "bbox": [ + 105, + 627, + 505, + 641 + ], + "spans": [ + { + "bbox": [ + 105, + 627, + 435, + 641 + ], + "score": 1.0, + "content": "Proof. We consider the QR decomposition of M. It is known that we can compose", + "type": "text" + }, + { + "bbox": [ + 435, + 628, + 448, + 639 + ], + "score": 0.3, + "content": "\\mathbf { M }", + "type": "inline_equation" + }, + { + "bbox": [ + 448, + 627, + 505, + 641 + ], + "score": 1.0, + "content": "into a product", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 638, + 505, + 651 + ], + "spans": [ + { + "bbox": [ + 105, + 638, + 117, + 651 + ], + "score": 1.0, + "content": "of", + "type": "text" + }, + { + "bbox": [ + 117, + 640, + 142, + 650 + ], + "score": 0.87, + "content": "n - 1", + "type": "inline_equation" + }, + { + "bbox": [ + 142, + 638, + 505, + 651 + ], + "score": 1.0, + "content": "Householder reflections and an orthogonal/unitary diagonal matrix (Householder, 1958).10", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 650, + 344, + 662 + ], + "spans": [ + { + "bbox": [ + 105, + 650, + 317, + 662 + ], + "score": 1.0, + "content": "From Lemma K.1, each Householder reflection is in", + "type": "text" + }, + { + "bbox": [ + 317, + 650, + 340, + 660 + ], + "score": 0.3, + "content": "o s B", + "type": "inline_equation" + }, + { + "bbox": [ + 340, + 650, + 344, + 662 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 35, + "bbox_fs": [ + 105, + 627, + 505, + 662 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 667, + 505, + 704 + ], + "lines": [ + { + "bbox": [ + 106, + 667, + 505, + 680 + ], + "spans": [ + { + "bbox": [ + 106, + 667, + 254, + 680 + ], + "score": 1.0, + "content": "To complete the proof, we argue that", + "type": "text" + }, + { + "bbox": [ + 255, + 667, + 265, + 677 + ], + "score": 0.55, + "content": "\\mathbf { R }", + "type": "inline_equation" + }, + { + "bbox": [ + 265, + 667, + 477, + 680 + ], + "score": 1.0, + "content": "can be folded into the rightmost butterfly matrix. Let", + "type": "text" + }, + { + "bbox": [ + 478, + 667, + 492, + 679 + ], + "score": 0.89, + "content": "\\mathbf { Q } _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 492, + 667, + 505, + 680 + ], + "score": 1.0, + "content": "be", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 103, + 675, + 507, + 696 + ], + "spans": [ + { + "bbox": [ + 103, + 675, + 267, + 696 + ], + "score": 1.0, + "content": "the rightmost butterfly factor matrix in", + "type": "text" + }, + { + "bbox": [ + 267, + 678, + 315, + 692 + ], + "score": 0.85, + "content": "\\mathbf { Q } ~ ( \\in \\widetilde { \\mathbf { B } } _ { n } ^ { ( n ) } )", + "type": "inline_equation" + }, + { + "bbox": [ + 315, + 675, + 416, + 696 + ], + "score": 1.0, + "content": ". Right multiplication of", + "type": "text" + }, + { + "bbox": [ + 417, + 680, + 431, + 692 + ], + "score": 0.92, + "content": "\\mathbf { Q } _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 432, + 675, + 445, + 696 + ], + "score": 1.0, + "content": "by", + "type": "text" + }, + { + "bbox": [ + 446, + 681, + 456, + 691 + ], + "score": 0.48, + "content": "\\mathbf { R }", + "type": "inline_equation" + }, + { + "bbox": [ + 456, + 675, + 507, + 696 + ], + "score": 1.0, + "content": "scales each", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 106, + 691, + 505, + 705 + ], + "spans": [ + { + "bbox": [ + 106, + 691, + 152, + 705 + ], + "score": 1.0, + "content": "columns of", + "type": "text" + }, + { + "bbox": [ + 153, + 692, + 167, + 703 + ], + "score": 0.88, + "content": "\\mathbf { Q } _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 167, + 691, + 203, + 705 + ], + "score": 1.0, + "content": "by some", + "type": "text" + }, + { + "bbox": [ + 203, + 692, + 228, + 702 + ], + "score": 0.9, + "content": "c \\in \\mathbb { C }", + "type": "inline_equation" + }, + { + "bbox": [ + 228, + 691, + 249, + 705 + ], + "score": 1.0, + "content": "with", + "type": "text" + }, + { + "bbox": [ + 249, + 692, + 284, + 704 + ], + "score": 0.92, + "content": "| | c | | = 1", + "type": "inline_equation" + }, + { + "bbox": [ + 284, + 691, + 288, + 705 + ], + "score": 1.0, + "content": "(", + "type": "text" + }, + { + "bbox": [ + 288, + 692, + 298, + 702 + ], + "score": 0.67, + "content": "\\mathbf { R }", + "type": "inline_equation" + }, + { + "bbox": [ + 298, + 691, + 505, + 705 + ], + "score": 1.0, + "content": "is unitary diagonal). This preserves both the sparsity", + "type": "text" + } + ], + "index": 39 + } + ], + "index": 38, + "bbox_fs": [ + 103, + 667, + 507, + 705 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 105, + 82, + 505, + 106 + ], + "lines": [ + { + "bbox": [ + 105, + 82, + 507, + 95 + ], + "spans": [ + { + "bbox": [ + 105, + 82, + 146, + 95 + ], + "score": 1.0, + "content": "pattern of", + "type": "text" + }, + { + "bbox": [ + 147, + 83, + 161, + 94 + ], + "score": 0.88, + "content": "\\mathbf { Q } _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 162, + 82, + 464, + 95 + ], + "score": 1.0, + "content": "and the orthogonality of its columns. Moreover, the norm of each column of", + "type": "text" + }, + { + "bbox": [ + 464, + 83, + 487, + 94 + ], + "score": 0.9, + "content": "\\mathbf { Q } _ { 1 } \\mathbf { R }", + "type": "inline_equation" + }, + { + "bbox": [ + 487, + 82, + 507, + 95 + ], + "score": 1.0, + "content": "is 1.", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 92, + 507, + 106 + ], + "spans": [ + { + "bbox": [ + 105, + 92, + 149, + 106 + ], + "score": 1.0, + "content": "Therefore,", + "type": "text" + }, + { + "bbox": [ + 150, + 94, + 173, + 105 + ], + "score": 0.9, + "content": "\\mathbf { Q } _ { 1 } \\mathbf { R }", + "type": "inline_equation" + }, + { + "bbox": [ + 173, + 92, + 340, + 106 + ], + "score": 1.0, + "content": "is an orthogonal butterfly factor matrix, so", + "type": "text" + }, + { + "bbox": [ + 340, + 93, + 440, + 106 + ], + "score": 0.91, + "content": "\\mathbf { M } = \\mathbf { Q } \\mathbf { R } \\in ( { \\mathcal { O } } B B ) ^ { n - 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 440, + 92, + 507, + 106 + ], + "score": 1.0, + "content": ", as desired.", + "type": "text" + } + ], + "index": 1 + } + ], + "index": 0.5 + }, + { + "type": "text", + "bbox": [ + 108, + 117, + 276, + 128 + ], + "lines": [ + { + "bbox": [ + 106, + 117, + 276, + 129 + ], + "spans": [ + { + "bbox": [ + 106, + 117, + 276, + 129 + ], + "score": 1.0, + "content": "We now return to the proof of Lemma K.1", + "type": "text" + } + ], + "index": 2 + } + ], + "index": 2 + }, + { + "type": "text", + "bbox": [ + 106, + 140, + 504, + 165 + ], + "lines": [ + { + "bbox": [ + 105, + 138, + 505, + 156 + ], + "spans": [ + { + "bbox": [ + 105, + 138, + 222, + 156 + ], + "score": 1.0, + "content": "Proof of Lemma K.1. Given", + "type": "text" + }, + { + "bbox": [ + 222, + 142, + 254, + 152 + ], + "score": 0.88, + "content": "\\mathbf { u } \\in \\mathbb { C } ^ { n }", + "type": "inline_equation" + }, + { + "bbox": [ + 255, + 138, + 259, + 156 + ], + "score": 0.0, + "content": "", + "type": "text" + }, + { + "bbox": [ + 259, + 144, + 267, + 151 + ], + "score": 0.67, + "content": "\\scriptstyle { \\dot { n } }", + "type": "inline_equation" + }, + { + "bbox": [ + 267, + 138, + 339, + 156 + ], + "score": 1.0, + "content": "a power of 2), let", + "type": "text" + }, + { + "bbox": [ + 339, + 140, + 505, + 154 + ], + "score": 0.76, + "content": "\\mathbf { u } _ { 0 } = \\mathbf { u } [ : n / 2 ] \\in \\mathbb { C } ^ { n / 2 } , \\mathbf { u } _ { 1 } = \\mathbf { u } [ n / 2 : ] \\in", + "type": "inline_equation" + } + ], + "index": 3 + }, + { + "bbox": [ + 107, + 151, + 288, + 166 + ], + "spans": [ + { + "bbox": [ + 107, + 153, + 128, + 164 + ], + "score": 0.88, + "content": "\\mathbb { C } ^ { n / 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 128, + 151, + 277, + 166 + ], + "score": 1.0, + "content": "denote the first and second halves of", + "type": "text" + }, + { + "bbox": [ + 277, + 156, + 285, + 164 + ], + "score": 0.32, + "content": "\\mathbf { u }", + "type": "inline_equation" + }, + { + "bbox": [ + 285, + 151, + 288, + 166 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 3.5 + }, + { + "type": "text", + "bbox": [ + 106, + 170, + 505, + 219 + ], + "lines": [ + { + "bbox": [ + 105, + 169, + 505, + 182 + ], + "spans": [ + { + "bbox": [ + 105, + 169, + 160, + 182 + ], + "score": 1.0, + "content": "To show that", + "type": "text" + }, + { + "bbox": [ + 161, + 171, + 205, + 181 + ], + "score": 0.91, + "content": "\\mathbf { H } \\in \\mathcal { O } B B", + "type": "inline_equation" + }, + { + "bbox": [ + 205, + 169, + 316, + 182 + ], + "score": 1.0, + "content": "with inner diagonal matrix", + "type": "text" + }, + { + "bbox": [ + 317, + 171, + 322, + 181 + ], + "score": 0.32, + "content": "\\mathbf { I }", + "type": "inline_equation" + }, + { + "bbox": [ + 323, + 169, + 505, + 182 + ], + "score": 1.0, + "content": ", we proceed by induction. The base case for", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 107, + 182, + 505, + 193 + ], + "spans": [ + { + "bbox": [ + 107, + 182, + 132, + 192 + ], + "score": 0.89, + "content": "n = 2", + "type": "inline_equation" + }, + { + "bbox": [ + 132, + 182, + 402, + 193 + ], + "score": 1.0, + "content": "is trivial. It suffices to show that there exist unitary butterfly factors", + "type": "text" + }, + { + "bbox": [ + 402, + 182, + 423, + 193 + ], + "score": 0.86, + "content": "\\mathbf { L } , \\mathbf { R }", + "type": "inline_equation" + }, + { + "bbox": [ + 424, + 182, + 505, + 193 + ], + "score": 1.0, + "content": "such that LHR has", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 192, + 429, + 219 + ], + "spans": [ + { + "bbox": [ + 105, + 192, + 144, + 216 + ], + "score": 1.0, + "content": "the form", + "type": "text" + }, + { + "bbox": [ + 144, + 192, + 277, + 219 + ], + "score": 0.89, + "content": "\\begin{array}{c} \\begin{array} { r l } { \\left[ { \\displaystyle { \\bf { I } } _ { n / 2 } - 2 { \\bf { v } } _ { 0 } { \\bf { v } } _ { 0 } ^ { * } } & { { \\bf { 0 } } } \\\\ { { \\bf { 0 } } } & { { \\bf { I } } _ { n / 2 } - 2 { \\bf { v } } _ { 1 } { \\bf { v } } _ { 1 } ^ { * } } \\end{array} \\right] } & { { } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 277, + 198, + 365, + 211 + ], + "score": 1.0, + "content": "for some unit vectors", + "type": "text" + }, + { + "bbox": [ + 366, + 198, + 424, + 211 + ], + "score": 0.91, + "content": "\\mathbf { v } _ { 0 } , \\mathbf { v } _ { 1 } \\in \\mathbb { C } ^ { n / 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 425, + 198, + 429, + 211 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 6 + }, + { + "type": "text", + "bbox": [ + 106, + 222, + 135, + 234 + ], + "lines": [ + { + "bbox": [ + 105, + 221, + 136, + 236 + ], + "spans": [ + { + "bbox": [ + 105, + 221, + 136, + 236 + ], + "score": 1.0, + "content": "Define", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 8 + }, + { + "type": "interline_equation", + "bbox": [ + 132, + 232, + 479, + 273 + ], + "lines": [ + { + "bbox": [ + 132, + 232, + 479, + 273 + ], + "spans": [ + { + "bbox": [ + 132, + 232, + 479, + 273 + ], + "score": 0.94, + "content": "\\left( \\mathbf { v } _ { 0 } [ i ] , \\mathbf { v } _ { 1 } [ i ] \\right) = \\left\\{ \\begin{array} { l l } { \\left( \\frac { \\mathbf { u } _ { 0 } [ i ] } { \\sqrt { | \\mathbf { u } _ { 0 } [ i ] | ^ { 2 } + | \\mathbf { u } _ { 1 } [ i ] | ^ { 2 } } } , \\frac { \\mathbf { u } _ { 1 } [ i ] } { \\sqrt { | \\mathbf { u } _ { 0 } [ i ] | ^ { 2 } + | \\mathbf { u } _ { 1 } [ i ] | ^ { 2 } } } \\right) } & { \\mathrm { ~ i f ~ } | \\mathbf { u } _ { 0 } [ i ] | ^ { 2 } + | \\mathbf { u } _ { 1 } [ i ] | ^ { 2 } \\neq 0 } \\\\ { ( 1 , 0 ) } & { \\mathrm { ~ o t h e r w i s e } } \\end{array} \\right. .", + "type": "interline_equation", + "image_path": "dab1a7d7eeb5ddd6790a51a75c9d0c076905ff45f6bf6e7282ee24cacc1386ba.jpg" + } + ] + } + ], + "index": 10, + "virtual_lines": [ + { + "bbox": [ + 132, + 232, + 479, + 245.66666666666666 + ], + "spans": [], + "index": 9 + }, + { + "bbox": [ + 132, + 245.66666666666666, + 479, + 259.3333333333333 + ], + "spans": [], + "index": 10 + }, + { + "bbox": [ + 132, + 259.3333333333333, + 479, + 273.0 + ], + "spans": [], + "index": 11 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 273, + 202, + 285 + ], + "lines": [ + { + "bbox": [ + 106, + 273, + 203, + 286 + ], + "spans": [ + { + "bbox": [ + 106, + 273, + 203, + 286 + ], + "score": 1.0, + "content": "It is easily checked that", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 12 + }, + { + "type": "interline_equation", + "bbox": [ + 205, + 285, + 406, + 333 + ], + "lines": [ + { + "bbox": [ + 205, + 285, + 406, + 333 + ], + "spans": [ + { + "bbox": [ + 205, + 285, + 406, + 333 + ], + "score": 0.91, + "content": "\\begin{array} { r l } & { \\mathbf { v } _ { 0 } [ i ] ^ { * } \\mathbf { v } _ { 0 } [ i ] + \\mathbf { v } _ { 1 } [ i ] ^ { * } \\mathbf { v } _ { 1 } [ i ] = 1 } \\\\ & { \\mathbf { v } _ { 0 } [ i ] ^ { * } \\mathbf { u } _ { 0 } [ i ] + \\mathbf { v } _ { 1 } [ i ] ^ { * } \\mathbf { u } _ { 1 } [ i ] = \\sqrt { | \\mathbf { u } _ { 0 } [ i ] | ^ { 2 } + | \\mathbf { u } _ { 1 } [ i ] | ^ { 2 } } . } \\\\ & { \\mathbf { v } _ { 1 } [ i ] \\mathbf { u } _ { 0 } [ i ] - \\mathbf { v } _ { 0 } [ i ] ^ { * } \\mathbf { u } _ { 1 } [ i ] = 0 } \\end{array}", + "type": "interline_equation", + "image_path": "17934c44a6bc21e7d3e38c4314de2ffa56b998c532c893fbd39c394a4069234c.jpg" + } + ] + } + ], + "index": 14, + "virtual_lines": [ + { + "bbox": [ + 205, + 285, + 406, + 301.0 + ], + "spans": [], + "index": 13 + }, + { + "bbox": [ + 205, + 301.0, + 406, + 317.0 + ], + "spans": [], + "index": 14 + }, + { + "bbox": [ + 205, + 317.0, + 406, + 333.0 + ], + "spans": [], + "index": 15 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 338, + 151, + 349 + ], + "lines": [ + { + "bbox": [ + 105, + 336, + 153, + 351 + ], + "spans": [ + { + "bbox": [ + 105, + 336, + 153, + 351 + ], + "score": 1.0, + "content": "We choose", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 16 + }, + { + "type": "interline_equation", + "bbox": [ + 241, + 345, + 369, + 373 + ], + "lines": [ + { + "bbox": [ + 241, + 345, + 369, + 373 + ], + "spans": [ + { + "bbox": [ + 241, + 345, + 369, + 373 + ], + "score": 0.93, + "content": "\\mathbf { L } = \\left[ \\begin{array} { c c } { \\mathrm { D i a g } ( \\mathbf { v } _ { 0 } ^ { * } ) } & { \\mathrm { D i a g } ( \\mathbf { v } _ { 1 } ^ { * } ) } \\\\ { \\mathrm { D i a g } ( \\mathbf { v } _ { 1 } ) } & { \\mathrm { D i a g } ( - \\mathbf { v } _ { 0 } ) } \\end{array} \\right]", + "type": "interline_equation", + "image_path": "cfd68d858526984b2fd0965d062dde71143e8ada7dee9824ea11b7517bd2259b.jpg" + } + ] + } + ], + "index": 17, + "virtual_lines": [ + { + "bbox": [ + 241, + 345, + 369, + 373 + ], + "spans": [], + "index": 17 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 379, + 505, + 419 + ], + "lines": [ + { + "bbox": [ + 105, + 379, + 505, + 405 + ], + "spans": [ + { + "bbox": [ + 105, + 384, + 124, + 399 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 124, + 386, + 185, + 398 + ], + "score": 0.62, + "content": "{ \\bf R } = { \\bf L } ^ { * } . { \\bf L } , { \\bf R }", + "type": "inline_equation" + }, + { + "bbox": [ + 185, + 384, + 387, + 399 + ], + "score": 1.0, + "content": "are (permuted) direct sums of blocks of the form", + "type": "text" + }, + { + "bbox": [ + 387, + 379, + 459, + 405 + ], + "score": 0.91, + "content": "\\begin{array} { r l } { \\left[ \\mathbf { v } _ { 0 } [ i ] ^ { * } } & { { } \\mathbf { v } _ { 1 } [ i ] ^ { * } \\right] } \\\\ { \\mathbf { v } _ { 1 } [ i ] } & { { } - \\mathbf { v } _ { 0 } [ i ] } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 459, + 384, + 505, + 399 + ], + "score": 1.0, + "content": ", which are", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 104, + 401, + 439, + 422 + ], + "spans": [ + { + "bbox": [ + 104, + 401, + 285, + 422 + ], + "score": 1.0, + "content": "orthogonal by construction (via (5)). Hence,", + "type": "text" + }, + { + "bbox": [ + 285, + 404, + 325, + 418 + ], + "score": 0.93, + "content": "\\mathbf { L } \\in \\widetilde { \\mathbf { B } } _ { n } ^ { ( n ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 325, + 401, + 343, + 422 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 344, + 405, + 397, + 419 + ], + "score": 0.91, + "content": "\\mathbf { R } \\in ( \\widetilde { \\mathbf { B } } ^ { * } ) _ { n } ^ { ( n ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 397, + 401, + 439, + 422 + ], + "score": 1.0, + "content": ". Further,", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 18.5 + }, + { + "type": "interline_equation", + "bbox": [ + 132, + 419, + 477, + 519 + ], + "lines": [ + { + "bbox": [ + 132, + 419, + 477, + 519 + ], + "spans": [ + { + "bbox": [ + 132, + 419, + 477, + 519 + ], + "score": 0.95, + "content": "\\begin{array} { r l } & { \\mathbf { L H R } = \\left[ \\mathrm { D i a g } ( \\mathbf { v } _ { 0 } ^ { * } ) \\quad \\mathrm { D i a g } ( \\mathbf { v } _ { 1 } ^ { * } ) \\right] \\left( \\mathbf { I } - 2 \\left[ \\mathbf { u } _ { 0 } \\right] \\left[ \\mathbf { u } _ { 0 } \\right] ^ { * } \\right) \\left[ \\mathrm { D i a g } ( \\mathbf { v } _ { 0 } ^ { * } ) \\quad \\mathrm { D i a g } ( \\mathbf { v } _ { 1 } ^ { * } ) \\right] ^ { * } } \\\\ & { \\qquad = \\mathbf { I } - 2 \\left[ \\mathrm { D i a g } ( \\mathbf { v } _ { 0 } ^ { * } ) \\quad \\mathrm { D i a g } ( - \\mathbf { v } _ { 0 } ) \\right] \\left[ \\mathbf { u } _ { 0 } \\right] \\left[ \\mathbf { u } _ { 0 } \\right] ^ { * } \\left[ \\mathrm { D i a g } ( \\mathbf { v } _ { 0 } ^ { * } ) \\quad \\mathrm { D i a g } ( - \\mathbf { v } _ { 0 } ) \\right] ^ { * } } \\\\ & { \\qquad = \\mathbf { I } - 2 \\left[ \\mathrm { D i a g } ( \\mathbf { v } _ { 1 } ) \\quad \\mathrm { D i a g } ( - \\mathbf { v } _ { 0 } ) \\right] \\left[ \\mathbf { u } _ { 1 } \\right] \\left[ \\mathbf { u } _ { 1 } \\right] ^ { * } \\left[ \\mathrm { D i a g } ( \\mathbf { v } _ { 0 } ^ { * } ) \\quad \\mathrm { D i a g } ( \\mathbf { v } _ { 1 } ^ { * } ) \\right] ^ { * } } \\\\ & { \\qquad = \\mathbf { I } - 2 \\underbrace { \\left[ \\mathbf { v } _ { 0 } ^ { * } \\circ \\mathbf { u } _ { 0 } + \\mathbf { v } _ { 1 } ^ { * } \\circ \\mathbf { u } _ { 1 } \\right] } _ { \\mathbf { W } } \\left[ \\underbrace { \\mathbf { v } _ { 0 } ^ { * } \\circ \\mathbf { u } _ { 0 } + \\mathbf { v } _ { 1 } ^ { * } \\circ \\mathbf { u } _ { 1 } } _ { \\mathbf { W } } \\right] ^ { * } , } \\end{array}", + "type": "interline_equation", + "image_path": "9f96645ba18326c50e66e5bbaa4589dafa71d2fcc6c14d05c9b802b7d4b4108c.jpg" + } + ] + } + ], + "index": 21, + "virtual_lines": [ + { + "bbox": [ + 132, + 419, + 477, + 452.3333333333333 + ], + "spans": [], + "index": 20 + }, + { + "bbox": [ + 132, + 452.3333333333333, + 477, + 485.66666666666663 + ], + "spans": [], + "index": 21 + }, + { + "bbox": [ + 132, + 485.66666666666663, + 477, + 519.0 + ], + "spans": [], + "index": 22 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 520, + 305, + 531 + ], + "lines": [ + { + "bbox": [ + 105, + 518, + 306, + 533 + ], + "spans": [ + { + "bbox": [ + 105, + 518, + 133, + 533 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 133, + 522, + 140, + 529 + ], + "score": 0.62, + "content": "\\circ", + "type": "inline_equation" + }, + { + "bbox": [ + 140, + 518, + 306, + 533 + ], + "score": 1.0, + "content": "denotes the Hadamard product. From (6)", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 23 + }, + { + "type": "interline_equation", + "bbox": [ + 212, + 532, + 398, + 561 + ], + "lines": [ + { + "bbox": [ + 212, + 532, + 398, + 561 + ], + "spans": [ + { + "bbox": [ + 212, + 532, + 398, + 561 + ], + "score": 0.93, + "content": "\\mathbf { w } [ i ] = \\left\\{ \\begin{array} { l l } { \\sqrt { | \\mathbf { u } _ { 0 } [ i ] | ^ { 2 } + | \\mathbf { u } _ { 1 } [ i ] | ^ { 2 } } } & { i \\in [ n / 2 ] } \\\\ { 0 } & { i \\in [ n / 2 : n ] } \\end{array} \\right.", + "type": "interline_equation", + "image_path": "a668234d32d753cdfbc43faa1543c8a3a8c6b8cdc7114e28aeb66753ac325c57.jpg" + } + ] + } + ], + "index": 24, + "virtual_lines": [ + { + "bbox": [ + 212, + 532, + 398, + 561 + ], + "spans": [], + "index": 24 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 564, + 351, + 576 + ], + "lines": [ + { + "bbox": [ + 105, + 562, + 350, + 578 + ], + "spans": [ + { + "bbox": [ + 105, + 562, + 264, + 578 + ], + "score": 1.0, + "content": "Denoting the first half of this vector by", + "type": "text" + }, + { + "bbox": [ + 264, + 563, + 311, + 576 + ], + "score": 0.92, + "content": "\\mathbf { w } _ { 0 } \\in \\mathbb { C } ^ { n / 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 311, + 562, + 350, + 578 + ], + "score": 1.0, + "content": ", we have", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 25 + }, + { + "type": "interline_equation", + "bbox": [ + 245, + 577, + 364, + 605 + ], + "lines": [ + { + "bbox": [ + 245, + 577, + 364, + 605 + ], + "spans": [ + { + "bbox": [ + 245, + 577, + 364, + 605 + ], + "score": 0.93, + "content": "\\mathbf { L H R } = \\left[ \\begin{array} { c c } { \\mathbf { I } - 2 \\mathbf { w } _ { 0 } \\mathbf { w } _ { 0 } ^ { * } } & { \\mathbf { 0 } } \\\\ { \\mathbf { 0 } } & { \\mathbf { I } } \\end{array} \\right] ,", + "type": "interline_equation", + "image_path": "f8972fb6a76a451dd7b4c661a4db147ea4f4157077150d7b044b6e93bb74647e.jpg" + } + ] + } + ], + "index": 26, + "virtual_lines": [ + { + "bbox": [ + 245, + 577, + 364, + 605 + ], + "spans": [], + "index": 26 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 605, + 341, + 618 + ], + "lines": [ + { + "bbox": [ + 105, + 604, + 342, + 619 + ], + "spans": [ + { + "bbox": [ + 105, + 604, + 133, + 619 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 133, + 605, + 214, + 618 + ], + "score": 0.93, + "content": "\\| \\mathbf { w } _ { 0 } \\| _ { 2 } = \\| \\mathbf { u } \\| _ { 2 } = 1", + "type": "inline_equation" + }, + { + "bbox": [ + 214, + 604, + 342, + 619 + ], + "score": 1.0, + "content": ". The result follows inductively.", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 27 + }, + { + "type": "text", + "bbox": [ + 107, + 628, + 504, + 651 + ], + "lines": [ + { + "bbox": [ + 105, + 628, + 506, + 641 + ], + "spans": [ + { + "bbox": [ + 105, + 628, + 506, + 641 + ], + "score": 1.0, + "content": "As an immediate corollary, we can use Singular Value Decomposition to obtain a factorization for an", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 640, + 200, + 651 + ], + "spans": [ + { + "bbox": [ + 106, + 640, + 143, + 651 + ], + "score": 1.0, + "content": "arbitrary", + "type": "text" + }, + { + "bbox": [ + 144, + 641, + 169, + 651 + ], + "score": 0.89, + "content": "n \\times n", + "type": "inline_equation" + }, + { + "bbox": [ + 169, + 640, + 200, + 651 + ], + "score": 1.0, + "content": "matrix.", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 28.5 + }, + { + "type": "text", + "bbox": [ + 107, + 653, + 505, + 676 + ], + "lines": [ + { + "bbox": [ + 105, + 649, + 506, + 668 + ], + "spans": [ + { + "bbox": [ + 105, + 649, + 267, + 668 + ], + "score": 1.0, + "content": "Corollary K.3. Let M be an arbitrary", + "type": "text" + }, + { + "bbox": [ + 268, + 654, + 293, + 664 + ], + "score": 0.88, + "content": "n \\times n", + "type": "inline_equation" + }, + { + "bbox": [ + 294, + 649, + 353, + 668 + ], + "score": 1.0, + "content": "matrix. Then,", + "type": "text" + }, + { + "bbox": [ + 353, + 652, + 427, + 665 + ], + "score": 0.93, + "content": "\\mathbf { M } \\in ( { \\mathcal { O } } B B ) ^ { 2 n - 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 428, + 649, + 506, + 668 + ], + "score": 1.0, + "content": ", where all but one", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 665, + 316, + 677 + ], + "spans": [ + { + "bbox": [ + 106, + 665, + 316, + 677 + ], + "score": 1.0, + "content": "matrix in the decomposition is orthogonal (unitary).", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 30.5 + }, + { + "type": "text", + "bbox": [ + 107, + 687, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 687, + 505, + 700 + ], + "spans": [ + { + "bbox": [ + 105, + 687, + 395, + 700 + ], + "score": 1.0, + "content": "Proof. By employing Singular Value Decomposition, we can decompose", + "type": "text" + }, + { + "bbox": [ + 395, + 688, + 408, + 698 + ], + "score": 0.33, + "content": "\\mathbf { M }", + "type": "inline_equation" + }, + { + "bbox": [ + 408, + 687, + 419, + 700 + ], + "score": 1.0, + "content": "as", + "type": "text" + }, + { + "bbox": [ + 419, + 687, + 474, + 698 + ], + "score": 0.91, + "content": "\\mathbf { M } = \\mathbf { U } \\pmb { \\Sigma } \\mathbf { V } ^ { * }", + "type": "inline_equation" + }, + { + "bbox": [ + 475, + 687, + 505, + 700 + ], + "score": 1.0, + "content": ", where", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 696, + 506, + 712 + ], + "spans": [ + { + "bbox": [ + 106, + 699, + 134, + 710 + ], + "score": 0.69, + "content": "\\mathbf { U } , \\mathbf { V } ^ { * }", + "type": "inline_equation" + }, + { + "bbox": [ + 135, + 696, + 219, + 712 + ], + "score": 1.0, + "content": "are orthogonal and", + "type": "text" + }, + { + "bbox": [ + 219, + 699, + 228, + 708 + ], + "score": 0.82, + "content": "\\pmb { \\Sigma }", + "type": "inline_equation" + }, + { + "bbox": [ + 229, + 696, + 336, + 712 + ], + "score": 1.0, + "content": "is diagonal. By Lemma", + "type": "text" + }, + { + "bbox": [ + 336, + 698, + 447, + 711 + ], + "score": 0.67, + "content": "\\mathbf { K } . 2 , \\mathbf { U } , \\bar { \\mathbf { V } } ^ { * } \\in ( \\mathcal { O B } \\mathcal { B } ) ^ { n - 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 447, + 696, + 506, + 712 + ], + "score": 1.0, + "content": ", and trivially", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 707, + 507, + 723 + ], + "spans": [ + { + "bbox": [ + 106, + 710, + 150, + 720 + ], + "score": 0.87, + "content": "\\pmb { \\Sigma } \\in \\mathcal { O } B B", + "type": "inline_equation" + }, + { + "bbox": [ + 150, + 707, + 186, + 723 + ], + "score": 1.0, + "content": ". Hence,", + "type": "text" + }, + { + "bbox": [ + 186, + 709, + 259, + 722 + ], + "score": 0.92, + "content": "\\mathbf { M } \\in ( \\mathcal { O } B B ) ^ { 2 n - 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 260, + 707, + 305, + 723 + ], + "score": 1.0, + "content": ". Note that", + "type": "text" + }, + { + "bbox": [ + 305, + 710, + 315, + 720 + ], + "score": 0.83, + "content": "\\pmb { \\Sigma }", + "type": "inline_equation" + }, + { + "bbox": [ + 315, + 707, + 507, + 723 + ], + "score": 1.0, + "content": "is the only matrix in the decomposition that is", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 720, + 504, + 734 + ], + "spans": [ + { + "bbox": [ + 105, + 720, + 208, + 734 + ], + "score": 1.0, + "content": "not orthogonal (unitary).", + "type": "text" + }, + { + "bbox": [ + 495, + 722, + 504, + 730 + ], + "score": 0.999, + "content": "□", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 33.5 + } + ], + "page_idx": 36, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 107, + 26, + 294, + 38 + ], + "lines": [ + { + "bbox": [ + 106, + 25, + 294, + 39 + ], + "spans": [ + { + "bbox": [ + 106, + 25, + 294, + 39 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2020", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 300, + 751, + 311, + 760 + ], + "lines": [ + { + "bbox": [ + 299, + 750, + 312, + 764 + ], + "spans": [ + { + "bbox": [ + 299, + 750, + 312, + 764 + ], + "score": 1.0, + "content": "37", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 495, + 605, + 505, + 616 + ], + "lines": [ + { + "bbox": [ + 496, + 607, + 505, + 617 + ], + "spans": [ + { + "bbox": [ + 496, + 607, + 505, + 617 + ], + "score": 0.998, + "content": "□", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "list", + "bbox": [ + 105, + 82, + 505, + 106 + ], + "lines": [ + { + "bbox": [ + 105, + 82, + 507, + 95 + ], + "spans": [ + { + "bbox": [ + 105, + 82, + 146, + 95 + ], + "score": 1.0, + "content": "pattern of", + "type": "text" + }, + { + "bbox": [ + 147, + 83, + 161, + 94 + ], + "score": 0.88, + "content": "\\mathbf { Q } _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 162, + 82, + 464, + 95 + ], + "score": 1.0, + "content": "and the orthogonality of its columns. Moreover, the norm of each column of", + "type": "text" + }, + { + "bbox": [ + 464, + 83, + 487, + 94 + ], + "score": 0.9, + "content": "\\mathbf { Q } _ { 1 } \\mathbf { R }", + "type": "inline_equation" + }, + { + "bbox": [ + 487, + 82, + 507, + 95 + ], + "score": 1.0, + "content": "is 1.", + "type": "text" + } + ], + "index": 0, + "is_list_end_line": true + }, + { + "bbox": [ + 105, + 92, + 507, + 106 + ], + "spans": [ + { + "bbox": [ + 105, + 92, + 149, + 106 + ], + "score": 1.0, + "content": "Therefore,", + "type": "text" + }, + { + "bbox": [ + 150, + 94, + 173, + 105 + ], + "score": 0.9, + "content": "\\mathbf { Q } _ { 1 } \\mathbf { R }", + "type": "inline_equation" + }, + { + "bbox": [ + 173, + 92, + 340, + 106 + ], + "score": 1.0, + "content": "is an orthogonal butterfly factor matrix, so", + "type": "text" + }, + { + "bbox": [ + 340, + 93, + 440, + 106 + ], + "score": 0.91, + "content": "\\mathbf { M } = \\mathbf { Q } \\mathbf { R } \\in ( { \\mathcal { O } } B B ) ^ { n - 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 440, + 92, + 507, + 106 + ], + "score": 1.0, + "content": ", as desired.", + "type": "text" + } + ], + "index": 1, + "is_list_start_line": true, + "is_list_end_line": true + } + ], + "index": 0.5, + "bbox_fs": [ + 105, + 82, + 507, + 106 + ] + }, + { + "type": "text", + "bbox": [ + 108, + 117, + 276, + 128 + ], + "lines": [ + { + "bbox": [ + 106, + 117, + 276, + 129 + ], + "spans": [ + { + "bbox": [ + 106, + 117, + 276, + 129 + ], + "score": 1.0, + "content": "We now return to the proof of Lemma K.1", + "type": "text" + } + ], + "index": 2 + } + ], + "index": 2, + "bbox_fs": [ + 106, + 117, + 276, + 129 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 140, + 504, + 165 + ], + "lines": [ + { + "bbox": [ + 105, + 138, + 505, + 156 + ], + "spans": [ + { + "bbox": [ + 105, + 138, + 222, + 156 + ], + "score": 1.0, + "content": "Proof of Lemma K.1. Given", + "type": "text" + }, + { + "bbox": [ + 222, + 142, + 254, + 152 + ], + "score": 0.88, + "content": "\\mathbf { u } \\in \\mathbb { C } ^ { n }", + "type": "inline_equation" + }, + { + "bbox": [ + 255, + 138, + 259, + 156 + ], + "score": 0.0, + "content": "", + "type": "text" + }, + { + "bbox": [ + 259, + 144, + 267, + 151 + ], + "score": 0.67, + "content": "\\scriptstyle { \\dot { n } }", + "type": "inline_equation" + }, + { + "bbox": [ + 267, + 138, + 339, + 156 + ], + "score": 1.0, + "content": "a power of 2), let", + "type": "text" + }, + { + "bbox": [ + 339, + 140, + 505, + 154 + ], + "score": 0.76, + "content": "\\mathbf { u } _ { 0 } = \\mathbf { u } [ : n / 2 ] \\in \\mathbb { C } ^ { n / 2 } , \\mathbf { u } _ { 1 } = \\mathbf { u } [ n / 2 : ] \\in", + "type": "inline_equation" + } + ], + "index": 3 + }, + { + "bbox": [ + 107, + 151, + 288, + 166 + ], + "spans": [ + { + "bbox": [ + 107, + 153, + 128, + 164 + ], + "score": 0.88, + "content": "\\mathbb { C } ^ { n / 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 128, + 151, + 277, + 166 + ], + "score": 1.0, + "content": "denote the first and second halves of", + "type": "text" + }, + { + "bbox": [ + 277, + 156, + 285, + 164 + ], + "score": 0.32, + "content": "\\mathbf { u }", + "type": "inline_equation" + }, + { + "bbox": [ + 285, + 151, + 288, + 166 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 3.5, + "bbox_fs": [ + 105, + 138, + 505, + 166 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 170, + 505, + 219 + ], + "lines": [ + { + "bbox": [ + 105, + 169, + 505, + 182 + ], + "spans": [ + { + "bbox": [ + 105, + 169, + 160, + 182 + ], + "score": 1.0, + "content": "To show that", + "type": "text" + }, + { + "bbox": [ + 161, + 171, + 205, + 181 + ], + "score": 0.91, + "content": "\\mathbf { H } \\in \\mathcal { O } B B", + "type": "inline_equation" + }, + { + "bbox": [ + 205, + 169, + 316, + 182 + ], + "score": 1.0, + "content": "with inner diagonal matrix", + "type": "text" + }, + { + "bbox": [ + 317, + 171, + 322, + 181 + ], + "score": 0.32, + "content": "\\mathbf { I }", + "type": "inline_equation" + }, + { + "bbox": [ + 323, + 169, + 505, + 182 + ], + "score": 1.0, + "content": ", we proceed by induction. The base case for", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 107, + 182, + 505, + 193 + ], + "spans": [ + { + "bbox": [ + 107, + 182, + 132, + 192 + ], + "score": 0.89, + "content": "n = 2", + "type": "inline_equation" + }, + { + "bbox": [ + 132, + 182, + 402, + 193 + ], + "score": 1.0, + "content": "is trivial. It suffices to show that there exist unitary butterfly factors", + "type": "text" + }, + { + "bbox": [ + 402, + 182, + 423, + 193 + ], + "score": 0.86, + "content": "\\mathbf { L } , \\mathbf { R }", + "type": "inline_equation" + }, + { + "bbox": [ + 424, + 182, + 505, + 193 + ], + "score": 1.0, + "content": "such that LHR has", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 192, + 429, + 219 + ], + "spans": [ + { + "bbox": [ + 105, + 192, + 144, + 216 + ], + "score": 1.0, + "content": "the form", + "type": "text" + }, + { + "bbox": [ + 144, + 192, + 277, + 219 + ], + "score": 0.89, + "content": "\\begin{array}{c} \\begin{array} { r l } { \\left[ { \\displaystyle { \\bf { I } } _ { n / 2 } - 2 { \\bf { v } } _ { 0 } { \\bf { v } } _ { 0 } ^ { * } } & { { \\bf { 0 } } } \\\\ { { \\bf { 0 } } } & { { \\bf { I } } _ { n / 2 } - 2 { \\bf { v } } _ { 1 } { \\bf { v } } _ { 1 } ^ { * } } \\end{array} \\right] } & { { } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 277, + 198, + 365, + 211 + ], + "score": 1.0, + "content": "for some unit vectors", + "type": "text" + }, + { + "bbox": [ + 366, + 198, + 424, + 211 + ], + "score": 0.91, + "content": "\\mathbf { v } _ { 0 } , \\mathbf { v } _ { 1 } \\in \\mathbb { C } ^ { n / 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 425, + 198, + 429, + 211 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 6, + "bbox_fs": [ + 105, + 169, + 505, + 219 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 222, + 135, + 234 + ], + "lines": [ + { + "bbox": [ + 105, + 221, + 136, + 236 + ], + "spans": [ + { + "bbox": [ + 105, + 221, + 136, + 236 + ], + "score": 1.0, + "content": "Define", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 8, + "bbox_fs": [ + 105, + 221, + 136, + 236 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 132, + 232, + 479, + 273 + ], + "lines": [ + { + "bbox": [ + 132, + 232, + 479, + 273 + ], + "spans": [ + { + "bbox": [ + 132, + 232, + 479, + 273 + ], + "score": 0.94, + "content": "\\left( \\mathbf { v } _ { 0 } [ i ] , \\mathbf { v } _ { 1 } [ i ] \\right) = \\left\\{ \\begin{array} { l l } { \\left( \\frac { \\mathbf { u } _ { 0 } [ i ] } { \\sqrt { | \\mathbf { u } _ { 0 } [ i ] | ^ { 2 } + | \\mathbf { u } _ { 1 } [ i ] | ^ { 2 } } } , \\frac { \\mathbf { u } _ { 1 } [ i ] } { \\sqrt { | \\mathbf { u } _ { 0 } [ i ] | ^ { 2 } + | \\mathbf { u } _ { 1 } [ i ] | ^ { 2 } } } \\right) } & { \\mathrm { ~ i f ~ } | \\mathbf { u } _ { 0 } [ i ] | ^ { 2 } + | \\mathbf { u } _ { 1 } [ i ] | ^ { 2 } \\neq 0 } \\\\ { ( 1 , 0 ) } & { \\mathrm { ~ o t h e r w i s e } } \\end{array} \\right. .", + "type": "interline_equation", + "image_path": "dab1a7d7eeb5ddd6790a51a75c9d0c076905ff45f6bf6e7282ee24cacc1386ba.jpg" + } + ] + } + ], + "index": 10, + "virtual_lines": [ + { + "bbox": [ + 132, + 232, + 479, + 245.66666666666666 + ], + "spans": [], + "index": 9 + }, + { + "bbox": [ + 132, + 245.66666666666666, + 479, + 259.3333333333333 + ], + "spans": [], + "index": 10 + }, + { + "bbox": [ + 132, + 259.3333333333333, + 479, + 273.0 + ], + "spans": [], + "index": 11 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 273, + 202, + 285 + ], + "lines": [ + { + "bbox": [ + 106, + 273, + 203, + 286 + ], + "spans": [ + { + "bbox": [ + 106, + 273, + 203, + 286 + ], + "score": 1.0, + "content": "It is easily checked that", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 12, + "bbox_fs": [ + 106, + 273, + 203, + 286 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 205, + 285, + 406, + 333 + ], + "lines": [ + { + "bbox": [ + 205, + 285, + 406, + 333 + ], + "spans": [ + { + "bbox": [ + 205, + 285, + 406, + 333 + ], + "score": 0.91, + "content": "\\begin{array} { r l } & { \\mathbf { v } _ { 0 } [ i ] ^ { * } \\mathbf { v } _ { 0 } [ i ] + \\mathbf { v } _ { 1 } [ i ] ^ { * } \\mathbf { v } _ { 1 } [ i ] = 1 } \\\\ & { \\mathbf { v } _ { 0 } [ i ] ^ { * } \\mathbf { u } _ { 0 } [ i ] + \\mathbf { v } _ { 1 } [ i ] ^ { * } \\mathbf { u } _ { 1 } [ i ] = \\sqrt { | \\mathbf { u } _ { 0 } [ i ] | ^ { 2 } + | \\mathbf { u } _ { 1 } [ i ] | ^ { 2 } } . } \\\\ & { \\mathbf { v } _ { 1 } [ i ] \\mathbf { u } _ { 0 } [ i ] - \\mathbf { v } _ { 0 } [ i ] ^ { * } \\mathbf { u } _ { 1 } [ i ] = 0 } \\end{array}", + "type": "interline_equation", + "image_path": "17934c44a6bc21e7d3e38c4314de2ffa56b998c532c893fbd39c394a4069234c.jpg" + } + ] + } + ], + "index": 14, + "virtual_lines": [ + { + "bbox": [ + 205, + 285, + 406, + 301.0 + ], + "spans": [], + "index": 13 + }, + { + "bbox": [ + 205, + 301.0, + 406, + 317.0 + ], + "spans": [], + "index": 14 + }, + { + "bbox": [ + 205, + 317.0, + 406, + 333.0 + ], + "spans": [], + "index": 15 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 338, + 151, + 349 + ], + "lines": [ + { + "bbox": [ + 105, + 336, + 153, + 351 + ], + "spans": [ + { + "bbox": [ + 105, + 336, + 153, + 351 + ], + "score": 1.0, + "content": "We choose", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 16, + "bbox_fs": [ + 105, + 336, + 153, + 351 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 241, + 345, + 369, + 373 + ], + "lines": [ + { + "bbox": [ + 241, + 345, + 369, + 373 + ], + "spans": [ + { + "bbox": [ + 241, + 345, + 369, + 373 + ], + "score": 0.93, + "content": "\\mathbf { L } = \\left[ \\begin{array} { c c } { \\mathrm { D i a g } ( \\mathbf { v } _ { 0 } ^ { * } ) } & { \\mathrm { D i a g } ( \\mathbf { v } _ { 1 } ^ { * } ) } \\\\ { \\mathrm { D i a g } ( \\mathbf { v } _ { 1 } ) } & { \\mathrm { D i a g } ( - \\mathbf { v } _ { 0 } ) } \\end{array} \\right]", + "type": "interline_equation", + "image_path": "cfd68d858526984b2fd0965d062dde71143e8ada7dee9824ea11b7517bd2259b.jpg" + } + ] + } + ], + "index": 17, + "virtual_lines": [ + { + "bbox": [ + 241, + 345, + 369, + 373 + ], + "spans": [], + "index": 17 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 379, + 505, + 419 + ], + "lines": [ + { + "bbox": [ + 105, + 379, + 505, + 405 + ], + "spans": [ + { + "bbox": [ + 105, + 384, + 124, + 399 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 124, + 386, + 185, + 398 + ], + "score": 0.62, + "content": "{ \\bf R } = { \\bf L } ^ { * } . { \\bf L } , { \\bf R }", + "type": "inline_equation" + }, + { + "bbox": [ + 185, + 384, + 387, + 399 + ], + "score": 1.0, + "content": "are (permuted) direct sums of blocks of the form", + "type": "text" + }, + { + "bbox": [ + 387, + 379, + 459, + 405 + ], + "score": 0.91, + "content": "\\begin{array} { r l } { \\left[ \\mathbf { v } _ { 0 } [ i ] ^ { * } } & { { } \\mathbf { v } _ { 1 } [ i ] ^ { * } \\right] } \\\\ { \\mathbf { v } _ { 1 } [ i ] } & { { } - \\mathbf { v } _ { 0 } [ i ] } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 459, + 384, + 505, + 399 + ], + "score": 1.0, + "content": ", which are", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 104, + 401, + 439, + 422 + ], + "spans": [ + { + "bbox": [ + 104, + 401, + 285, + 422 + ], + "score": 1.0, + "content": "orthogonal by construction (via (5)). Hence,", + "type": "text" + }, + { + "bbox": [ + 285, + 404, + 325, + 418 + ], + "score": 0.93, + "content": "\\mathbf { L } \\in \\widetilde { \\mathbf { B } } _ { n } ^ { ( n ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 325, + 401, + 343, + 422 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 344, + 405, + 397, + 419 + ], + "score": 0.91, + "content": "\\mathbf { R } \\in ( \\widetilde { \\mathbf { B } } ^ { * } ) _ { n } ^ { ( n ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 397, + 401, + 439, + 422 + ], + "score": 1.0, + "content": ". Further,", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 18.5, + "bbox_fs": [ + 104, + 379, + 505, + 422 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 132, + 419, + 477, + 519 + ], + "lines": [ + { + "bbox": [ + 132, + 419, + 477, + 519 + ], + "spans": [ + { + "bbox": [ + 132, + 419, + 477, + 519 + ], + "score": 0.95, + "content": "\\begin{array} { r l } & { \\mathbf { L H R } = \\left[ \\mathrm { D i a g } ( \\mathbf { v } _ { 0 } ^ { * } ) \\quad \\mathrm { D i a g } ( \\mathbf { v } _ { 1 } ^ { * } ) \\right] \\left( \\mathbf { I } - 2 \\left[ \\mathbf { u } _ { 0 } \\right] \\left[ \\mathbf { u } _ { 0 } \\right] ^ { * } \\right) \\left[ \\mathrm { D i a g } ( \\mathbf { v } _ { 0 } ^ { * } ) \\quad \\mathrm { D i a g } ( \\mathbf { v } _ { 1 } ^ { * } ) \\right] ^ { * } } \\\\ & { \\qquad = \\mathbf { I } - 2 \\left[ \\mathrm { D i a g } ( \\mathbf { v } _ { 0 } ^ { * } ) \\quad \\mathrm { D i a g } ( - \\mathbf { v } _ { 0 } ) \\right] \\left[ \\mathbf { u } _ { 0 } \\right] \\left[ \\mathbf { u } _ { 0 } \\right] ^ { * } \\left[ \\mathrm { D i a g } ( \\mathbf { v } _ { 0 } ^ { * } ) \\quad \\mathrm { D i a g } ( - \\mathbf { v } _ { 0 } ) \\right] ^ { * } } \\\\ & { \\qquad = \\mathbf { I } - 2 \\left[ \\mathrm { D i a g } ( \\mathbf { v } _ { 1 } ) \\quad \\mathrm { D i a g } ( - \\mathbf { v } _ { 0 } ) \\right] \\left[ \\mathbf { u } _ { 1 } \\right] \\left[ \\mathbf { u } _ { 1 } \\right] ^ { * } \\left[ \\mathrm { D i a g } ( \\mathbf { v } _ { 0 } ^ { * } ) \\quad \\mathrm { D i a g } ( \\mathbf { v } _ { 1 } ^ { * } ) \\right] ^ { * } } \\\\ & { \\qquad = \\mathbf { I } - 2 \\underbrace { \\left[ \\mathbf { v } _ { 0 } ^ { * } \\circ \\mathbf { u } _ { 0 } + \\mathbf { v } _ { 1 } ^ { * } \\circ \\mathbf { u } _ { 1 } \\right] } _ { \\mathbf { W } } \\left[ \\underbrace { \\mathbf { v } _ { 0 } ^ { * } \\circ \\mathbf { u } _ { 0 } + \\mathbf { v } _ { 1 } ^ { * } \\circ \\mathbf { u } _ { 1 } } _ { \\mathbf { W } } \\right] ^ { * } , } \\end{array}", + "type": "interline_equation", + "image_path": "9f96645ba18326c50e66e5bbaa4589dafa71d2fcc6c14d05c9b802b7d4b4108c.jpg" + } + ] + } + ], + "index": 21, + "virtual_lines": [ + { + "bbox": [ + 132, + 419, + 477, + 452.3333333333333 + ], + "spans": [], + "index": 20 + }, + { + "bbox": [ + 132, + 452.3333333333333, + 477, + 485.66666666666663 + ], + "spans": [], + "index": 21 + }, + { + "bbox": [ + 132, + 485.66666666666663, + 477, + 519.0 + ], + "spans": [], + "index": 22 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 520, + 305, + 531 + ], + "lines": [ + { + "bbox": [ + 105, + 518, + 306, + 533 + ], + "spans": [ + { + "bbox": [ + 105, + 518, + 133, + 533 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 133, + 522, + 140, + 529 + ], + "score": 0.62, + "content": "\\circ", + "type": "inline_equation" + }, + { + "bbox": [ + 140, + 518, + 306, + 533 + ], + "score": 1.0, + "content": "denotes the Hadamard product. From (6)", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 23, + "bbox_fs": [ + 105, + 518, + 306, + 533 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 212, + 532, + 398, + 561 + ], + "lines": [ + { + "bbox": [ + 212, + 532, + 398, + 561 + ], + "spans": [ + { + "bbox": [ + 212, + 532, + 398, + 561 + ], + "score": 0.93, + "content": "\\mathbf { w } [ i ] = \\left\\{ \\begin{array} { l l } { \\sqrt { | \\mathbf { u } _ { 0 } [ i ] | ^ { 2 } + | \\mathbf { u } _ { 1 } [ i ] | ^ { 2 } } } & { i \\in [ n / 2 ] } \\\\ { 0 } & { i \\in [ n / 2 : n ] } \\end{array} \\right.", + "type": "interline_equation", + "image_path": "a668234d32d753cdfbc43faa1543c8a3a8c6b8cdc7114e28aeb66753ac325c57.jpg" + } + ] + } + ], + "index": 24, + "virtual_lines": [ + { + "bbox": [ + 212, + 532, + 398, + 561 + ], + "spans": [], + "index": 24 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 564, + 351, + 576 + ], + "lines": [ + { + "bbox": [ + 105, + 562, + 350, + 578 + ], + "spans": [ + { + "bbox": [ + 105, + 562, + 264, + 578 + ], + "score": 1.0, + "content": "Denoting the first half of this vector by", + "type": "text" + }, + { + "bbox": [ + 264, + 563, + 311, + 576 + ], + "score": 0.92, + "content": "\\mathbf { w } _ { 0 } \\in \\mathbb { C } ^ { n / 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 311, + 562, + 350, + 578 + ], + "score": 1.0, + "content": ", we have", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 25, + "bbox_fs": [ + 105, + 562, + 350, + 578 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 245, + 577, + 364, + 605 + ], + "lines": [ + { + "bbox": [ + 245, + 577, + 364, + 605 + ], + "spans": [ + { + "bbox": [ + 245, + 577, + 364, + 605 + ], + "score": 0.93, + "content": "\\mathbf { L H R } = \\left[ \\begin{array} { c c } { \\mathbf { I } - 2 \\mathbf { w } _ { 0 } \\mathbf { w } _ { 0 } ^ { * } } & { \\mathbf { 0 } } \\\\ { \\mathbf { 0 } } & { \\mathbf { I } } \\end{array} \\right] ,", + "type": "interline_equation", + "image_path": "f8972fb6a76a451dd7b4c661a4db147ea4f4157077150d7b044b6e93bb74647e.jpg" + } + ] + } + ], + "index": 26, + "virtual_lines": [ + { + "bbox": [ + 245, + 577, + 364, + 605 + ], + "spans": [], + "index": 26 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 605, + 341, + 618 + ], + "lines": [ + { + "bbox": [ + 105, + 604, + 342, + 619 + ], + "spans": [ + { + "bbox": [ + 105, + 604, + 133, + 619 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 133, + 605, + 214, + 618 + ], + "score": 0.93, + "content": "\\| \\mathbf { w } _ { 0 } \\| _ { 2 } = \\| \\mathbf { u } \\| _ { 2 } = 1", + "type": "inline_equation" + }, + { + "bbox": [ + 214, + 604, + 342, + 619 + ], + "score": 1.0, + "content": ". The result follows inductively.", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 27, + "bbox_fs": [ + 105, + 604, + 342, + 619 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 628, + 504, + 651 + ], + "lines": [ + { + "bbox": [ + 105, + 628, + 506, + 641 + ], + "spans": [ + { + "bbox": [ + 105, + 628, + 506, + 641 + ], + "score": 1.0, + "content": "As an immediate corollary, we can use Singular Value Decomposition to obtain a factorization for an", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 640, + 200, + 651 + ], + "spans": [ + { + "bbox": [ + 106, + 640, + 143, + 651 + ], + "score": 1.0, + "content": "arbitrary", + "type": "text" + }, + { + "bbox": [ + 144, + 641, + 169, + 651 + ], + "score": 0.89, + "content": "n \\times n", + "type": "inline_equation" + }, + { + "bbox": [ + 169, + 640, + 200, + 651 + ], + "score": 1.0, + "content": "matrix.", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 28.5, + "bbox_fs": [ + 105, + 628, + 506, + 651 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 653, + 505, + 676 + ], + "lines": [ + { + "bbox": [ + 105, + 649, + 506, + 668 + ], + "spans": [ + { + "bbox": [ + 105, + 649, + 267, + 668 + ], + "score": 1.0, + "content": "Corollary K.3. Let M be an arbitrary", + "type": "text" + }, + { + "bbox": [ + 268, + 654, + 293, + 664 + ], + "score": 0.88, + "content": "n \\times n", + "type": "inline_equation" + }, + { + "bbox": [ + 294, + 649, + 353, + 668 + ], + "score": 1.0, + "content": "matrix. Then,", + "type": "text" + }, + { + "bbox": [ + 353, + 652, + 427, + 665 + ], + "score": 0.93, + "content": "\\mathbf { M } \\in ( { \\mathcal { O } } B B ) ^ { 2 n - 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 428, + 649, + 506, + 668 + ], + "score": 1.0, + "content": ", where all but one", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 665, + 316, + 677 + ], + "spans": [ + { + "bbox": [ + 106, + 665, + 316, + 677 + ], + "score": 1.0, + "content": "matrix in the decomposition is orthogonal (unitary).", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 30.5, + "bbox_fs": [ + 105, + 649, + 506, + 677 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 687, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 687, + 505, + 700 + ], + "spans": [ + { + "bbox": [ + 105, + 687, + 395, + 700 + ], + "score": 1.0, + "content": "Proof. By employing Singular Value Decomposition, we can decompose", + "type": "text" + }, + { + "bbox": [ + 395, + 688, + 408, + 698 + ], + "score": 0.33, + "content": "\\mathbf { M }", + "type": "inline_equation" + }, + { + "bbox": [ + 408, + 687, + 419, + 700 + ], + "score": 1.0, + "content": "as", + "type": "text" + }, + { + "bbox": [ + 419, + 687, + 474, + 698 + ], + "score": 0.91, + "content": "\\mathbf { M } = \\mathbf { U } \\pmb { \\Sigma } \\mathbf { V } ^ { * }", + "type": "inline_equation" + }, + { + "bbox": [ + 475, + 687, + 505, + 700 + ], + "score": 1.0, + "content": ", where", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 696, + 506, + 712 + ], + "spans": [ + { + "bbox": [ + 106, + 699, + 134, + 710 + ], + "score": 0.69, + "content": "\\mathbf { U } , \\mathbf { V } ^ { * }", + "type": "inline_equation" + }, + { + "bbox": [ + 135, + 696, + 219, + 712 + ], + "score": 1.0, + "content": "are orthogonal and", + "type": "text" + }, + { + "bbox": [ + 219, + 699, + 228, + 708 + ], + "score": 0.82, + "content": "\\pmb { \\Sigma }", + "type": "inline_equation" + }, + { + "bbox": [ + 229, + 696, + 336, + 712 + ], + "score": 1.0, + "content": "is diagonal. By Lemma", + "type": "text" + }, + { + "bbox": [ + 336, + 698, + 447, + 711 + ], + "score": 0.67, + "content": "\\mathbf { K } . 2 , \\mathbf { U } , \\bar { \\mathbf { V } } ^ { * } \\in ( \\mathcal { O B } \\mathcal { B } ) ^ { n - 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 447, + 696, + 506, + 712 + ], + "score": 1.0, + "content": ", and trivially", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 707, + 507, + 723 + ], + "spans": [ + { + "bbox": [ + 106, + 710, + 150, + 720 + ], + "score": 0.87, + "content": "\\pmb { \\Sigma } \\in \\mathcal { O } B B", + "type": "inline_equation" + }, + { + "bbox": [ + 150, + 707, + 186, + 723 + ], + "score": 1.0, + "content": ". Hence,", + "type": "text" + }, + { + "bbox": [ + 186, + 709, + 259, + 722 + ], + "score": 0.92, + "content": "\\mathbf { M } \\in ( \\mathcal { O } B B ) ^ { 2 n - 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 260, + 707, + 305, + 723 + ], + "score": 1.0, + "content": ". Note that", + "type": "text" + }, + { + "bbox": [ + 305, + 710, + 315, + 720 + ], + "score": 0.83, + "content": "\\pmb { \\Sigma }", + "type": "inline_equation" + }, + { + "bbox": [ + 315, + 707, + 507, + 723 + ], + "score": 1.0, + "content": "is the only matrix in the decomposition that is", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 720, + 504, + 734 + ], + "spans": [ + { + "bbox": [ + 105, + 720, + 208, + 734 + ], + "score": 1.0, + "content": "not orthogonal (unitary).", + "type": "text" + }, + { + "bbox": [ + 495, + 722, + 504, + 730 + ], + "score": 0.999, + "content": "□", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 33.5, + "bbox_fs": [ + 105, + 687, + 507, + 734 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "title", + "bbox": [ + 108, + 82, + 208, + 94 + ], + "lines": [ + { + "bbox": [ + 105, + 81, + 209, + 96 + ], + "spans": [ + { + "bbox": [ + 105, + 81, + 209, + 96 + ], + "score": 1.0, + "content": "K.3 CONSTRUCTIONS", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "text", + "bbox": [ + 106, + 102, + 505, + 169 + ], + "lines": [ + { + "bbox": [ + 106, + 102, + 505, + 115 + ], + "spans": [ + { + "bbox": [ + 106, + 102, + 233, + 115 + ], + "score": 1.0, + "content": "We show that we can construct", + "type": "text" + }, + { + "bbox": [ + 233, + 105, + 239, + 113 + ], + "score": 0.43, + "content": "s", + "type": "inline_equation" + }, + { + "bbox": [ + 239, + 102, + 505, + 115 + ], + "score": 1.0, + "content": "-sparse matrices in the OBB hierarchy with the same width as the", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 107, + 114, + 505, + 127 + ], + "spans": [ + { + "bbox": [ + 107, + 114, + 126, + 125 + ], + "score": 0.86, + "content": "B B ^ { * }", + "type": "inline_equation" + }, + { + "bbox": [ + 126, + 114, + 505, + 127 + ], + "score": 1.0, + "content": "hierarchy. The proof follows a structure to that of Theorem 3. We begin by arguing about", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 104, + 125, + 506, + 138 + ], + "spans": [ + { + "bbox": [ + 104, + 125, + 506, + 138 + ], + "score": 1.0, + "content": "permutation and step matrices, then using the same factorization to argue that matrices with at most", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 106, + 135, + 506, + 148 + ], + "spans": [ + { + "bbox": [ + 106, + 137, + 114, + 146 + ], + "score": 0.67, + "content": "n", + "type": "inline_equation" + }, + { + "bbox": [ + 114, + 135, + 205, + 148 + ], + "score": 1.0, + "content": "NNZ are contained in", + "type": "text" + }, + { + "bbox": [ + 206, + 135, + 236, + 148 + ], + "score": 0.92, + "content": "( B B ^ { * } ) ^ { 4 }", + "type": "inline_equation" + }, + { + "bbox": [ + 237, + 135, + 506, + 148 + ], + "score": 1.0, + "content": ". Then, we will appeal to a modified sum closure lemma to extend", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 106, + 147, + 505, + 159 + ], + "spans": [ + { + "bbox": [ + 106, + 147, + 249, + 159 + ], + "score": 1.0, + "content": "the argument to matrices of general", + "type": "text" + }, + { + "bbox": [ + 249, + 149, + 255, + 157 + ], + "score": 0.58, + "content": "s", + "type": "inline_equation" + }, + { + "bbox": [ + 255, + 147, + 505, + 159 + ], + "score": 1.0, + "content": "NNZ. Similar to Appendix F, we can use these results to place", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 157, + 469, + 171 + ], + "spans": [ + { + "bbox": [ + 105, + 157, + 403, + 171 + ], + "score": 1.0, + "content": "all matrices with low-depth circuits for matrix vector multiplication in the", + "type": "text" + }, + { + "bbox": [ + 403, + 158, + 426, + 168 + ], + "score": 0.25, + "content": "\\ O B B", + "type": "inline_equation" + }, + { + "bbox": [ + 427, + 157, + 469, + 171 + ], + "score": 1.0, + "content": "hierarchy.", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 3.5 + }, + { + "type": "title", + "bbox": [ + 108, + 181, + 210, + 193 + ], + "lines": [ + { + "bbox": [ + 105, + 180, + 211, + 194 + ], + "spans": [ + { + "bbox": [ + 105, + 180, + 211, + 194 + ], + "score": 1.0, + "content": "K.3.1 PERMUTATIONS", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 7 + }, + { + "type": "text", + "bbox": [ + 109, + 200, + 504, + 223 + ], + "lines": [ + { + "bbox": [ + 106, + 199, + 506, + 214 + ], + "spans": [ + { + "bbox": [ + 106, + 199, + 506, + 214 + ], + "score": 1.0, + "content": "We begin by presenting the argument that permutations are included in OBB as a corollary to", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 210, + 156, + 224 + ], + "spans": [ + { + "bbox": [ + 106, + 210, + 156, + 224 + ], + "score": 1.0, + "content": "Theorem 2.", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 8.5 + }, + { + "type": "text", + "bbox": [ + 106, + 225, + 366, + 239 + ], + "lines": [ + { + "bbox": [ + 106, + 225, + 366, + 239 + ], + "spans": [ + { + "bbox": [ + 106, + 226, + 323, + 239 + ], + "score": 1.0, + "content": "Corollary K.4. Let P be a permutation matrix. Then", + "type": "text" + }, + { + "bbox": [ + 323, + 225, + 362, + 237 + ], + "score": 0.92, + "content": "\\mathbf { P } \\in \\widetilde { B } \\widetilde { B } ^ { * }", + "type": "inline_equation" + }, + { + "bbox": [ + 362, + 226, + 366, + 239 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 10 + }, + { + "type": "text", + "bbox": [ + 107, + 250, + 506, + 284 + ], + "lines": [ + { + "bbox": [ + 105, + 250, + 506, + 263 + ], + "spans": [ + { + "bbox": [ + 105, + 250, + 506, + 263 + ], + "score": 1.0, + "content": "Proof. We appeal to the decomposition from Theorem 2, noting that all butterfly factor matrices", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 262, + 507, + 274 + ], + "spans": [ + { + "bbox": [ + 105, + 262, + 507, + 274 + ], + "score": 1.0, + "content": "constructed in the proofs of Lemmas G.3 and G.1 are permutation matrices, and thus are orthogonal.", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 272, + 506, + 285 + ], + "spans": [ + { + "bbox": [ + 105, + 272, + 136, + 285 + ], + "score": 1.0, + "content": "Hence,", + "type": "text" + }, + { + "bbox": [ + 137, + 273, + 180, + 283 + ], + "score": 0.89, + "content": "\\mathbf { P } \\in { \\mathcal { O } } B B", + "type": "inline_equation" + }, + { + "bbox": [ + 181, + 272, + 330, + 285 + ], + "score": 1.0, + "content": "where the inner diagonal matrix is I.", + "type": "text" + }, + { + "bbox": [ + 494, + 273, + 506, + 284 + ], + "score": 0.999, + "content": "□", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 12 + }, + { + "type": "text", + "bbox": [ + 105, + 297, + 473, + 310 + ], + "lines": [ + { + "bbox": [ + 105, + 296, + 474, + 311 + ], + "spans": [ + { + "bbox": [ + 105, + 297, + 451, + 311 + ], + "score": 1.0, + "content": "Similarly, the construction of Lemma G.6 also show that permutations are included in", + "type": "text" + }, + { + "bbox": [ + 451, + 296, + 470, + 309 + ], + "score": 0.88, + "content": "\\widetilde { B } ^ { * } \\widetilde { B }", + "type": "inline_equation" + }, + { + "bbox": [ + 471, + 297, + 474, + 311 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 14 + }, + { + "type": "text", + "bbox": [ + 107, + 314, + 366, + 327 + ], + "lines": [ + { + "bbox": [ + 106, + 313, + 366, + 328 + ], + "spans": [ + { + "bbox": [ + 106, + 314, + 188, + 328 + ], + "score": 1.0, + "content": "Corollary K.5. Let", + "type": "text" + }, + { + "bbox": [ + 188, + 316, + 197, + 325 + ], + "score": 0.39, + "content": "\\mathbf { P }", + "type": "inline_equation" + }, + { + "bbox": [ + 198, + 314, + 323, + 328 + ], + "score": 1.0, + "content": "be a permutation matrix. Then", + "type": "text" + }, + { + "bbox": [ + 323, + 313, + 363, + 326 + ], + "score": 0.91, + "content": "\\mathbf { P } \\in \\widetilde { B } ^ { * } \\widetilde { B }", + "type": "inline_equation" + }, + { + "bbox": [ + 363, + 314, + 366, + 328 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 15 + }, + { + "type": "text", + "bbox": [ + 107, + 336, + 505, + 358 + ], + "lines": [ + { + "bbox": [ + 105, + 335, + 506, + 350 + ], + "spans": [ + { + "bbox": [ + 105, + 335, + 322, + 350 + ], + "score": 1.0, + "content": "To prove the containment of sparse matrices within the", + "type": "text" + }, + { + "bbox": [ + 322, + 337, + 345, + 347 + ], + "score": 0.43, + "content": "o s B", + "type": "inline_equation" + }, + { + "bbox": [ + 346, + 335, + 506, + 350 + ], + "score": 1.0, + "content": "hierarchy, we make use of the following", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 347, + 140, + 360 + ], + "spans": [ + { + "bbox": [ + 105, + 347, + 140, + 360 + ], + "score": 1.0, + "content": "lemma.", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 16.5 + }, + { + "type": "text", + "bbox": [ + 105, + 362, + 504, + 384 + ], + "lines": [ + { + "bbox": [ + 106, + 361, + 505, + 374 + ], + "spans": [ + { + "bbox": [ + 106, + 361, + 180, + 374 + ], + "score": 1.0, + "content": "Lemma K.6. Let", + "type": "text" + }, + { + "bbox": [ + 180, + 363, + 189, + 372 + ], + "score": 0.58, + "content": "\\mathbf { P }", + "type": "inline_equation" + }, + { + "bbox": [ + 190, + 361, + 309, + 374 + ], + "score": 1.0, + "content": "be a permutation matrix and", + "type": "text" + }, + { + "bbox": [ + 309, + 363, + 320, + 372 + ], + "score": 0.52, + "content": "\\mathbf { D }", + "type": "inline_equation" + }, + { + "bbox": [ + 320, + 361, + 505, + 374 + ], + "score": 1.0, + "content": "a diagonal matrix. Then there exist diagonal", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 373, + 233, + 385 + ], + "spans": [ + { + "bbox": [ + 105, + 373, + 143, + 385 + ], + "score": 1.0, + "content": "matrices", + "type": "text" + }, + { + "bbox": [ + 144, + 374, + 156, + 384 + ], + "score": 0.86, + "content": "\\mathbf { D ^ { \\prime } }", + "type": "inline_equation" + }, + { + "bbox": [ + 156, + 373, + 174, + 385 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 175, + 374, + 190, + 384 + ], + "score": 0.87, + "content": "\\mathbf { D } ^ { \\prime \\prime }", + "type": "inline_equation" + }, + { + "bbox": [ + 190, + 373, + 233, + 385 + ], + "score": 1.0, + "content": "such that:", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 18.5 + }, + { + "type": "interline_equation", + "bbox": [ + 231, + 389, + 379, + 403 + ], + "lines": [ + { + "bbox": [ + 231, + 389, + 379, + 403 + ], + "spans": [ + { + "bbox": [ + 231, + 389, + 379, + 403 + ], + "score": 0.85, + "content": "\\mathbf { P D } = \\mathbf { D } ^ { \\prime } \\mathbf { P } \\qquad \\mathbf { D P } = \\mathbf { P D } ^ { \\prime \\prime } .", + "type": "interline_equation", + "image_path": "fe8c4adc1543d5353b474daacb64f8be5e56c9e4bae6f964697a824a5c0bcdee.jpg" + } + ] + } + ], + "index": 20, + "virtual_lines": [ + { + "bbox": [ + 231, + 389, + 379, + 403 + ], + "spans": [], + "index": 20 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 415, + 345, + 428 + ], + "lines": [ + { + "bbox": [ + 105, + 411, + 341, + 432 + ], + "spans": [ + { + "bbox": [ + 105, + 411, + 152, + 432 + ], + "score": 1.0, + "content": "Proof. Let", + "type": "text" + }, + { + "bbox": [ + 152, + 418, + 159, + 425 + ], + "score": 0.8, + "content": "\\sigma", + "type": "inline_equation" + }, + { + "bbox": [ + 160, + 411, + 276, + 432 + ], + "score": 1.0, + "content": "be the permutation such that", + "type": "text" + }, + { + "bbox": [ + 276, + 415, + 341, + 429 + ], + "score": 0.92, + "content": "\\mathbf { P } [ i , j ] = \\delta _ { i , \\sigma ( j ) }", + "type": "inline_equation" + } + ], + "index": 21 + } + ], + "index": 21 + }, + { + "type": "text", + "bbox": [ + 106, + 432, + 366, + 446 + ], + "lines": [ + { + "bbox": [ + 105, + 431, + 367, + 447 + ], + "spans": [ + { + "bbox": [ + 105, + 431, + 135, + 447 + ], + "score": 1.0, + "content": "Define", + "type": "text" + }, + { + "bbox": [ + 136, + 433, + 149, + 444 + ], + "score": 0.88, + "content": "\\mathbf { D ^ { \\prime } }", + "type": "inline_equation" + }, + { + "bbox": [ + 149, + 431, + 187, + 447 + ], + "score": 1.0, + "content": "such that", + "type": "text" + }, + { + "bbox": [ + 188, + 433, + 289, + 446 + ], + "score": 0.92, + "content": "\\mathbf { D } ^ { \\prime } [ \\sigma ( j ) , \\sigma ( j ) ] = \\mathbf { D } [ j , j ]", + "type": "inline_equation" + }, + { + "bbox": [ + 289, + 431, + 327, + 447 + ], + "score": 1.0, + "content": ". Then, if", + "type": "text" + }, + { + "bbox": [ + 327, + 433, + 363, + 446 + ], + "score": 0.92, + "content": "i = \\sigma ( j )", + "type": "inline_equation" + }, + { + "bbox": [ + 363, + 431, + 367, + 447 + ], + "score": 1.0, + "content": ":", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 22 + }, + { + "type": "interline_equation", + "bbox": [ + 125, + 450, + 483, + 465 + ], + "lines": [ + { + "bbox": [ + 125, + 450, + 483, + 465 + ], + "spans": [ + { + "bbox": [ + 125, + 450, + 483, + 465 + ], + "score": 0.9, + "content": "\\begin{array} { r } { ( \\mathbf { P D } ) [ i , j ] = \\mathbf { P } [ i , j ] \\mathbf { D } [ j , j ] = \\mathbf { D } ^ { \\prime } [ \\sigma ( j ) , \\sigma ( j ) ] \\mathbf { P } [ \\sigma ( j ) , j ] = ( \\mathbf { D } ^ { \\prime } \\mathbf { P } ) [ \\sigma ( j ) , j ] = ( \\mathbf { D } ^ { \\prime } \\mathbf { P } ) [ i , j ] . } \\end{array}", + "type": "interline_equation", + "image_path": "90be0a73847e56d57f8128affbb600568993047e9c5010abca684d5e3e906638.jpg" + } + ] + } + ], + "index": 23, + "virtual_lines": [ + { + "bbox": [ + 125, + 450, + 483, + 465 + ], + "spans": [], + "index": 23 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 468, + 431, + 482 + ], + "lines": [ + { + "bbox": [ + 106, + 468, + 431, + 483 + ], + "spans": [ + { + "bbox": [ + 106, + 468, + 161, + 483 + ], + "score": 1.0, + "content": "Otherwise, if", + "type": "text" + }, + { + "bbox": [ + 161, + 469, + 197, + 482 + ], + "score": 0.93, + "content": "i \\neq \\sigma ( j )", + "type": "inline_equation" + }, + { + "bbox": [ + 198, + 468, + 221, + 483 + ], + "score": 1.0, + "content": ", then", + "type": "text" + }, + { + "bbox": [ + 221, + 469, + 342, + 482 + ], + "score": 0.9, + "content": "( { \\bf P D } ) [ i , j ] = 0 = ( { \\bf D ^ { \\prime } P } ) [ i , j ]", + "type": "inline_equation" + }, + { + "bbox": [ + 342, + 468, + 375, + 483 + ], + "score": 1.0, + "content": ". Hence,", + "type": "text" + }, + { + "bbox": [ + 376, + 469, + 426, + 480 + ], + "score": 0.87, + "content": "\\mathbf { P D } = \\mathbf { D } ^ { \\prime } \\mathbf { P }", + "type": "inline_equation" + }, + { + "bbox": [ + 427, + 468, + 431, + 483 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 24 + }, + { + "type": "text", + "bbox": [ + 106, + 486, + 505, + 509 + ], + "lines": [ + { + "bbox": [ + 104, + 484, + 507, + 501 + ], + "spans": [ + { + "bbox": [ + 104, + 484, + 137, + 501 + ], + "score": 1.0, + "content": "Define", + "type": "text" + }, + { + "bbox": [ + 137, + 486, + 153, + 496 + ], + "score": 0.89, + "content": "\\mathbf { D } ^ { \\prime \\prime }", + "type": "inline_equation" + }, + { + "bbox": [ + 153, + 484, + 197, + 501 + ], + "score": 1.0, + "content": "such that", + "type": "text" + }, + { + "bbox": [ + 198, + 486, + 305, + 499 + ], + "score": 0.94, + "content": "{ \\bf D } ^ { \\prime \\prime } [ j , j ] = { \\bf D } [ \\sigma ( j ) , \\sigma ( j ) ]", + "type": "inline_equation" + }, + { + "bbox": [ + 306, + 484, + 507, + 501 + ], + "score": 1.0, + "content": ". An analogous argument to above shows that", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 496, + 502, + 508 + ], + "spans": [ + { + "bbox": [ + 106, + 497, + 159, + 508 + ], + "score": 0.89, + "content": "\\mathbf { D P } = \\mathbf { P D ^ { \\prime \\prime } }", + "type": "inline_equation" + }, + { + "bbox": [ + 159, + 496, + 163, + 508 + ], + "score": 1.0, + "content": ".", + "type": "text" + }, + { + "bbox": [ + 497, + 500, + 502, + 506 + ], + "score": 0.0, + "content": "", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 25.5 + }, + { + "type": "title", + "bbox": [ + 107, + 521, + 212, + 533 + ], + "lines": [ + { + "bbox": [ + 105, + 520, + 213, + 534 + ], + "spans": [ + { + "bbox": [ + 105, + 520, + 213, + 534 + ], + "score": 1.0, + "content": "K.3.2 STEP MATRICES", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 27 + }, + { + "type": "text", + "bbox": [ + 106, + 540, + 504, + 563 + ], + "lines": [ + { + "bbox": [ + 105, + 539, + 505, + 553 + ], + "spans": [ + { + "bbox": [ + 105, + 539, + 132, + 553 + ], + "score": 1.0, + "content": "In the", + "type": "text" + }, + { + "bbox": [ + 132, + 541, + 151, + 551 + ], + "score": 0.89, + "content": "B B ^ { * }", + "type": "inline_equation" + }, + { + "bbox": [ + 152, + 539, + 505, + 553 + ], + "score": 1.0, + "content": "hierarchy (Lemma I.2), we were able to show that horizontal step matrices are butterfly", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 551, + 372, + 564 + ], + "spans": [ + { + "bbox": [ + 105, + 551, + 306, + 564 + ], + "score": 1.0, + "content": "matrices. Here, we present a similar result for the", + "type": "text" + }, + { + "bbox": [ + 306, + 552, + 329, + 562 + ], + "score": 0.47, + "content": "\\ O B B", + "type": "inline_equation" + }, + { + "bbox": [ + 330, + 551, + 372, + 564 + ], + "score": 1.0, + "content": "hierarchy.", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 28.5 + }, + { + "type": "text", + "bbox": [ + 106, + 566, + 504, + 590 + ], + "lines": [ + { + "bbox": [ + 105, + 565, + 506, + 579 + ], + "spans": [ + { + "bbox": [ + 105, + 565, + 179, + 579 + ], + "score": 1.0, + "content": "Lemma K.7. Let", + "type": "text" + }, + { + "bbox": [ + 180, + 566, + 190, + 576 + ], + "score": 0.42, + "content": "\\mathbf { H }", + "type": "inline_equation" + }, + { + "bbox": [ + 190, + 565, + 215, + 579 + ], + "score": 1.0, + "content": "be an", + "type": "text" + }, + { + "bbox": [ + 215, + 568, + 241, + 577 + ], + "score": 0.88, + "content": "n \\times n", + "type": "inline_equation" + }, + { + "bbox": [ + 241, + 565, + 434, + 579 + ], + "score": 1.0, + "content": "horizontal step matrix. Then we can decompose", + "type": "text" + }, + { + "bbox": [ + 434, + 567, + 475, + 577 + ], + "score": 0.86, + "content": "{ \\bf H } = { \\bf D } { \\bf O }", + "type": "inline_equation" + }, + { + "bbox": [ + 475, + 565, + 506, + 579 + ], + "score": 1.0, + "content": ", where", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 577, + 250, + 591 + ], + "spans": [ + { + "bbox": [ + 106, + 579, + 117, + 590 + ], + "score": 0.35, + "content": "\\mathbf { D }", + "type": "inline_equation" + }, + { + "bbox": [ + 117, + 578, + 218, + 591 + ], + "score": 1.0, + "content": "is a diagonal matrix and", + "type": "text" + }, + { + "bbox": [ + 218, + 577, + 247, + 590 + ], + "score": 0.9, + "content": "\\mathbf { O } \\in \\widetilde { B }", + "type": "inline_equation" + }, + { + "bbox": [ + 247, + 578, + 250, + 591 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 30.5 + }, + { + "type": "text", + "bbox": [ + 107, + 603, + 504, + 626 + ], + "lines": [ + { + "bbox": [ + 105, + 602, + 505, + 617 + ], + "spans": [ + { + "bbox": [ + 105, + 602, + 505, + 617 + ], + "score": 1.0, + "content": "Proof. Throughout the proof, we make reference to the original horizontal step matrix construction", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 614, + 242, + 626 + ], + "spans": [ + { + "bbox": [ + 105, + 614, + 242, + 626 + ], + "score": 1.0, + "content": "given in Lemma I.2 and its proof.", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 32.5 + }, + { + "type": "text", + "bbox": [ + 107, + 630, + 505, + 676 + ], + "lines": [ + { + "bbox": [ + 105, + 630, + 506, + 644 + ], + "spans": [ + { + "bbox": [ + 105, + 630, + 250, + 644 + ], + "score": 1.0, + "content": "To begin, we show that an arbitrary", + "type": "text" + }, + { + "bbox": [ + 251, + 630, + 284, + 642 + ], + "score": 0.92, + "content": "2 ^ { k } \\times 2 ^ { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 284, + 630, + 347, + 644 + ], + "score": 1.0, + "content": "butterfly factor", + "type": "text" + }, + { + "bbox": [ + 347, + 631, + 366, + 643 + ], + "score": 0.91, + "content": "\\mathbf { H } _ { 2 ^ { k } }", + "type": "inline_equation" + }, + { + "bbox": [ + 366, + 630, + 465, + 644 + ], + "score": 1.0, + "content": "in the decomposition of", + "type": "text" + }, + { + "bbox": [ + 465, + 631, + 476, + 641 + ], + "score": 0.39, + "content": "\\mathbf { H }", + "type": "inline_equation" + }, + { + "bbox": [ + 476, + 630, + 506, + 644 + ], + "score": 1.0, + "content": "can be", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 642, + 505, + 654 + ], + "spans": [ + { + "bbox": [ + 105, + 642, + 505, + 654 + ], + "score": 1.0, + "content": "expressed as the product of a diagonal matrix and an orthogonal butterfly factor. Since a butterfly", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 653, + 506, + 666 + ], + "spans": [ + { + "bbox": [ + 105, + 654, + 199, + 666 + ], + "score": 1.0, + "content": "factor is direct sum of", + "type": "text" + }, + { + "bbox": [ + 199, + 654, + 223, + 664 + ], + "score": 0.91, + "content": "2 \\times 2", + "type": "inline_equation" + }, + { + "bbox": [ + 223, + 654, + 384, + 666 + ], + "score": 1.0, + "content": "matrices, there is a permutation matrix", + "type": "text" + }, + { + "bbox": [ + 385, + 653, + 402, + 665 + ], + "score": 0.9, + "content": "\\mathbf { P } _ { 2 ^ { k } }", + "type": "inline_equation" + }, + { + "bbox": [ + 402, + 654, + 506, + 666 + ], + "score": 1.0, + "content": "such that conjugation of", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 106, + 663, + 399, + 678 + ], + "spans": [ + { + "bbox": [ + 106, + 664, + 125, + 676 + ], + "score": 0.89, + "content": "\\mathbf { H } _ { 2 ^ { k } }", + "type": "inline_equation" + }, + { + "bbox": [ + 125, + 663, + 139, + 678 + ], + "score": 1.0, + "content": "by", + "type": "text" + }, + { + "bbox": [ + 139, + 664, + 157, + 676 + ], + "score": 0.9, + "content": "\\mathbf { P } _ { 2 ^ { k } }", + "type": "inline_equation" + }, + { + "bbox": [ + 157, + 663, + 278, + 678 + ], + "score": 1.0, + "content": "gives a block diagonal matrix", + "type": "text" + }, + { + "bbox": [ + 279, + 664, + 297, + 677 + ], + "score": 0.92, + "content": "\\mathbf { H } _ { 2 ^ { k } } ^ { \\prime }", + "type": "inline_equation" + }, + { + "bbox": [ + 297, + 663, + 310, + 678 + ], + "score": 1.0, + "content": "of", + "type": "text" + }, + { + "bbox": [ + 310, + 664, + 343, + 677 + ], + "score": 0.91, + "content": "{ \\frac { n } { 2 } } \\ 2 \\times 2", + "type": "inline_equation" + }, + { + "bbox": [ + 343, + 663, + 399, + 678 + ], + "score": 1.0, + "content": "matrices, i.e.", + "type": "text" + } + ], + "index": 37 + } + ], + "index": 35.5 + }, + { + "type": "interline_equation", + "bbox": [ + 262, + 682, + 349, + 696 + ], + "lines": [ + { + "bbox": [ + 262, + 682, + 349, + 696 + ], + "spans": [ + { + "bbox": [ + 262, + 682, + 349, + 696 + ], + "score": 0.91, + "content": "\\mathbf { P } _ { 2 ^ { k } } \\mathbf { H } _ { 2 ^ { k } } \\mathbf { P } _ { 2 ^ { k } } ^ { * } = \\mathbf { H } _ { 2 ^ { k } } ^ { \\prime } .", + "type": "interline_equation", + "image_path": "a4cb9b1d7f35443fb39d0293b250e9717c463e164f18db57e633ea12a96fa94e.jpg" + } + ] + } + ], + "index": 38, + "virtual_lines": [ + { + "bbox": [ + 262, + 682, + 349, + 696 + ], + "spans": [], + "index": 38 + } + ] + }, + { + "type": "text", + "bbox": [ + 105, + 700, + 419, + 713 + ], + "lines": [ + { + "bbox": [ + 105, + 700, + 421, + 715 + ], + "spans": [ + { + "bbox": [ + 105, + 700, + 297, + 715 + ], + "score": 1.0, + "content": "(See Figure 10 for an illustration.) Specifically,", + "type": "text" + }, + { + "bbox": [ + 297, + 702, + 314, + 713 + ], + "score": 0.9, + "content": "\\mathbf { P } _ { 2 ^ { k } }", + "type": "inline_equation" + }, + { + "bbox": [ + 314, + 700, + 421, + 715 + ], + "score": 1.0, + "content": "is the permutation where:", + "type": "text" + } + ], + "index": 39 + } + ], + "index": 39 + }, + { + "type": "interline_equation", + "bbox": [ + 211, + 718, + 399, + 735 + ], + "lines": [ + { + "bbox": [ + 211, + 718, + 399, + 735 + ], + "spans": [ + { + "bbox": [ + 211, + 718, + 399, + 735 + ], + "score": 0.89, + "content": "{ \\bf P } _ { s } [ 2 i , : ] = { \\bf e } _ { i } ^ { T } { \\bf \\mathrm { ~ ~ { ~ \\cal ~ ~ { ~ P ~ } ~ } ~ } } _ { s } [ 2 i + 1 , : ] = { \\bf e } _ { i + \\frac { n } { 2 } } ^ { T } .", + "type": "interline_equation", + "image_path": "2f7c163aae1a9c1f7639cc3e31b77728c067fe128716b6f52400cb4c059c2759.jpg" + } + ] + } + ], + "index": 40, + "virtual_lines": [ + { + "bbox": [ + 211, + 718, + 399, + 735 + ], + "spans": [], + "index": 40 + } + ] + } + ], + "page_idx": 37, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 107, + 26, + 294, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 25, + 294, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 25, + 294, + 38 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2020", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 300, + 751, + 311, + 760 + ], + "lines": [ + { + "bbox": [ + 298, + 750, + 313, + 763 + ], + "spans": [ + { + "bbox": [ + 298, + 750, + 313, + 763 + ], + "score": 1.0, + "content": "38", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "title", + "bbox": [ + 108, + 82, + 208, + 94 + ], + "lines": [ + { + "bbox": [ + 105, + 81, + 209, + 96 + ], + "spans": [ + { + "bbox": [ + 105, + 81, + 209, + 96 + ], + "score": 1.0, + "content": "K.3 CONSTRUCTIONS", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "text", + "bbox": [ + 106, + 102, + 505, + 169 + ], + "lines": [ + { + "bbox": [ + 106, + 102, + 505, + 115 + ], + "spans": [ + { + "bbox": [ + 106, + 102, + 233, + 115 + ], + "score": 1.0, + "content": "We show that we can construct", + "type": "text" + }, + { + "bbox": [ + 233, + 105, + 239, + 113 + ], + "score": 0.43, + "content": "s", + "type": "inline_equation" + }, + { + "bbox": [ + 239, + 102, + 505, + 115 + ], + "score": 1.0, + "content": "-sparse matrices in the OBB hierarchy with the same width as the", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 107, + 114, + 505, + 127 + ], + "spans": [ + { + "bbox": [ + 107, + 114, + 126, + 125 + ], + "score": 0.86, + "content": "B B ^ { * }", + "type": "inline_equation" + }, + { + "bbox": [ + 126, + 114, + 505, + 127 + ], + "score": 1.0, + "content": "hierarchy. The proof follows a structure to that of Theorem 3. We begin by arguing about", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 104, + 125, + 506, + 138 + ], + "spans": [ + { + "bbox": [ + 104, + 125, + 506, + 138 + ], + "score": 1.0, + "content": "permutation and step matrices, then using the same factorization to argue that matrices with at most", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 106, + 135, + 506, + 148 + ], + "spans": [ + { + "bbox": [ + 106, + 137, + 114, + 146 + ], + "score": 0.67, + "content": "n", + "type": "inline_equation" + }, + { + "bbox": [ + 114, + 135, + 205, + 148 + ], + "score": 1.0, + "content": "NNZ are contained in", + "type": "text" + }, + { + "bbox": [ + 206, + 135, + 236, + 148 + ], + "score": 0.92, + "content": "( B B ^ { * } ) ^ { 4 }", + "type": "inline_equation" + }, + { + "bbox": [ + 237, + 135, + 506, + 148 + ], + "score": 1.0, + "content": ". Then, we will appeal to a modified sum closure lemma to extend", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 106, + 147, + 505, + 159 + ], + "spans": [ + { + "bbox": [ + 106, + 147, + 249, + 159 + ], + "score": 1.0, + "content": "the argument to matrices of general", + "type": "text" + }, + { + "bbox": [ + 249, + 149, + 255, + 157 + ], + "score": 0.58, + "content": "s", + "type": "inline_equation" + }, + { + "bbox": [ + 255, + 147, + 505, + 159 + ], + "score": 1.0, + "content": "NNZ. Similar to Appendix F, we can use these results to place", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 157, + 469, + 171 + ], + "spans": [ + { + "bbox": [ + 105, + 157, + 403, + 171 + ], + "score": 1.0, + "content": "all matrices with low-depth circuits for matrix vector multiplication in the", + "type": "text" + }, + { + "bbox": [ + 403, + 158, + 426, + 168 + ], + "score": 0.25, + "content": "\\ O B B", + "type": "inline_equation" + }, + { + "bbox": [ + 427, + 157, + 469, + 171 + ], + "score": 1.0, + "content": "hierarchy.", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 3.5, + "bbox_fs": [ + 104, + 102, + 506, + 171 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 181, + 210, + 193 + ], + "lines": [ + { + "bbox": [ + 105, + 180, + 211, + 194 + ], + "spans": [ + { + "bbox": [ + 105, + 180, + 211, + 194 + ], + "score": 1.0, + "content": "K.3.1 PERMUTATIONS", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 7 + }, + { + "type": "text", + "bbox": [ + 109, + 200, + 504, + 223 + ], + "lines": [ + { + "bbox": [ + 106, + 199, + 506, + 214 + ], + "spans": [ + { + "bbox": [ + 106, + 199, + 506, + 214 + ], + "score": 1.0, + "content": "We begin by presenting the argument that permutations are included in OBB as a corollary to", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 210, + 156, + 224 + ], + "spans": [ + { + "bbox": [ + 106, + 210, + 156, + 224 + ], + "score": 1.0, + "content": "Theorem 2.", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 8.5, + "bbox_fs": [ + 106, + 199, + 506, + 224 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 225, + 366, + 239 + ], + "lines": [ + { + "bbox": [ + 106, + 225, + 366, + 239 + ], + "spans": [ + { + "bbox": [ + 106, + 226, + 323, + 239 + ], + "score": 1.0, + "content": "Corollary K.4. Let P be a permutation matrix. Then", + "type": "text" + }, + { + "bbox": [ + 323, + 225, + 362, + 237 + ], + "score": 0.92, + "content": "\\mathbf { P } \\in \\widetilde { B } \\widetilde { B } ^ { * }", + "type": "inline_equation" + }, + { + "bbox": [ + 362, + 226, + 366, + 239 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 10, + "bbox_fs": [ + 106, + 225, + 366, + 239 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 250, + 506, + 284 + ], + "lines": [ + { + "bbox": [ + 105, + 250, + 506, + 263 + ], + "spans": [ + { + "bbox": [ + 105, + 250, + 506, + 263 + ], + "score": 1.0, + "content": "Proof. We appeal to the decomposition from Theorem 2, noting that all butterfly factor matrices", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 262, + 507, + 274 + ], + "spans": [ + { + "bbox": [ + 105, + 262, + 507, + 274 + ], + "score": 1.0, + "content": "constructed in the proofs of Lemmas G.3 and G.1 are permutation matrices, and thus are orthogonal.", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 272, + 506, + 285 + ], + "spans": [ + { + "bbox": [ + 105, + 272, + 136, + 285 + ], + "score": 1.0, + "content": "Hence,", + "type": "text" + }, + { + "bbox": [ + 137, + 273, + 180, + 283 + ], + "score": 0.89, + "content": "\\mathbf { P } \\in { \\mathcal { O } } B B", + "type": "inline_equation" + }, + { + "bbox": [ + 181, + 272, + 330, + 285 + ], + "score": 1.0, + "content": "where the inner diagonal matrix is I.", + "type": "text" + }, + { + "bbox": [ + 494, + 273, + 506, + 284 + ], + "score": 0.999, + "content": "□", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 12, + "bbox_fs": [ + 105, + 250, + 507, + 285 + ] + }, + { + "type": "text", + "bbox": [ + 105, + 297, + 473, + 310 + ], + "lines": [ + { + "bbox": [ + 105, + 296, + 474, + 311 + ], + "spans": [ + { + "bbox": [ + 105, + 297, + 451, + 311 + ], + "score": 1.0, + "content": "Similarly, the construction of Lemma G.6 also show that permutations are included in", + "type": "text" + }, + { + "bbox": [ + 451, + 296, + 470, + 309 + ], + "score": 0.88, + "content": "\\widetilde { B } ^ { * } \\widetilde { B }", + "type": "inline_equation" + }, + { + "bbox": [ + 471, + 297, + 474, + 311 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 14, + "bbox_fs": [ + 105, + 296, + 474, + 311 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 314, + 366, + 327 + ], + "lines": [ + { + "bbox": [ + 106, + 313, + 366, + 328 + ], + "spans": [ + { + "bbox": [ + 106, + 314, + 188, + 328 + ], + "score": 1.0, + "content": "Corollary K.5. Let", + "type": "text" + }, + { + "bbox": [ + 188, + 316, + 197, + 325 + ], + "score": 0.39, + "content": "\\mathbf { P }", + "type": "inline_equation" + }, + { + "bbox": [ + 198, + 314, + 323, + 328 + ], + "score": 1.0, + "content": "be a permutation matrix. Then", + "type": "text" + }, + { + "bbox": [ + 323, + 313, + 363, + 326 + ], + "score": 0.91, + "content": "\\mathbf { P } \\in \\widetilde { B } ^ { * } \\widetilde { B }", + "type": "inline_equation" + }, + { + "bbox": [ + 363, + 314, + 366, + 328 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 15, + "bbox_fs": [ + 106, + 313, + 366, + 328 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 336, + 505, + 358 + ], + "lines": [ + { + "bbox": [ + 105, + 335, + 506, + 350 + ], + "spans": [ + { + "bbox": [ + 105, + 335, + 322, + 350 + ], + "score": 1.0, + "content": "To prove the containment of sparse matrices within the", + "type": "text" + }, + { + "bbox": [ + 322, + 337, + 345, + 347 + ], + "score": 0.43, + "content": "o s B", + "type": "inline_equation" + }, + { + "bbox": [ + 346, + 335, + 506, + 350 + ], + "score": 1.0, + "content": "hierarchy, we make use of the following", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 347, + 140, + 360 + ], + "spans": [ + { + "bbox": [ + 105, + 347, + 140, + 360 + ], + "score": 1.0, + "content": "lemma.", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 16.5, + "bbox_fs": [ + 105, + 335, + 506, + 360 + ] + }, + { + "type": "text", + "bbox": [ + 105, + 362, + 504, + 384 + ], + "lines": [ + { + "bbox": [ + 106, + 361, + 505, + 374 + ], + "spans": [ + { + "bbox": [ + 106, + 361, + 180, + 374 + ], + "score": 1.0, + "content": "Lemma K.6. Let", + "type": "text" + }, + { + "bbox": [ + 180, + 363, + 189, + 372 + ], + "score": 0.58, + "content": "\\mathbf { P }", + "type": "inline_equation" + }, + { + "bbox": [ + 190, + 361, + 309, + 374 + ], + "score": 1.0, + "content": "be a permutation matrix and", + "type": "text" + }, + { + "bbox": [ + 309, + 363, + 320, + 372 + ], + "score": 0.52, + "content": "\\mathbf { D }", + "type": "inline_equation" + }, + { + "bbox": [ + 320, + 361, + 505, + 374 + ], + "score": 1.0, + "content": "a diagonal matrix. Then there exist diagonal", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 373, + 233, + 385 + ], + "spans": [ + { + "bbox": [ + 105, + 373, + 143, + 385 + ], + "score": 1.0, + "content": "matrices", + "type": "text" + }, + { + "bbox": [ + 144, + 374, + 156, + 384 + ], + "score": 0.86, + "content": "\\mathbf { D ^ { \\prime } }", + "type": "inline_equation" + }, + { + "bbox": [ + 156, + 373, + 174, + 385 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 175, + 374, + 190, + 384 + ], + "score": 0.87, + "content": "\\mathbf { D } ^ { \\prime \\prime }", + "type": "inline_equation" + }, + { + "bbox": [ + 190, + 373, + 233, + 385 + ], + "score": 1.0, + "content": "such that:", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 18.5, + "bbox_fs": [ + 105, + 361, + 505, + 385 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 231, + 389, + 379, + 403 + ], + "lines": [ + { + "bbox": [ + 231, + 389, + 379, + 403 + ], + "spans": [ + { + "bbox": [ + 231, + 389, + 379, + 403 + ], + "score": 0.85, + "content": "\\mathbf { P D } = \\mathbf { D } ^ { \\prime } \\mathbf { P } \\qquad \\mathbf { D P } = \\mathbf { P D } ^ { \\prime \\prime } .", + "type": "interline_equation", + "image_path": "fe8c4adc1543d5353b474daacb64f8be5e56c9e4bae6f964697a824a5c0bcdee.jpg" + } + ] + } + ], + "index": 20, + "virtual_lines": [ + { + "bbox": [ + 231, + 389, + 379, + 403 + ], + "spans": [], + "index": 20 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 415, + 345, + 428 + ], + "lines": [ + { + "bbox": [ + 105, + 411, + 341, + 432 + ], + "spans": [ + { + "bbox": [ + 105, + 411, + 152, + 432 + ], + "score": 1.0, + "content": "Proof. Let", + "type": "text" + }, + { + "bbox": [ + 152, + 418, + 159, + 425 + ], + "score": 0.8, + "content": "\\sigma", + "type": "inline_equation" + }, + { + "bbox": [ + 160, + 411, + 276, + 432 + ], + "score": 1.0, + "content": "be the permutation such that", + "type": "text" + }, + { + "bbox": [ + 276, + 415, + 341, + 429 + ], + "score": 0.92, + "content": "\\mathbf { P } [ i , j ] = \\delta _ { i , \\sigma ( j ) }", + "type": "inline_equation" + } + ], + "index": 21 + } + ], + "index": 21, + "bbox_fs": [ + 105, + 411, + 341, + 432 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 432, + 366, + 446 + ], + "lines": [ + { + "bbox": [ + 105, + 431, + 367, + 447 + ], + "spans": [ + { + "bbox": [ + 105, + 431, + 135, + 447 + ], + "score": 1.0, + "content": "Define", + "type": "text" + }, + { + "bbox": [ + 136, + 433, + 149, + 444 + ], + "score": 0.88, + "content": "\\mathbf { D ^ { \\prime } }", + "type": "inline_equation" + }, + { + "bbox": [ + 149, + 431, + 187, + 447 + ], + "score": 1.0, + "content": "such that", + "type": "text" + }, + { + "bbox": [ + 188, + 433, + 289, + 446 + ], + "score": 0.92, + "content": "\\mathbf { D } ^ { \\prime } [ \\sigma ( j ) , \\sigma ( j ) ] = \\mathbf { D } [ j , j ]", + "type": "inline_equation" + }, + { + "bbox": [ + 289, + 431, + 327, + 447 + ], + "score": 1.0, + "content": ". Then, if", + "type": "text" + }, + { + "bbox": [ + 327, + 433, + 363, + 446 + ], + "score": 0.92, + "content": "i = \\sigma ( j )", + "type": "inline_equation" + }, + { + "bbox": [ + 363, + 431, + 367, + 447 + ], + "score": 1.0, + "content": ":", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 22, + "bbox_fs": [ + 105, + 431, + 367, + 447 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 125, + 450, + 483, + 465 + ], + "lines": [ + { + "bbox": [ + 125, + 450, + 483, + 465 + ], + "spans": [ + { + "bbox": [ + 125, + 450, + 483, + 465 + ], + "score": 0.9, + "content": "\\begin{array} { r } { ( \\mathbf { P D } ) [ i , j ] = \\mathbf { P } [ i , j ] \\mathbf { D } [ j , j ] = \\mathbf { D } ^ { \\prime } [ \\sigma ( j ) , \\sigma ( j ) ] \\mathbf { P } [ \\sigma ( j ) , j ] = ( \\mathbf { D } ^ { \\prime } \\mathbf { P } ) [ \\sigma ( j ) , j ] = ( \\mathbf { D } ^ { \\prime } \\mathbf { P } ) [ i , j ] . } \\end{array}", + "type": "interline_equation", + "image_path": "90be0a73847e56d57f8128affbb600568993047e9c5010abca684d5e3e906638.jpg" + } + ] + } + ], + "index": 23, + "virtual_lines": [ + { + "bbox": [ + 125, + 450, + 483, + 465 + ], + "spans": [], + "index": 23 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 468, + 431, + 482 + ], + "lines": [ + { + "bbox": [ + 106, + 468, + 431, + 483 + ], + "spans": [ + { + "bbox": [ + 106, + 468, + 161, + 483 + ], + "score": 1.0, + "content": "Otherwise, if", + "type": "text" + }, + { + "bbox": [ + 161, + 469, + 197, + 482 + ], + "score": 0.93, + "content": "i \\neq \\sigma ( j )", + "type": "inline_equation" + }, + { + "bbox": [ + 198, + 468, + 221, + 483 + ], + "score": 1.0, + "content": ", then", + "type": "text" + }, + { + "bbox": [ + 221, + 469, + 342, + 482 + ], + "score": 0.9, + "content": "( { \\bf P D } ) [ i , j ] = 0 = ( { \\bf D ^ { \\prime } P } ) [ i , j ]", + "type": "inline_equation" + }, + { + "bbox": [ + 342, + 468, + 375, + 483 + ], + "score": 1.0, + "content": ". Hence,", + "type": "text" + }, + { + "bbox": [ + 376, + 469, + 426, + 480 + ], + "score": 0.87, + "content": "\\mathbf { P D } = \\mathbf { D } ^ { \\prime } \\mathbf { P }", + "type": "inline_equation" + }, + { + "bbox": [ + 427, + 468, + 431, + 483 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 24, + "bbox_fs": [ + 106, + 468, + 431, + 483 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 486, + 505, + 509 + ], + "lines": [ + { + "bbox": [ + 104, + 484, + 507, + 501 + ], + "spans": [ + { + "bbox": [ + 104, + 484, + 137, + 501 + ], + "score": 1.0, + "content": "Define", + "type": "text" + }, + { + "bbox": [ + 137, + 486, + 153, + 496 + ], + "score": 0.89, + "content": "\\mathbf { D } ^ { \\prime \\prime }", + "type": "inline_equation" + }, + { + "bbox": [ + 153, + 484, + 197, + 501 + ], + "score": 1.0, + "content": "such that", + "type": "text" + }, + { + "bbox": [ + 198, + 486, + 305, + 499 + ], + "score": 0.94, + "content": "{ \\bf D } ^ { \\prime \\prime } [ j , j ] = { \\bf D } [ \\sigma ( j ) , \\sigma ( j ) ]", + "type": "inline_equation" + }, + { + "bbox": [ + 306, + 484, + 507, + 501 + ], + "score": 1.0, + "content": ". An analogous argument to above shows that", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 496, + 502, + 508 + ], + "spans": [ + { + "bbox": [ + 106, + 497, + 159, + 508 + ], + "score": 0.89, + "content": "\\mathbf { D P } = \\mathbf { P D ^ { \\prime \\prime } }", + "type": "inline_equation" + }, + { + "bbox": [ + 159, + 496, + 163, + 508 + ], + "score": 1.0, + "content": ".", + "type": "text" + }, + { + "bbox": [ + 497, + 500, + 502, + 506 + ], + "score": 0.0, + "content": "", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 25.5, + "bbox_fs": [ + 104, + 484, + 507, + 508 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 521, + 212, + 533 + ], + "lines": [ + { + "bbox": [ + 105, + 520, + 213, + 534 + ], + "spans": [ + { + "bbox": [ + 105, + 520, + 213, + 534 + ], + "score": 1.0, + "content": "K.3.2 STEP MATRICES", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 27 + }, + { + "type": "text", + "bbox": [ + 106, + 540, + 504, + 563 + ], + "lines": [ + { + "bbox": [ + 105, + 539, + 505, + 553 + ], + "spans": [ + { + "bbox": [ + 105, + 539, + 132, + 553 + ], + "score": 1.0, + "content": "In the", + "type": "text" + }, + { + "bbox": [ + 132, + 541, + 151, + 551 + ], + "score": 0.89, + "content": "B B ^ { * }", + "type": "inline_equation" + }, + { + "bbox": [ + 152, + 539, + 505, + 553 + ], + "score": 1.0, + "content": "hierarchy (Lemma I.2), we were able to show that horizontal step matrices are butterfly", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 551, + 372, + 564 + ], + "spans": [ + { + "bbox": [ + 105, + 551, + 306, + 564 + ], + "score": 1.0, + "content": "matrices. Here, we present a similar result for the", + "type": "text" + }, + { + "bbox": [ + 306, + 552, + 329, + 562 + ], + "score": 0.47, + "content": "\\ O B B", + "type": "inline_equation" + }, + { + "bbox": [ + 330, + 551, + 372, + 564 + ], + "score": 1.0, + "content": "hierarchy.", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 28.5, + "bbox_fs": [ + 105, + 539, + 505, + 564 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 566, + 504, + 590 + ], + "lines": [ + { + "bbox": [ + 105, + 565, + 506, + 579 + ], + "spans": [ + { + "bbox": [ + 105, + 565, + 179, + 579 + ], + "score": 1.0, + "content": "Lemma K.7. Let", + "type": "text" + }, + { + "bbox": [ + 180, + 566, + 190, + 576 + ], + "score": 0.42, + "content": "\\mathbf { H }", + "type": "inline_equation" + }, + { + "bbox": [ + 190, + 565, + 215, + 579 + ], + "score": 1.0, + "content": "be an", + "type": "text" + }, + { + "bbox": [ + 215, + 568, + 241, + 577 + ], + "score": 0.88, + "content": "n \\times n", + "type": "inline_equation" + }, + { + "bbox": [ + 241, + 565, + 434, + 579 + ], + "score": 1.0, + "content": "horizontal step matrix. Then we can decompose", + "type": "text" + }, + { + "bbox": [ + 434, + 567, + 475, + 577 + ], + "score": 0.86, + "content": "{ \\bf H } = { \\bf D } { \\bf O }", + "type": "inline_equation" + }, + { + "bbox": [ + 475, + 565, + 506, + 579 + ], + "score": 1.0, + "content": ", where", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 577, + 250, + 591 + ], + "spans": [ + { + "bbox": [ + 106, + 579, + 117, + 590 + ], + "score": 0.35, + "content": "\\mathbf { D }", + "type": "inline_equation" + }, + { + "bbox": [ + 117, + 578, + 218, + 591 + ], + "score": 1.0, + "content": "is a diagonal matrix and", + "type": "text" + }, + { + "bbox": [ + 218, + 577, + 247, + 590 + ], + "score": 0.9, + "content": "\\mathbf { O } \\in \\widetilde { B }", + "type": "inline_equation" + }, + { + "bbox": [ + 247, + 578, + 250, + 591 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 30.5, + "bbox_fs": [ + 105, + 565, + 506, + 591 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 603, + 504, + 626 + ], + "lines": [ + { + "bbox": [ + 105, + 602, + 505, + 617 + ], + "spans": [ + { + "bbox": [ + 105, + 602, + 505, + 617 + ], + "score": 1.0, + "content": "Proof. Throughout the proof, we make reference to the original horizontal step matrix construction", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 614, + 242, + 626 + ], + "spans": [ + { + "bbox": [ + 105, + 614, + 242, + 626 + ], + "score": 1.0, + "content": "given in Lemma I.2 and its proof.", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 32.5, + "bbox_fs": [ + 105, + 602, + 505, + 626 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 630, + 505, + 676 + ], + "lines": [ + { + "bbox": [ + 105, + 630, + 506, + 644 + ], + "spans": [ + { + "bbox": [ + 105, + 630, + 250, + 644 + ], + "score": 1.0, + "content": "To begin, we show that an arbitrary", + "type": "text" + }, + { + "bbox": [ + 251, + 630, + 284, + 642 + ], + "score": 0.92, + "content": "2 ^ { k } \\times 2 ^ { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 284, + 630, + 347, + 644 + ], + "score": 1.0, + "content": "butterfly factor", + "type": "text" + }, + { + "bbox": [ + 347, + 631, + 366, + 643 + ], + "score": 0.91, + "content": "\\mathbf { H } _ { 2 ^ { k } }", + "type": "inline_equation" + }, + { + "bbox": [ + 366, + 630, + 465, + 644 + ], + "score": 1.0, + "content": "in the decomposition of", + "type": "text" + }, + { + "bbox": [ + 465, + 631, + 476, + 641 + ], + "score": 0.39, + "content": "\\mathbf { H }", + "type": "inline_equation" + }, + { + "bbox": [ + 476, + 630, + 506, + 644 + ], + "score": 1.0, + "content": "can be", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 642, + 505, + 654 + ], + "spans": [ + { + "bbox": [ + 105, + 642, + 505, + 654 + ], + "score": 1.0, + "content": "expressed as the product of a diagonal matrix and an orthogonal butterfly factor. Since a butterfly", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 653, + 506, + 666 + ], + "spans": [ + { + "bbox": [ + 105, + 654, + 199, + 666 + ], + "score": 1.0, + "content": "factor is direct sum of", + "type": "text" + }, + { + "bbox": [ + 199, + 654, + 223, + 664 + ], + "score": 0.91, + "content": "2 \\times 2", + "type": "inline_equation" + }, + { + "bbox": [ + 223, + 654, + 384, + 666 + ], + "score": 1.0, + "content": "matrices, there is a permutation matrix", + "type": "text" + }, + { + "bbox": [ + 385, + 653, + 402, + 665 + ], + "score": 0.9, + "content": "\\mathbf { P } _ { 2 ^ { k } }", + "type": "inline_equation" + }, + { + "bbox": [ + 402, + 654, + 506, + 666 + ], + "score": 1.0, + "content": "such that conjugation of", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 106, + 663, + 399, + 678 + ], + "spans": [ + { + "bbox": [ + 106, + 664, + 125, + 676 + ], + "score": 0.89, + "content": "\\mathbf { H } _ { 2 ^ { k } }", + "type": "inline_equation" + }, + { + "bbox": [ + 125, + 663, + 139, + 678 + ], + "score": 1.0, + "content": "by", + "type": "text" + }, + { + "bbox": [ + 139, + 664, + 157, + 676 + ], + "score": 0.9, + "content": "\\mathbf { P } _ { 2 ^ { k } }", + "type": "inline_equation" + }, + { + "bbox": [ + 157, + 663, + 278, + 678 + ], + "score": 1.0, + "content": "gives a block diagonal matrix", + "type": "text" + }, + { + "bbox": [ + 279, + 664, + 297, + 677 + ], + "score": 0.92, + "content": "\\mathbf { H } _ { 2 ^ { k } } ^ { \\prime }", + "type": "inline_equation" + }, + { + "bbox": [ + 297, + 663, + 310, + 678 + ], + "score": 1.0, + "content": "of", + "type": "text" + }, + { + "bbox": [ + 310, + 664, + 343, + 677 + ], + "score": 0.91, + "content": "{ \\frac { n } { 2 } } \\ 2 \\times 2", + "type": "inline_equation" + }, + { + "bbox": [ + 343, + 663, + 399, + 678 + ], + "score": 1.0, + "content": "matrices, i.e.", + "type": "text" + } + ], + "index": 37 + } + ], + "index": 35.5, + "bbox_fs": [ + 105, + 630, + 506, + 678 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 262, + 682, + 349, + 696 + ], + "lines": [ + { + "bbox": [ + 262, + 682, + 349, + 696 + ], + "spans": [ + { + "bbox": [ + 262, + 682, + 349, + 696 + ], + "score": 0.91, + "content": "\\mathbf { P } _ { 2 ^ { k } } \\mathbf { H } _ { 2 ^ { k } } \\mathbf { P } _ { 2 ^ { k } } ^ { * } = \\mathbf { H } _ { 2 ^ { k } } ^ { \\prime } .", + "type": "interline_equation", + "image_path": "a4cb9b1d7f35443fb39d0293b250e9717c463e164f18db57e633ea12a96fa94e.jpg" + } + ] + } + ], + "index": 38, + "virtual_lines": [ + { + "bbox": [ + 262, + 682, + 349, + 696 + ], + "spans": [], + "index": 38 + } + ] + }, + { + "type": "text", + "bbox": [ + 105, + 700, + 419, + 713 + ], + "lines": [ + { + "bbox": [ + 105, + 700, + 421, + 715 + ], + "spans": [ + { + "bbox": [ + 105, + 700, + 297, + 715 + ], + "score": 1.0, + "content": "(See Figure 10 for an illustration.) Specifically,", + "type": "text" + }, + { + "bbox": [ + 297, + 702, + 314, + 713 + ], + "score": 0.9, + "content": "\\mathbf { P } _ { 2 ^ { k } }", + "type": "inline_equation" + }, + { + "bbox": [ + 314, + 700, + 421, + 715 + ], + "score": 1.0, + "content": "is the permutation where:", + "type": "text" + } + ], + "index": 39 + } + ], + "index": 39, + "bbox_fs": [ + 105, + 700, + 421, + 715 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 211, + 718, + 399, + 735 + ], + "lines": [ + { + "bbox": [ + 211, + 718, + 399, + 735 + ], + "spans": [ + { + "bbox": [ + 211, + 718, + 399, + 735 + ], + "score": 0.89, + "content": "{ \\bf P } _ { s } [ 2 i , : ] = { \\bf e } _ { i } ^ { T } { \\bf \\mathrm { ~ ~ { ~ \\cal ~ ~ { ~ P ~ } ~ } ~ } } _ { s } [ 2 i + 1 , : ] = { \\bf e } _ { i + \\frac { n } { 2 } } ^ { T } .", + "type": "interline_equation", + "image_path": "2f7c163aae1a9c1f7639cc3e31b77728c067fe128716b6f52400cb4c059c2759.jpg" + } + ] + } + ], + "index": 40, + "virtual_lines": [ + { + "bbox": [ + 211, + 718, + 399, + 735 + ], + "spans": [], + "index": 40 + } + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "image", + "bbox": [ + 108, + 88, + 546, + 228 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 108, + 88, + 546, + 228 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 108, + 88, + 546, + 228 + ], + "spans": [ + { + "bbox": [ + 108, + 88, + 546, + 228 + ], + "score": 0.957, + "type": "image", + "image_path": "a4607d3febb414e4ea1a9f3500d9dc6ca268a2509f0b7cd81b00a8b90f87d6e6.jpg" + } + ] + } + ], + "index": 1, + "virtual_lines": [ + { + "bbox": [ + 108, + 88, + 546, + 134.66666666666666 + ], + "spans": [], + "index": 0 + }, + { + "bbox": [ + 108, + 134.66666666666666, + 546, + 181.33333333333331 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 108, + 181.33333333333331, + 546, + 227.99999999999997 + ], + "spans": [], + "index": 2 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 225, + 245, + 385, + 258 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 223, + 243, + 385, + 260 + ], + "spans": [ + { + "bbox": [ + 223, + 243, + 370, + 260 + ], + "score": 1.0, + "content": "Figure 10: Block diagonalization of", + "type": "text" + }, + { + "bbox": [ + 371, + 246, + 385, + 257 + ], + "score": 0.82, + "content": "{ \\bf { H } } _ { 8 }", + "type": "inline_equation" + } + ], + "index": 3 + } + ], + "index": 3 + } + ], + "index": 2.0 + }, + { + "type": "text", + "bbox": [ + 106, + 276, + 506, + 321 + ], + "lines": [ + { + "bbox": [ + 106, + 276, + 506, + 288 + ], + "spans": [ + { + "bbox": [ + 106, + 276, + 213, + 288 + ], + "score": 1.0, + "content": "We argue that each of these", + "type": "text" + }, + { + "bbox": [ + 213, + 277, + 233, + 287 + ], + "score": 0.88, + "content": "2 \\times 2", + "type": "inline_equation" + }, + { + "bbox": [ + 234, + 276, + 506, + 288 + ], + "score": 1.0, + "content": "blocks can be decomposed into a diagonal matrix times an orthogonal", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 106, + 288, + 505, + 298 + ], + "spans": [ + { + "bbox": [ + 106, + 288, + 505, + 298 + ], + "score": 1.0, + "content": "matrix. Note that the butterfly factor matrices constructed in the proof of Lemma I.2 each have at", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 299, + 506, + 310 + ], + "spans": [ + { + "bbox": [ + 105, + 299, + 506, + 310 + ], + "score": 1.0, + "content": "most one non-zero entry per column. Hence, there are 4 cases to consider. Note that matrices with at", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 310, + 335, + 321 + ], + "spans": [ + { + "bbox": [ + 105, + 310, + 335, + 321 + ], + "score": 1.0, + "content": "most one non-zero entry are exhausted by Cases 1 and 2.", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 5.5 + }, + { + "type": "interline_equation", + "bbox": [ + 124, + 324, + 254, + 370 + ], + "lines": [ + { + "bbox": [ + 124, + 324, + 254, + 370 + ], + "spans": [ + { + "bbox": [ + 124, + 324, + 254, + 370 + ], + "score": 0.91, + "content": "\\mathfrak { e } \\ 1 : \\left[ 0 \\atop 0 \\right] = \\underbrace { \\left[ 0 \\begin{array} { l l } { 0 } & { 0 } \\\\ { 0 } & { b } \\end{array} \\right] } _ { \\textbf { D } } \\underbrace { \\left[ 0 \\begin{array} { l l } { 1 } & { 0 } \\\\ { 0 } & { 1 } \\end{array} \\right] } _ { \\textbf { O } }", + "type": "interline_equation", + "image_path": "09b0b65bd89e123e7438424fe136eb0b9b11d952dedecd55f05ad3381597b1c7.jpg" + 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"score": 1.0, + "content": "rotation matrix, which is commonly known to be orthogonal.", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 538, + 505, + 551 + ], + "spans": [ + { + "bbox": [ + 106, + 539, + 401, + 551 + ], + "score": 1.0, + "content": "Assume that we perform the above decomposition on all of the blocks of", + "type": "text" + }, + { + "bbox": [ + 401, + 538, + 419, + 551 + ], + "score": 0.92, + "content": "\\mathbf { H } _ { 2 ^ { k } } ^ { \\prime }", + "type": "inline_equation" + }, + { + "bbox": [ + 419, + 539, + 505, + 551 + ], + "score": 1.0, + "content": "in parallel, therefore", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 549, + 267, + 563 + ], + "spans": [ + { + "bbox": [ + 105, + 549, + 151, + 561 + ], + "score": 1.0, + "content": "expressing", + "type": "text" + }, + { + "bbox": [ + 151, + 549, + 207, + 563 + ], + "score": 0.93, + "content": "\\mathbf { H } _ { 2 ^ { k } } ^ { \\prime } \\doteq \\mathbf { D } ^ { \\prime } \\mathbf { O } ^ { \\prime }", + "type": "inline_equation" + }, + { + "bbox": [ + 207, + 549, + 267, + 561 + ], + "score": 1.0, + "content": ". We now have", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 18 + }, + { + "type": "interline_equation", + "bbox": [ + 259, + 564, + 351, + 578 + ], + "lines": [ + { + "bbox": [ + 259, + 564, + 351, + 578 + ], + "spans": [ + { + "bbox": [ + 259, + 564, + 351, + 578 + ], + "score": 0.91, + "content": "{ \\bf H } _ { 2 ^ { k } } = { \\bf P } _ { 2 ^ { k } } ^ { * } { \\bf D } ^ { \\prime } { \\bf O } ^ { \\prime } { \\bf P } _ { 2 ^ { k } } .", + "type": "interline_equation", + "image_path": "89a88ce963a6e85b3c2811f0c74bd30d0508fbae2288d51f86f5118bce0d4745.jpg" + } + ] + } + ], + "index": 20, + "virtual_lines": [ + { + "bbox": [ + 259, + 564, + 351, + 578 + ], + "spans": [], + "index": 20 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 580, + 262, + 592 + ], + "lines": [ + { + "bbox": [ + 106, + 579, + 262, + 593 + ], + "spans": [ + { + "bbox": [ + 106, + 579, + 262, + 593 + ], + "score": 1.0, + "content": "By Lemma K.6, we can rewrite this as", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 21 + }, + { + "type": "interline_equation", + "bbox": [ + 258, + 593, + 352, + 608 + ], + "lines": [ + { + "bbox": [ + 258, + 593, + 352, + 608 + ], + "spans": [ + { + "bbox": [ + 258, + 593, + 352, + 608 + ], + "score": 0.92, + "content": "{ \\bf H } _ { 2 ^ { k } } = { \\bf D } ^ { \\prime \\prime } { \\bf P } _ { 2 ^ { k } } ^ { * } { \\bf O } ^ { \\prime } { \\bf P } _ { 2 ^ { k } } .", + "type": "interline_equation", + "image_path": "cf9a8c7c3b1979c8c7b0b2ac95af55918af29275013ff34084fa276a7eea87fc.jpg" + } + ] + } + ], + "index": 22, + "virtual_lines": [ + { + "bbox": [ + 258, + 593, + 352, + 608 + ], + "spans": [], + "index": 22 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 609, + 506, + 645 + ], + "lines": [ + { + "bbox": [ + 105, + 608, + 507, + 624 + ], + "spans": [ + { + "bbox": [ + 105, + 608, + 145, + 624 + ], + "score": 1.0, + "content": "Note that", + "type": "text" + }, + { + "bbox": [ + 146, + 609, + 191, + 623 + ], + "score": 0.93, + "content": "\\mathbf { P } _ { 2 ^ { k } } ^ { * } \\mathbf { O } ^ { \\prime } \\mathbf { P } _ { 2 ^ { k } }", + "type": "inline_equation" + }, + { + "bbox": [ + 191, + 608, + 507, + 624 + ], + "score": 1.0, + "content": "is the product of three orthogonal matrices, and thus orthogonal. Additionally,", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 621, + 506, + 636 + ], + "spans": [ + { + "bbox": [ + 105, + 621, + 185, + 636 + ], + "score": 1.0, + "content": "the construction of", + "type": "text" + }, + { + "bbox": [ + 185, + 623, + 202, + 634 + ], + "score": 0.89, + "content": "\\mathbf { P } _ { 2 ^ { k } }", + "type": "inline_equation" + }, + { + "bbox": [ + 202, + 621, + 254, + 636 + ], + "score": 1.0, + "content": "ensures that", + "type": "text" + }, + { + "bbox": [ + 254, + 622, + 300, + 635 + ], + "score": 0.93, + "content": "\\mathbf { P } _ { 2 ^ { k } } ^ { * } \\mathbf { O } ^ { \\prime } \\mathbf { P } _ { 2 ^ { k } }", + "type": "inline_equation" + }, + { + "bbox": [ + 300, + 621, + 414, + 636 + ], + "score": 1.0, + "content": "is butterfly factor.11 Hence,", + "type": "text" + }, + { + "bbox": [ + 414, + 622, + 432, + 634 + ], + "score": 0.89, + "content": "\\mathbf { H } _ { 2 ^ { k } }", + "type": "inline_equation" + }, + { + "bbox": [ + 433, + 621, + 506, + 636 + ], + "score": 1.0, + "content": "can be expressed", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 633, + 431, + 645 + ], + "spans": [ + { + "bbox": [ + 105, + 633, + 431, + 645 + ], + "score": 1.0, + "content": "as the product of a diagonal matrix and an orthogonal butterfly factor, as desired.", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 24 + }, + { + "type": "text", + "bbox": [ + 106, + 650, + 505, + 692 + ], + "lines": [ + { + "bbox": [ + 106, + 649, + 505, + 663 + ], + "spans": [ + { + "bbox": [ + 106, + 649, + 505, + 663 + ], + "score": 1.0, + "content": "Now, we show that this decomposition of butterfly factors implies Lemma K.7. By performing this", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 101, + 659, + 503, + 684 + ], + "spans": [ + { + "bbox": [ + 101, + 659, + 482, + 684 + ], + "score": 1.0, + "content": "decomposition in parallel on each butterfly factor, we conclude that any butterfly factor matrix", + "type": "text" + }, + { + "bbox": [ + 482, + 661, + 503, + 677 + ], + "score": 0.91, + "content": "\\mathbf { \\bar { H } } _ { 2 ^ { k } } ^ { ( n ) }", + "type": "inline_equation" + } + ], + "index": 27 + }, + { + "bbox": [ + 102, + 671, + 305, + 698 + ], + "spans": [ + { + "bbox": [ + 102, + 671, + 117, + 698 + ], + "score": 1.0, + "content": "of", + "type": "text" + }, + { + "bbox": [ + 118, + 679, + 128, + 689 + ], + "score": 0.4, + "content": "\\mathbf { H }", + "type": "inline_equation" + }, + { + "bbox": [ + 128, + 671, + 220, + 698 + ], + "score": 1.0, + "content": "can be decomposed as", + "type": "text" + }, + { + "bbox": [ + 240, + 672, + 297, + 695 + ], + "score": 1.0, + "content": ") = D2k O(n)2k .", + "type": "text" + }, + { + "bbox": [ + 293, + 675, + 305, + 689 + ], + "score": 1.0, + "content": "12", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 27 + } + ], + "page_idx": 38, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 106, + 697, + 504, + 732 + ], + "lines": [ + { + "bbox": [ + 114, + 695, + 506, + 713 + ], + "spans": [ + { + "bbox": [ + 114, + 695, + 181, + 713 + ], + "score": 1.0, + "content": "11Conjugation by", + "type": "text" + }, + { + "bbox": [ + 181, + 699, + 198, + 710 + ], + "score": 0.89, + "content": "\\mathbf { P } _ { 2 ^ { k } }", + "type": "inline_equation" + }, + { + "bbox": [ + 198, + 695, + 288, + 713 + ], + "score": 1.0, + "content": "is an isomorphism from", + "type": "text" + }, + { + "bbox": [ + 288, + 698, + 318, + 708 + ], + "score": 0.89, + "content": "2 ^ { k } \\times 2 ^ { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 318, + 695, + 506, + 713 + ], + "score": 1.0, + "content": "butterfly factors onto block diagonal matrices with", + "type": "text" + } + ] + }, + { + "bbox": [ + 106, + 704, + 507, + 734 + ], + "spans": [ + { + "bbox": [ + 106, + 708, + 126, + 720 + ], + "score": 0.81, + "content": "2 ^ { k - 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 126, + 704, + 129, + 734 + ], + "score": 1.0, + "content": ", N", + "type": "text" + }, + { + "bbox": [ + 129, + 710, + 151, + 720 + ], + "score": 0.83, + "content": "2 \\times 2", + "type": "inline_equation" + }, + { + "bbox": [ + 151, + 704, + 275, + 734 + ], + "score": 1.0, + "content": "blocks. Therefore, conjugation by at a block diagonal matrix compos", + "type": "text" + }, + { + "bbox": [ + 276, + 709, + 321, + 722 + ], + "score": 0.92, + "content": "\\mathbf { P } _ { 2 ^ { k } } ^ { - 1 } = \\mathbf { P } _ { 2 ^ { k } } ^ { * }", + "type": "inline_equation" + }, + { + "bbox": [ + 322, + 704, + 507, + 734 + ], + "score": 1.0, + "content": "maps a block diagonal matrix to a butterfly factor.nal matrices is, itself, orthogonal.", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 107, + 27, + 293, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 26, + 294, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 26, + 294, + 38 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2020", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 300, + 751, + 311, + 760 + ], + "lines": [ + { + "bbox": [ + 298, + 750, + 313, + 763 + ], + "spans": [ + { + "bbox": [ + 298, + 750, + 313, + 763 + ], + "score": 1.0, + "content": "39", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "image", + "bbox": [ + 108, + 88, + 546, + 228 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 108, + 88, + 546, + 228 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 108, + 88, + 546, + 228 + ], + "spans": [ + { + "bbox": [ + 108, + 88, + 546, + 228 + ], + "score": 0.957, + "type": "image", + "image_path": "a4607d3febb414e4ea1a9f3500d9dc6ca268a2509f0b7cd81b00a8b90f87d6e6.jpg" + } + ] + } + ], + "index": 1, + "virtual_lines": [ + { + "bbox": [ + 108, + 88, + 546, + 134.66666666666666 + ], + "spans": [], + "index": 0 + }, + { + "bbox": [ + 108, + 134.66666666666666, + 546, + 181.33333333333331 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 108, + 181.33333333333331, + 546, + 227.99999999999997 + ], + "spans": [], + "index": 2 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 225, + 245, + 385, + 258 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 223, + 243, + 385, + 260 + ], + "spans": [ + { + "bbox": [ + 223, + 243, + 370, + 260 + ], + "score": 1.0, + "content": "Figure 10: Block diagonalization of", + "type": "text" + }, + { + "bbox": [ + 371, + 246, + 385, + 257 + ], + "score": 0.82, + "content": "{ \\bf { H } } _ { 8 }", + "type": "inline_equation" + } + ], + "index": 3 + } + ], + "index": 3 + } + ], + "index": 2.0 + }, + { + "type": "text", + "bbox": [ + 106, + 276, + 506, + 321 + ], + "lines": [ + { + "bbox": [ + 106, + 276, + 506, + 288 + ], + "spans": [ + { + "bbox": [ + 106, + 276, + 213, + 288 + ], + "score": 1.0, + "content": "We argue that each of these", + "type": "text" + }, + { + "bbox": [ + 213, + 277, + 233, + 287 + ], + "score": 0.88, + "content": "2 \\times 2", + "type": "inline_equation" + }, + { + "bbox": [ + 234, + 276, + 506, + 288 + ], + "score": 1.0, + "content": "blocks can be decomposed into a diagonal matrix times an orthogonal", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 106, + 288, + 505, + 298 + ], + "spans": [ + { + "bbox": [ + 106, + 288, + 505, + 298 + ], + "score": 1.0, + "content": "matrix. Note that the butterfly factor matrices constructed in the proof of Lemma I.2 each have at", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 299, + 506, + 310 + ], + "spans": [ + { + "bbox": [ + 105, + 299, + 506, + 310 + ], + "score": 1.0, + "content": "most one non-zero entry per column. Hence, there are 4 cases to consider. Note that matrices with at", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 310, + 335, + 321 + ], + "spans": [ + { + "bbox": [ + 105, + 310, + 335, + 321 + ], + "score": 1.0, + "content": "most one non-zero entry are exhausted by Cases 1 and 2.", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 5.5, + "bbox_fs": [ + 105, + 276, + 506, + 321 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 124, + 324, + 254, + 370 + ], + "lines": [ + { + "bbox": [ + 124, + 324, + 254, + 370 + ], + "spans": [ + { + "bbox": [ + 124, + 324, + 254, + 370 + ], + "score": 0.91, + "content": "\\mathfrak { e } \\ 1 : \\left[ 0 \\atop 0 \\right] = \\underbrace { \\left[ 0 \\begin{array} { l l } { 0 } & { 0 } \\\\ { 0 } & { b } \\end{array} \\right] } _ { \\textbf { D } } \\underbrace { \\left[ 0 \\begin{array} { l l } { 1 } & { 0 } \\\\ { 0 } & { 1 } \\end{array} \\right] } _ { \\textbf { O } }", + "type": "interline_equation", + "image_path": 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"inline_equation" + }, + { + "bbox": [ + 250, + 527, + 506, + 540 + ], + "score": 1.0, + "content": "rotation matrix, which is commonly known to be orthogonal.", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 538, + 505, + 551 + ], + "spans": [ + { + "bbox": [ + 106, + 539, + 401, + 551 + ], + "score": 1.0, + "content": "Assume that we perform the above decomposition on all of the blocks of", + "type": "text" + }, + { + "bbox": [ + 401, + 538, + 419, + 551 + ], + "score": 0.92, + "content": "\\mathbf { H } _ { 2 ^ { k } } ^ { \\prime }", + "type": "inline_equation" + }, + { + "bbox": [ + 419, + 539, + 505, + 551 + ], + "score": 1.0, + "content": "in parallel, therefore", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 549, + 267, + 563 + ], + "spans": [ + { + "bbox": [ + 105, + 549, + 151, + 561 + ], + "score": 1.0, + "content": "expressing", + "type": "text" + }, + { + "bbox": [ + 151, + 549, + 207, + 563 + ], + "score": 0.93, + "content": "\\mathbf { H } _ { 2 ^ { k } } ^ { \\prime } \\doteq \\mathbf { D } ^ { \\prime } \\mathbf { O } ^ { \\prime }", + "type": "inline_equation" + }, + { + "bbox": [ + 207, + 549, + 267, + 561 + ], + "score": 1.0, + "content": ". We now have", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 18, + "bbox_fs": [ + 105, + 527, + 506, + 563 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 259, + 564, + 351, + 578 + ], + "lines": [ + { + "bbox": [ + 259, + 564, + 351, + 578 + ], + "spans": [ + { + "bbox": [ + 259, + 564, + 351, + 578 + ], + "score": 0.91, + "content": "{ \\bf H } _ { 2 ^ { k } } = { \\bf P } _ { 2 ^ { k } } ^ { * } { \\bf D } ^ { \\prime } { \\bf O } ^ { \\prime } { \\bf P } _ { 2 ^ { k } } .", + "type": "interline_equation", + "image_path": "89a88ce963a6e85b3c2811f0c74bd30d0508fbae2288d51f86f5118bce0d4745.jpg" + } + ] + } + ], + "index": 20, + "virtual_lines": [ + { + "bbox": [ + 259, + 564, + 351, + 578 + ], + "spans": [], + "index": 20 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 580, + 262, + 592 + ], + "lines": [ + { + "bbox": [ + 106, + 579, + 262, + 593 + ], + "spans": [ + { + "bbox": [ + 106, + 579, + 262, + 593 + ], + "score": 1.0, + "content": "By Lemma K.6, we can rewrite this as", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 21, + "bbox_fs": [ + 106, + 579, + 262, + 593 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 258, + 593, + 352, + 608 + ], + "lines": [ + { + "bbox": [ + 258, + 593, + 352, + 608 + ], + "spans": [ + { + "bbox": [ + 258, + 593, + 352, + 608 + ], + "score": 0.92, + "content": "{ \\bf H } _ { 2 ^ { k } } = { \\bf D } ^ { \\prime \\prime } { \\bf P } _ { 2 ^ { k } } ^ { * } { \\bf O } ^ { \\prime } { \\bf P } _ { 2 ^ { k } } .", + "type": "interline_equation", + "image_path": "cf9a8c7c3b1979c8c7b0b2ac95af55918af29275013ff34084fa276a7eea87fc.jpg" + } + ] + } + ], + "index": 22, + "virtual_lines": [ + { + "bbox": [ + 258, + 593, + 352, + 608 + ], + "spans": [], + "index": 22 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 609, + 506, + 645 + ], + "lines": [ + { + "bbox": [ + 105, + 608, + 507, + 624 + ], + "spans": [ + { + "bbox": [ + 105, + 608, + 145, + 624 + ], + "score": 1.0, + "content": "Note that", + "type": "text" + }, + { + "bbox": [ + 146, + 609, + 191, + 623 + ], + "score": 0.93, + "content": "\\mathbf { P } _ { 2 ^ { k } } ^ { * } \\mathbf { O } ^ { \\prime } \\mathbf { P } _ { 2 ^ { k } }", + "type": "inline_equation" + }, + { + "bbox": [ + 191, + 608, + 507, + 624 + ], + "score": 1.0, + "content": "is the product of three orthogonal matrices, and thus orthogonal. Additionally,", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 621, + 506, + 636 + ], + "spans": [ + { + "bbox": [ + 105, + 621, + 185, + 636 + ], + "score": 1.0, + "content": "the construction of", + "type": "text" + }, + { + "bbox": [ + 185, + 623, + 202, + 634 + ], + "score": 0.89, + "content": "\\mathbf { P } _ { 2 ^ { k } }", + "type": "inline_equation" + }, + { + "bbox": [ + 202, + 621, + 254, + 636 + ], + "score": 1.0, + "content": "ensures that", + "type": "text" + }, + { + "bbox": [ + 254, + 622, + 300, + 635 + ], + "score": 0.93, + "content": "\\mathbf { P } _ { 2 ^ { k } } ^ { * } \\mathbf { O } ^ { \\prime } \\mathbf { P } _ { 2 ^ { k } }", + "type": "inline_equation" + }, + { + "bbox": [ + 300, + 621, + 414, + 636 + ], + "score": 1.0, + "content": "is butterfly factor.11 Hence,", + "type": "text" + }, + { + "bbox": [ + 414, + 622, + 432, + 634 + ], + "score": 0.89, + "content": "\\mathbf { H } _ { 2 ^ { k } }", + "type": "inline_equation" + }, + { + "bbox": [ + 433, + 621, + 506, + 636 + ], + "score": 1.0, + "content": "can be expressed", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 633, + 431, + 645 + ], + "spans": [ + { + "bbox": [ + 105, + 633, + 431, + 645 + ], + "score": 1.0, + "content": "as the product of a diagonal matrix and an orthogonal butterfly factor, as desired.", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 24, + "bbox_fs": [ + 105, + 608, + 507, + 645 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 650, + 505, + 692 + ], + "lines": [ + { + "bbox": [ + 106, + 649, + 505, + 663 + ], + "spans": [ + { + "bbox": [ + 106, + 649, + 505, + 663 + ], + "score": 1.0, + "content": "Now, we show that this decomposition of butterfly factors implies Lemma K.7. By performing this", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 101, + 659, + 503, + 684 + ], + "spans": [ + { + "bbox": [ + 101, + 659, + 482, + 684 + ], + "score": 1.0, + "content": "decomposition in parallel on each butterfly factor, we conclude that any butterfly factor matrix", + "type": "text" + }, + { + "bbox": [ + 482, + 661, + 503, + 677 + ], + "score": 0.91, + "content": "\\mathbf { \\bar { H } } _ { 2 ^ { k } } ^ { ( n ) }", + "type": "inline_equation" + } + ], + "index": 27 + }, + { + "bbox": [ + 102, + 671, + 305, + 698 + ], + "spans": [ + { + "bbox": [ + 102, + 671, + 117, + 698 + ], + "score": 1.0, + "content": "of", + "type": "text" + }, + { + "bbox": [ + 118, + 679, + 128, + 689 + ], + "score": 0.4, + "content": "\\mathbf { H }", + "type": "inline_equation" + }, + { + "bbox": [ + 128, + 671, + 220, + 698 + ], + "score": 1.0, + "content": "can be decomposed as", + "type": "text" + }, + { + "bbox": [ + 240, + 672, + 297, + 695 + ], + "score": 1.0, + "content": ") = D2k O(n)2k .", + "type": "text" + }, + { + "bbox": [ + 293, + 675, + 305, + 689 + ], + "score": 1.0, + "content": "12", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 27, + "bbox_fs": [ + 101, + 649, + 505, + 698 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 106, + 82, + 505, + 127 + ], + "lines": [ + { + "bbox": [ + 106, + 83, + 505, + 95 + ], + "spans": [ + { + "bbox": [ + 106, + 83, + 282, + 95 + ], + "score": 1.0, + "content": "We complete the argument by induction on", + "type": "text" + }, + { + "bbox": [ + 282, + 85, + 289, + 92 + ], + "score": 0.65, + "content": "n", + "type": "inline_equation" + }, + { + "bbox": [ + 290, + 83, + 352, + 95 + ], + "score": 1.0, + "content": ". The base case", + "type": "text" + }, + { + "bbox": [ + 352, + 83, + 378, + 93 + ], + "score": 0.91, + "content": "n = 2", + "type": "inline_equation" + }, + { + "bbox": [ + 378, + 83, + 505, + 95 + ], + "score": 1.0, + "content": "holds by the observation about", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 93, + 505, + 106 + ], + "spans": [ + { + "bbox": [ + 105, + 93, + 411, + 106 + ], + "score": 1.0, + "content": "butterfly factor matrices above. Assume that any horizontal step matrix of size", + "type": "text" + }, + { + "bbox": [ + 412, + 94, + 436, + 106 + ], + "score": 0.91, + "content": "{ \\frac { n } { 2 } } \\times { \\frac { n } { 2 } }", + "type": "inline_equation" + }, + { + "bbox": [ + 437, + 93, + 505, + 106 + ], + "score": 1.0, + "content": "can be expressed", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 105, + 505, + 117 + ], + "spans": [ + { + "bbox": [ + 105, + 105, + 416, + 117 + ], + "score": 1.0, + "content": "as a diagonal matrix times an orthogonal butterfly matrix. Now, consider the", + "type": "text" + }, + { + "bbox": [ + 416, + 106, + 442, + 115 + ], + "score": 0.89, + "content": "n \\times n", + "type": "inline_equation" + }, + { + "bbox": [ + 442, + 105, + 505, + 117 + ], + "score": 1.0, + "content": "horizontal step", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 115, + 315, + 128 + ], + "spans": [ + { + "bbox": [ + 105, + 115, + 135, + 128 + ], + "score": 1.0, + "content": "matrix", + "type": "text" + }, + { + "bbox": [ + 135, + 115, + 145, + 125 + ], + "score": 0.46, + "content": "\\mathbf { H }", + "type": "inline_equation" + }, + { + "bbox": [ + 146, + 115, + 222, + 128 + ], + "score": 1.0, + "content": ". From Lemma I.2,", + "type": "text" + }, + { + "bbox": [ + 222, + 115, + 233, + 125 + ], + "score": 0.44, + "content": "\\mathbf { H }", + "type": "inline_equation" + }, + { + "bbox": [ + 233, + 115, + 315, + 128 + ], + "score": 1.0, + "content": "can be expressed as", + "type": "text" + } + ], + "index": 3 + } + ], + "index": 1.5 + }, + { + "type": "interline_equation", + "bbox": [ + 256, + 132, + 354, + 160 + ], + "lines": [ + { + "bbox": [ + 256, + 132, + 354, + 160 + ], + "spans": [ + { + "bbox": [ + 256, + 132, + 354, + 160 + ], + "score": 0.94, + "content": "\\mathbf { H } = \\mathbf { B } _ { n } ^ { ( n ) } \\left[ \\begin{array} { c c } { \\mathbf { H } _ { 1 } } & { 0 } \\\\ { 0 } & { \\mathbf { H } _ { 2 } } \\end{array} \\right] ,", + "type": "interline_equation", + "image_path": "f416386e5bb8c0718c80ca863428e5a987c30b37d2bf57e1c04266c4a695868d.jpg" + } + ] + } + ], + "index": 4, + "virtual_lines": [ + { + "bbox": [ + 256, + 132, + 354, + 160 + ], + "spans": [], + "index": 4 + } + ] + }, + { + "type": "text", + "bbox": [ + 105, + 165, + 428, + 178 + ], + "lines": [ + { + "bbox": [ + 105, + 164, + 429, + 181 + ], + "spans": [ + { + "bbox": [ + 105, + 164, + 133, + 181 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 133, + 166, + 165, + 177 + ], + "score": 0.9, + "content": "\\mathbf { H } _ { 1 } , \\mathbf { H } _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 166, + 164, + 182, + 181 + ], + "score": 1.0, + "content": "are", + "type": "text" + }, + { + "bbox": [ + 182, + 166, + 209, + 178 + ], + "score": 0.93, + "content": "{ \\frac { n } { 2 } } \\times { \\frac { n } { 2 } }", + "type": "inline_equation" + }, + { + "bbox": [ + 209, + 164, + 429, + 181 + ], + "score": 1.0, + "content": "horizontal step matrices. By our inductive hypothesis,", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 5 + }, + { + "type": "interline_equation", + "bbox": [ + 250, + 183, + 360, + 212 + ], + "lines": [ + { + "bbox": [ + 250, + 183, + 360, + 212 + ], + "spans": [ + { + "bbox": [ + 250, + 183, + 360, + 212 + ], + "score": 0.94, + "content": "\\mathbf { H } = \\mathbf { B } _ { n } ^ { ( n ) } \\mathbf { D } _ { 1 } \\left[ \\begin{array} { c c } { \\mathbf { O } _ { 1 } } & { 0 } \\\\ { 0 } & { \\mathbf { O } _ { 2 } } \\end{array} \\right] ,", + "type": "interline_equation", + "image_path": "136728a2ff3ccef05087ea5f6c333207cca10b3291a4750d2d1d162a6f2c4119.jpg" + } + ] + } + ], + "index": 6.5, + "virtual_lines": [ + { + "bbox": [ + 250, + 183, + 360, + 197.5 + ], + "spans": [], + "index": 6 + }, + { + "bbox": [ + 250, + 197.5, + 360, + 212.0 + ], + "spans": [], + "index": 7 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 219, + 506, + 247 + ], + "lines": [ + { + "bbox": [ + 103, + 216, + 509, + 237 + ], + "spans": [ + { + "bbox": [ + 103, + 216, + 133, + 237 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 133, + 221, + 148, + 232 + ], + "score": 0.88, + "content": "\\mathbf { D } _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 148, + 216, + 212, + 237 + ], + "score": 1.0, + "content": "is diagonal and", + "type": "text" + }, + { + "bbox": [ + 213, + 221, + 244, + 232 + ], + "score": 0.83, + "content": "\\mathbf { O } _ { 1 } , \\mathbf { O } _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 244, + 216, + 261, + 237 + ], + "score": 1.0, + "content": "are", + "type": "text" + }, + { + "bbox": [ + 261, + 221, + 288, + 233 + ], + "score": 0.94, + "content": "{ \\frac { n } { 2 } } \\times { \\frac { n } { 2 } }", + "type": "inline_equation" + }, + { + "bbox": [ + 289, + 216, + 336, + 237 + ], + "score": 1.0, + "content": "matrices in", + "type": "text" + }, + { + "bbox": [ + 337, + 218, + 345, + 230 + ], + "score": 0.83, + "content": "\\widetilde { B }", + "type": "inline_equation" + }, + { + "bbox": [ + 345, + 216, + 389, + 237 + ], + "score": 1.0, + "content": ". However,", + "type": "text" + }, + { + "bbox": [ + 390, + 218, + 424, + 232 + ], + "score": 0.93, + "content": "{ \\bf B } _ { n } ^ { ( n ) } { \\bf D } _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 425, + 216, + 509, + 237 + ], + "score": 1.0, + "content": "is a butterfly factor,", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 104, + 230, + 329, + 250 + ], + "spans": [ + { + "bbox": [ + 104, + 230, + 242, + 250 + ], + "score": 1.0, + "content": "and therefore can be expressed as", + "type": "text" + }, + { + "bbox": [ + 242, + 233, + 278, + 246 + ], + "score": 0.92, + "content": "\\mathbf { D } _ { n } \\mathbf { O } _ { n } ^ { ( n ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 278, + 230, + 329, + 250 + ], + "score": 1.0, + "content": ". Therefore,", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 8.5 + }, + { + "type": "interline_equation", + "bbox": [ + 232, + 252, + 378, + 280 + ], + "lines": [ + { + "bbox": [ + 232, + 252, + 378, + 280 + ], + "spans": [ + { + "bbox": [ + 232, + 252, + 378, + 280 + ], + "score": 0.94, + "content": "{ \\bf H } = { \\bf D } _ { n } { \\bf O } _ { n } ^ { ( n ) } \\left[ \\begin{array} { c c } { { \\bf O } _ { 1 } } & { 0 } \\\\ { 0 } & { { \\bf O } _ { 2 } } \\end{array} \\right] = { \\bf D } _ { n } { \\bf O } ,", + "type": "interline_equation", + "image_path": "df417c49b7d774cb2d559d0f61e552401521f85e7e356657ff6094a2d734c2ed.jpg" + } + ] + } + ], + "index": 10, + "virtual_lines": [ + { + "bbox": [ + 232, + 252, + 378, + 280 + ], + "spans": [], + "index": 10 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 287, + 202, + 300 + ], + "lines": [ + { + "bbox": [ + 105, + 285, + 203, + 302 + ], + "spans": [ + { + "bbox": [ + 105, + 285, + 126, + 302 + ], + "score": 1.0, + "content": "with", + "type": "text" + }, + { + "bbox": [ + 127, + 286, + 155, + 299 + ], + "score": 0.9, + "content": "\\mathbf { O } \\in \\widetilde { B }", + "type": "inline_equation" + }, + { + "bbox": [ + 156, + 285, + 203, + 302 + ], + "score": 1.0, + "content": ", as desired.", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 11 + }, + { + "type": "text", + "bbox": [ + 107, + 314, + 504, + 337 + ], + "lines": [ + { + "bbox": [ + 105, + 314, + 506, + 327 + ], + "spans": [ + { + "bbox": [ + 105, + 314, + 169, + 327 + ], + "score": 1.0, + "content": "Just as with the", + "type": "text" + }, + { + "bbox": [ + 169, + 315, + 189, + 325 + ], + "score": 0.89, + "content": "B B ^ { * }", + "type": "inline_equation" + }, + { + "bbox": [ + 189, + 314, + 506, + 327 + ], + "score": 1.0, + "content": "hierarchy, the decomposition of vertical step matrices falls out as an immediate", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 325, + 286, + 338 + ], + "spans": [ + { + "bbox": [ + 106, + 325, + 286, + 338 + ], + "score": 1.0, + "content": "corollary to the horizontal step matrix proof.", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 12.5 + }, + { + "type": "text", + "bbox": [ + 106, + 340, + 505, + 365 + ], + "lines": [ + { + "bbox": [ + 106, + 339, + 504, + 354 + ], + "spans": [ + { + "bbox": [ + 106, + 339, + 188, + 354 + ], + "score": 1.0, + "content": "Corollary K.8. Let", + "type": "text" + }, + { + "bbox": [ + 189, + 341, + 198, + 351 + ], + "score": 0.47, + "content": "\\mathbf { V }", + "type": "inline_equation" + }, + { + "bbox": [ + 199, + 339, + 401, + 354 + ], + "score": 1.0, + "content": "be a vertical step matrix. Then we can decompose", + "type": "text" + }, + { + "bbox": [ + 402, + 341, + 447, + 351 + ], + "score": 0.9, + "content": "\\mathbf { V } = \\mathbf { O } ^ { * } \\mathbf { D }", + "type": "inline_equation" + }, + { + "bbox": [ + 447, + 339, + 477, + 354 + ], + "score": 1.0, + "content": ", where", + "type": "text" + }, + { + "bbox": [ + 477, + 341, + 487, + 351 + ], + "score": 0.61, + "content": "\\mathbf { D }", + "type": "inline_equation" + }, + { + "bbox": [ + 488, + 339, + 497, + 354 + ], + "score": 1.0, + "content": "is", + "type": "text" + }, + { + "bbox": [ + 497, + 343, + 504, + 351 + ], + "score": 0.32, + "content": "a", + "type": "inline_equation" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 351, + 232, + 365 + ], + "spans": [ + { + "bbox": [ + 105, + 352, + 190, + 365 + ], + "score": 1.0, + "content": "diagonal matrix and", + "type": "text" + }, + { + "bbox": [ + 191, + 351, + 228, + 363 + ], + "score": 0.92, + "content": "\\mathbf { O } ^ { * } \\in \\widetilde { B } ^ { * }", + "type": "inline_equation" + }, + { + "bbox": [ + 228, + 352, + 232, + 365 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 14.5 + }, + { + "type": "title", + "bbox": [ + 107, + 377, + 223, + 389 + ], + "lines": [ + { + "bbox": [ + 105, + 376, + 224, + 390 + ], + "spans": [ + { + "bbox": [ + 105, + 376, + 224, + 390 + ], + "score": 1.0, + "content": "K.3.3 SPARSE MATRICES", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 16 + }, + { + "type": "text", + "bbox": [ + 106, + 397, + 506, + 430 + ], + "lines": [ + { + "bbox": [ + 105, + 396, + 506, + 410 + ], + "spans": [ + { + "bbox": [ + 105, + 396, + 506, + 410 + ], + "score": 1.0, + "content": "Now that we have argued about the decomposition of permutation and step matrices in the OBB", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 408, + 504, + 420 + ], + "spans": [ + { + "bbox": [ + 105, + 408, + 496, + 420 + ], + "score": 1.0, + "content": "hierarchy, we can leverage the construction from Lemma I.1 to argue about matrices with at most", + "type": "text" + }, + { + "bbox": [ + 497, + 410, + 504, + 418 + ], + "score": 0.7, + "content": "n", + "type": "inline_equation" + } + ], + "index": 18 + }, + { + "bbox": [ + 104, + 417, + 133, + 432 + ], + "spans": [ + { + "bbox": [ + 104, + 417, + 133, + 432 + ], + "score": 1.0, + "content": "NNZ.", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 18 + }, + { + "type": "text", + "bbox": [ + 106, + 433, + 441, + 446 + ], + "lines": [ + { + "bbox": [ + 106, + 433, + 442, + 447 + ], + "spans": [ + { + "bbox": [ + 106, + 433, + 221, + 447 + ], + "score": 1.0, + "content": "Corollary K.9. Let S be an", + "type": "text" + }, + { + "bbox": [ + 221, + 436, + 247, + 445 + ], + "score": 0.88, + "content": "n \\times n", + "type": "inline_equation" + }, + { + "bbox": [ + 247, + 433, + 384, + 447 + ], + "score": 1.0, + "content": "matrix with at most n NNZ. Then,", + "type": "text" + }, + { + "bbox": [ + 385, + 433, + 438, + 446 + ], + "score": 0.89, + "content": "\\mathbf { S } \\in ( \\mathcal { O } B B ) ^ { 4 }", + "type": "inline_equation" + }, + { + "bbox": [ + 439, + 433, + 442, + 447 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 20 + }, + { + "type": "text", + "bbox": [ + 105, + 459, + 504, + 483 + ], + "lines": [ + { + "bbox": [ + 105, + 458, + 506, + 474 + ], + "spans": [ + { + "bbox": [ + 105, + 458, + 506, + 474 + ], + "score": 1.0, + "content": "Proof. We use the construction from Lemma I.1, along with Lemma K.7 and Corollary K.8, to", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 471, + 161, + 484 + ], + "spans": [ + { + "bbox": [ + 105, + 471, + 161, + 484 + ], + "score": 1.0, + "content": "express S as:", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 21.5 + }, + { + "type": "interline_equation", + "bbox": [ + 223, + 488, + 387, + 521 + ], + "lines": [ + { + "bbox": [ + 223, + 488, + 387, + 521 + ], + "spans": [ + { + "bbox": [ + 223, + 488, + 387, + 521 + ], + "score": 0.93, + "content": "\\mathbf { S } = \\underbrace { \\mathbf { O } _ { 1 } \\mathbf { O } _ { 1 } ^ { \\prime } } _ { \\mathbf { P } _ { 1 } } \\underbrace { \\mathbf { D } _ { 2 } \\mathbf { O } _ { 2 } } _ { \\mathbf { H } } \\underbrace { \\mathbf { O } _ { 3 } \\mathbf { O } _ { 3 } ^ { \\prime } } _ { \\mathbf { P } _ { 2 } } \\underbrace { \\mathbf { O } _ { 4 } ^ { \\prime } \\mathbf { D } _ { 4 } } _ { \\mathbf { V } } \\underbrace { \\mathbf { O } _ { 5 } \\mathbf { O } _ { 5 } ^ { \\prime } } _ { \\mathbf { P } _ { 3 } } ,", + "type": "interline_equation", + "image_path": "f4ddc3bdf875d54dacd9d6aaa9113a9957824b1947c2e065861b9fefe4c697cf.jpg" + } + ] + } + ], + "index": 23.5, + "virtual_lines": [ + { + "bbox": [ + 223, + 488, + 387, + 504.5 + ], + "spans": [], + "index": 23 + }, + { + "bbox": [ + 223, + 504.5, + 387, + 521.0 + ], + "spans": [], + "index": 24 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 527, + 506, + 567 + ], + "lines": [ + { + "bbox": [ + 105, + 527, + 507, + 543 + ], + "spans": [ + { + "bbox": [ + 105, + 527, + 145, + 543 + ], + "score": 1.0, + "content": "with each", + "type": "text" + }, + { + "bbox": [ + 145, + 527, + 178, + 541 + ], + "score": 0.92, + "content": "\\mathbf { O } _ { i } \\in \\widetilde { B }", + "type": "inline_equation" + }, + { + "bbox": [ + 178, + 527, + 200, + 543 + ], + "score": 1.0, + "content": ", each", + "type": "text" + }, + { + "bbox": [ + 200, + 527, + 238, + 542 + ], + "score": 0.93, + "content": "\\mathbf { O } _ { j } ^ { \\prime } \\in \\widetilde { B } ^ { * }", + "type": "inline_equation" + }, + { + "bbox": [ + 238, + 527, + 277, + 543 + ], + "score": 1.0, + "content": ", and each", + "type": "text" + }, + { + "bbox": [ + 277, + 529, + 292, + 541 + ], + "score": 0.89, + "content": "\\mathbf { D } _ { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 292, + 527, + 355, + 543 + ], + "score": 1.0, + "content": "diagonal. Since", + "type": "text" + }, + { + "bbox": [ + 355, + 529, + 369, + 540 + ], + "score": 0.89, + "content": "\\mathbf { P } _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 369, + 527, + 507, + 543 + ], + "score": 1.0, + "content": "is a permutation, by Corollary K.5,", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 104, + 541, + 507, + 559 + ], + "spans": [ + { + "bbox": [ + 104, + 541, + 180, + 559 + ], + "score": 1.0, + "content": "we can write it as", + "type": "text" + }, + { + "bbox": [ + 181, + 542, + 211, + 556 + ], + "score": 0.92, + "content": "\\tilde { \\bf { O } } _ { 3 } ^ { ' } \\tilde { \\bf { O } } _ { 3 }", + "type": "inline_equation" + }, + { + "bbox": [ + 211, + 541, + 250, + 559 + ], + "score": 1.0, + "content": "for some", + "type": "text" + }, + { + "bbox": [ + 251, + 542, + 292, + 556 + ], + "score": 0.93, + "content": "\\tilde { \\mathbf { O } _ { 3 } } ^ { \\prime } \\in \\widetilde { B } ^ { * }", + "type": "inline_equation" + }, + { + "bbox": [ + 292, + 541, + 311, + 559 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 311, + 542, + 345, + 556 + ], + "score": 0.93, + "content": "\\tilde { \\mathbf { O } } _ { 3 } \\in \\tilde { B }", + "type": "inline_equation" + }, + { + "bbox": [ + 345, + 541, + 441, + 559 + ], + "score": 1.0, + "content": ". Moreover, noting that", + "type": "text" + }, + { + "bbox": [ + 441, + 545, + 455, + 557 + ], + "score": 0.91, + "content": "\\mathbf { O } _ { 1 } ^ { \\prime }", + "type": "inline_equation" + }, + { + "bbox": [ + 456, + 541, + 474, + 559 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 474, + 545, + 489, + 556 + ], + "score": 0.89, + "content": "\\mathbf { O } _ { 5 }", + "type": "inline_equation" + }, + { + "bbox": [ + 489, + 541, + 507, + 559 + ], + "score": 1.0, + "content": "are", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 555, + 354, + 569 + ], + "spans": [ + { + "bbox": [ + 105, + 555, + 331, + 569 + ], + "score": 1.0, + "content": "permutations, we make use of Lemma K.6 to re-express", + "type": "text" + }, + { + "bbox": [ + 332, + 557, + 340, + 566 + ], + "score": 0.42, + "content": "\\mathbf { S }", + "type": "inline_equation" + }, + { + "bbox": [ + 340, + 555, + 354, + 569 + ], + "score": 1.0, + "content": "as:", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 26 + }, + { + "type": "interline_equation", + "bbox": [ + 222, + 572, + 388, + 608 + ], + "lines": [ + { + "bbox": [ + 222, + 572, + 388, + 608 + ], + "spans": [ + { + "bbox": [ + 222, + 572, + 388, + 608 + ], + "score": 0.93, + "content": "\\mathbf { S } = \\underbrace { \\mathbf { O } _ { 1 } \\mathbf { D } _ { 2 } ^ { \\prime } \\mathbf { O } _ { 1 } ^ { \\prime } } _ { \\mathbf { M } _ { 1 } } \\underbrace { \\mathbf { O } _ { 2 } \\tilde { \\mathbf { O } _ { 3 } } } _ { \\mathbf { M } _ { 2 } } \\underbrace { \\mathbf { \\tilde { O } _ { 3 } O _ { 4 } ^ { \\prime } } } _ { \\mathbf { M } _ { 3 } } \\underbrace { \\mathbf { O } _ { 5 } \\mathbf { D } _ { 4 } ^ { \\prime } \\mathbf { O } _ { 5 } ^ { \\prime } } _ { \\mathbf { M } _ { 4 } } .", + "type": "interline_equation", + "image_path": "798b4deb3104d6592027cdd284c5e253530e2eea6db0a3a2033a500f4a83478f.jpg" + } + ] + } + ], + "index": 28.5, + "virtual_lines": [ + { + "bbox": [ + 222, + 572, + 388, + 590.0 + ], + "spans": [], + "index": 28 + }, + { + "bbox": [ + 222, + 590.0, + 388, + 608.0 + ], + "spans": [], + "index": 29 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 614, + 353, + 627 + ], + "lines": [ + { + "bbox": [ + 106, + 614, + 353, + 628 + ], + "spans": [ + { + "bbox": [ + 106, + 614, + 166, + 628 + ], + "score": 1.0, + "content": "Note that each", + "type": "text" + }, + { + "bbox": [ + 167, + 615, + 217, + 626 + ], + "score": 0.89, + "content": "\\mathbf { M } _ { \\ell } \\in \\mathcal { O B B }", + "type": "inline_equation" + }, + { + "bbox": [ + 217, + 614, + 251, + 628 + ], + "score": 1.0, + "content": ". Hence,", + "type": "text" + }, + { + "bbox": [ + 251, + 614, + 305, + 627 + ], + "score": 0.93, + "content": "\\mathbf { S } \\in ( \\mathcal { O } B B ) ^ { 4 }", + "type": "inline_equation" + }, + { + "bbox": [ + 305, + 614, + 353, + 628 + ], + "score": 1.0, + "content": ", as desired.", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 30 + }, + { + "type": "text", + "bbox": [ + 107, + 640, + 506, + 675 + ], + "lines": [ + { + "bbox": [ + 106, + 641, + 505, + 654 + ], + "spans": [ + { + "bbox": [ + 106, + 641, + 505, + 654 + ], + "score": 1.0, + "content": "Just as in Appendix I, we would like to extend this orthogonal-based construction to capture matrices", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 652, + 505, + 665 + ], + "spans": [ + { + "bbox": [ + 105, + 652, + 505, + 665 + ], + "score": 1.0, + "content": "of general sparsity. To accomplish this, we introduce an addition closure lemma analogous to", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 661, + 257, + 676 + ], + "spans": [ + { + "bbox": [ + 105, + 661, + 190, + 676 + ], + "score": 1.0, + "content": "Lemma K.10 for the", + "type": "text" + }, + { + "bbox": [ + 191, + 663, + 214, + 674 + ], + "score": 0.46, + "content": "\\ O B B", + "type": "inline_equation" + }, + { + "bbox": [ + 214, + 661, + 257, + 676 + ], + "score": 1.0, + "content": "hierarchy.", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 32 + }, + { + "type": "text", + "bbox": [ + 107, + 677, + 478, + 692 + ], + "lines": [ + { + "bbox": [ + 103, + 673, + 477, + 695 + ], + "spans": [ + { + "bbox": [ + 103, + 673, + 184, + 695 + ], + "score": 1.0, + "content": "Lemma K.10. Let", + "type": "text" + }, + { + "bbox": [ + 185, + 678, + 237, + 690 + ], + "score": 0.91, + "content": "\\mathbf { A } _ { 1 } , \\ldots , \\mathbf { A } _ { m }", + "type": "inline_equation" + }, + { + "bbox": [ + 237, + 673, + 250, + 695 + ], + "score": 1.0, + "content": "be", + "type": "text" + }, + { + "bbox": [ + 250, + 678, + 275, + 689 + ], + "score": 0.88, + "content": "k \\times k", + "type": "inline_equation" + }, + { + "bbox": [ + 275, + 673, + 324, + 695 + ], + "score": 1.0, + "content": "matrices in", + "type": "text" + }, + { + "bbox": [ + 324, + 678, + 360, + 691 + ], + "score": 0.78, + "content": "( O B B ) _ { e } ^ { w }", + "type": "inline_equation" + }, + { + "bbox": [ + 361, + 673, + 382, + 695 + ], + "score": 1.0, + "content": "then", + "type": "text" + }, + { + "bbox": [ + 382, + 677, + 477, + 691 + ], + "score": 0.52, + "content": "\\begin{array} { r } { \\sum _ { i = 1 } ^ { m } \\mathbf { A } _ { i } \\in ( \\mathcal { O } B \\mathcal { B } ) _ { 4 e } ^ { m w } . } \\end{array}", + "type": "inline_equation" + } + ], + "index": 34 + } + ], + "index": 34 + }, + { + "type": "text", + "bbox": [ + 106, + 699, + 454, + 712 + ], + "lines": [ + { + "bbox": [ + 105, + 697, + 456, + 714 + ], + "spans": [ + { + "bbox": [ + 105, + 697, + 456, + 714 + ], + "score": 1.0, + "content": "With Lemma K.10, we arrive at the following Corollary on general orthogonal sparsity.", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 35 + }, + { + "type": "text", + "bbox": [ + 106, + 715, + 430, + 733 + ], + "lines": [ + { + "bbox": [ + 104, + 713, + 430, + 735 + ], + "spans": [ + { + "bbox": [ + 104, + 713, + 226, + 735 + ], + "score": 1.0, + "content": "Corollary K.11. Let S be an", + "type": "text" + }, + { + "bbox": [ + 227, + 722, + 252, + 731 + ], + "score": 0.86, + "content": "n \\times n", + "type": "inline_equation" + }, + { + "bbox": [ + 252, + 713, + 357, + 735 + ], + "score": 1.0, + "content": "matrix with s NNZ. 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The base case", + "type": "text" + }, + { + "bbox": [ + 352, + 83, + 378, + 93 + ], + "score": 0.91, + "content": "n = 2", + "type": "inline_equation" + }, + { + "bbox": [ + 378, + 83, + 505, + 95 + ], + "score": 1.0, + "content": "holds by the observation about", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 93, + 505, + 106 + ], + "spans": [ + { + "bbox": [ + 105, + 93, + 411, + 106 + ], + "score": 1.0, + "content": "butterfly factor matrices above. Assume that any horizontal step matrix of size", + "type": "text" + }, + { + "bbox": [ + 412, + 94, + 436, + 106 + ], + "score": 0.91, + "content": "{ \\frac { n } { 2 } } \\times { \\frac { n } { 2 } }", + "type": "inline_equation" + }, + { + "bbox": [ + 437, + 93, + 505, + 106 + ], + "score": 1.0, + "content": "can be expressed", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 105, + 505, + 117 + ], + "spans": [ + { + "bbox": [ + 105, + 105, + 416, + 117 + ], + "score": 1.0, + "content": "as a diagonal matrix times an orthogonal butterfly matrix. 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By our inductive hypothesis,", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 5, + "bbox_fs": [ + 105, + 164, + 429, + 181 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 250, + 183, + 360, + 212 + ], + "lines": [ + { + "bbox": [ + 250, + 183, + 360, + 212 + ], + "spans": [ + { + "bbox": [ + 250, + 183, + 360, + 212 + ], + "score": 0.94, + "content": "\\mathbf { H } = \\mathbf { B } _ { n } ^ { ( n ) } \\mathbf { D } _ { 1 } \\left[ \\begin{array} { c c } { \\mathbf { O } _ { 1 } } & { 0 } \\\\ { 0 } & { \\mathbf { O } _ { 2 } } \\end{array} \\right] ,", + "type": "interline_equation", + "image_path": "136728a2ff3ccef05087ea5f6c333207cca10b3291a4750d2d1d162a6f2c4119.jpg" + } + ] + } + ], + "index": 6.5, + "virtual_lines": [ + { + "bbox": [ + 250, + 183, + 360, + 197.5 + ], + "spans": [], + "index": 6 + }, + { + "bbox": [ + 250, + 197.5, + 360, + 212.0 + ], + "spans": [], + "index": 7 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 219, + 506, + 247 + ], + "lines": [ + { + "bbox": [ + 103, + 216, + 509, + 237 + ], + "spans": [ + { + "bbox": [ + 103, + 216, + 133, + 237 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 133, + 221, + 148, + 232 + ], + "score": 0.88, + "content": "\\mathbf { D } _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 148, + 216, + 212, + 237 + ], + "score": 1.0, + "content": "is diagonal and", + "type": "text" + }, + { + "bbox": [ + 213, + 221, + 244, + 232 + ], + "score": 0.83, + "content": "\\mathbf { O } _ { 1 } , \\mathbf { O } _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 244, + 216, + 261, + 237 + ], + "score": 1.0, + "content": "are", + "type": "text" + }, + { + "bbox": [ + 261, + 221, + 288, + 233 + ], + "score": 0.94, + "content": "{ \\frac { n } { 2 } } \\times { \\frac { n } { 2 } }", + "type": "inline_equation" + }, + { + "bbox": [ + 289, + 216, + 336, + 237 + ], + "score": 1.0, + "content": "matrices in", + "type": "text" + }, + { + "bbox": [ + 337, + 218, + 345, + 230 + ], + "score": 0.83, + "content": "\\widetilde { B }", + "type": "inline_equation" + }, + { + "bbox": [ + 345, + 216, + 389, + 237 + ], + "score": 1.0, + "content": ". However,", + "type": "text" + }, + { + "bbox": [ + 390, + 218, + 424, + 232 + ], + "score": 0.93, + "content": "{ \\bf B } _ { n } ^ { ( n ) } { \\bf D } _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 425, + 216, + 509, + 237 + ], + "score": 1.0, + "content": "is a butterfly factor,", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 104, + 230, + 329, + 250 + ], + "spans": [ + { + "bbox": [ + 104, + 230, + 242, + 250 + ], + "score": 1.0, + "content": "and therefore can be expressed as", + "type": "text" + }, + { + "bbox": [ + 242, + 233, + 278, + 246 + ], + "score": 0.92, + "content": "\\mathbf { D } _ { n } \\mathbf { O } _ { n } ^ { ( n ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 278, + 230, + 329, + 250 + ], + "score": 1.0, + "content": ". Therefore,", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 8.5, + "bbox_fs": [ + 103, + 216, + 509, + 250 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 232, + 252, + 378, + 280 + ], + "lines": [ + { + "bbox": [ + 232, + 252, + 378, + 280 + ], + "spans": [ + { + "bbox": [ + 232, + 252, + 378, + 280 + ], + "score": 0.94, + "content": "{ \\bf H } = { \\bf D } _ { n } { \\bf O } _ { n } ^ { ( n ) } \\left[ \\begin{array} { c c } { { \\bf O } _ { 1 } } & { 0 } \\\\ { 0 } & { { \\bf O } _ { 2 } } \\end{array} \\right] = { \\bf D } _ { n } { \\bf O } ,", + "type": "interline_equation", + "image_path": "df417c49b7d774cb2d559d0f61e552401521f85e7e356657ff6094a2d734c2ed.jpg" + } + ] + } + ], + "index": 10, + "virtual_lines": [ + { + "bbox": [ + 232, + 252, + 378, + 280 + ], + "spans": [], + "index": 10 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 287, + 202, + 300 + ], + "lines": [ + { + "bbox": [ + 105, + 285, + 203, + 302 + ], + "spans": [ + { + "bbox": [ + 105, + 285, + 126, + 302 + ], + "score": 1.0, + "content": "with", + "type": "text" + }, + { + "bbox": [ + 127, + 286, + 155, + 299 + ], + "score": 0.9, + "content": "\\mathbf { O } \\in \\widetilde { B }", + "type": "inline_equation" + }, + { + "bbox": [ + 156, + 285, + 203, + 302 + ], + "score": 1.0, + "content": ", as desired.", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 11, + "bbox_fs": [ + 105, + 285, + 203, + 302 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 314, + 504, + 337 + ], + "lines": [ + { + "bbox": [ + 105, + 314, + 506, + 327 + ], + "spans": [ + { + "bbox": [ + 105, + 314, + 169, + 327 + ], + "score": 1.0, + "content": "Just as with the", + "type": "text" + }, + { + "bbox": [ + 169, + 315, + 189, + 325 + ], + "score": 0.89, + "content": "B B ^ { * }", + "type": "inline_equation" + }, + { + "bbox": [ + 189, + 314, + 506, + 327 + ], + "score": 1.0, + "content": "hierarchy, the decomposition of vertical step matrices falls out as an immediate", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 325, + 286, + 338 + ], + "spans": [ + { + "bbox": [ + 106, + 325, + 286, + 338 + ], + "score": 1.0, + "content": "corollary to the horizontal step matrix proof.", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 12.5, + "bbox_fs": [ + 105, + 314, + 506, + 338 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 340, + 505, + 365 + ], + "lines": [ + { + "bbox": [ + 106, + 339, + 504, + 354 + ], + "spans": [ + { + "bbox": [ + 106, + 339, + 188, + 354 + ], + "score": 1.0, + "content": "Corollary K.8. Let", + "type": "text" + }, + { + "bbox": [ + 189, + 341, + 198, + 351 + ], + "score": 0.47, + "content": "\\mathbf { V }", + "type": "inline_equation" + }, + { + "bbox": [ + 199, + 339, + 401, + 354 + ], + "score": 1.0, + "content": "be a vertical step matrix. Then we can decompose", + "type": "text" + }, + { + "bbox": [ + 402, + 341, + 447, + 351 + ], + "score": 0.9, + "content": "\\mathbf { V } = \\mathbf { O } ^ { * } \\mathbf { D }", + "type": "inline_equation" + }, + { + "bbox": [ + 447, + 339, + 477, + 354 + ], + "score": 1.0, + "content": ", where", + "type": "text" + }, + { + "bbox": [ + 477, + 341, + 487, + 351 + ], + "score": 0.61, + "content": "\\mathbf { D }", + "type": "inline_equation" + }, + { + "bbox": [ + 488, + 339, + 497, + 354 + ], + "score": 1.0, + "content": "is", + "type": "text" + }, + { + "bbox": [ + 497, + 343, + 504, + 351 + ], + "score": 0.32, + "content": "a", + "type": "inline_equation" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 351, + 232, + 365 + ], + "spans": [ + { + "bbox": [ + 105, + 352, + 190, + 365 + ], + "score": 1.0, + "content": "diagonal matrix and", + "type": "text" + }, + { + "bbox": [ + 191, + 351, + 228, + 363 + ], + "score": 0.92, + "content": "\\mathbf { O } ^ { * } \\in \\widetilde { B } ^ { * }", + "type": "inline_equation" + }, + { + "bbox": [ + 228, + 352, + 232, + 365 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 14.5, + "bbox_fs": [ + 105, + 339, + 504, + 365 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 377, + 223, + 389 + ], + "lines": [ + { + "bbox": [ + 105, + 376, + 224, + 390 + ], + "spans": [ + { + "bbox": [ + 105, + 376, + 224, + 390 + ], + "score": 1.0, + "content": "K.3.3 SPARSE MATRICES", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 16 + }, + { + "type": "text", + "bbox": [ + 106, + 397, + 506, + 430 + ], + "lines": [ + { + "bbox": [ + 105, + 396, + 506, + 410 + ], + "spans": [ + { + "bbox": [ + 105, + 396, + 506, + 410 + ], + "score": 1.0, + "content": "Now that we have argued about the decomposition of permutation and step matrices in the OBB", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 408, + 504, + 420 + ], + "spans": [ + { + "bbox": [ + 105, + 408, + 496, + 420 + ], + "score": 1.0, + "content": "hierarchy, we can leverage the construction from Lemma I.1 to argue about matrices with at most", + "type": "text" + }, + { + "bbox": [ + 497, + 410, + 504, + 418 + ], + "score": 0.7, + "content": "n", + "type": "inline_equation" + } + ], + "index": 18 + }, + { + "bbox": [ + 104, + 417, + 133, + 432 + ], + "spans": [ + { + "bbox": [ + 104, + 417, + 133, + 432 + ], + "score": 1.0, + "content": "NNZ.", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 18, + "bbox_fs": [ + 104, + 396, + 506, + 432 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 433, + 441, + 446 + ], + "lines": [ + { + "bbox": [ + 106, + 433, + 442, + 447 + ], + "spans": [ + { + "bbox": [ + 106, + 433, + 221, + 447 + ], + "score": 1.0, + "content": "Corollary K.9. Let S be an", + "type": "text" + }, + { + "bbox": [ + 221, + 436, + 247, + 445 + ], + "score": 0.88, + "content": "n \\times n", + "type": "inline_equation" + }, + { + "bbox": [ + 247, + 433, + 384, + 447 + ], + "score": 1.0, + "content": "matrix with at most n NNZ. Then,", + "type": "text" + }, + { + "bbox": [ + 385, + 433, + 438, + 446 + ], + "score": 0.89, + "content": "\\mathbf { S } \\in ( \\mathcal { O } B B ) ^ { 4 }", + "type": "inline_equation" + }, + { + "bbox": [ + 439, + 433, + 442, + 447 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 20, + "bbox_fs": [ + 106, + 433, + 442, + 447 + ] + }, + { + "type": "text", + "bbox": [ + 105, + 459, + 504, + 483 + ], + "lines": [ + { + "bbox": [ + 105, + 458, + 506, + 474 + ], + "spans": [ + { + "bbox": [ + 105, + 458, + 506, + 474 + ], + "score": 1.0, + "content": "Proof. We use the construction from Lemma I.1, along with Lemma K.7 and Corollary K.8, to", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 471, + 161, + 484 + ], + "spans": [ + { + "bbox": [ + 105, + 471, + 161, + 484 + ], + "score": 1.0, + "content": "express S as:", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 21.5, + "bbox_fs": [ + 105, + 458, + 506, + 484 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 223, + 488, + 387, + 521 + ], + "lines": [ + { + "bbox": [ + 223, + 488, + 387, + 521 + ], + "spans": [ + { + "bbox": [ + 223, + 488, + 387, + 521 + ], + "score": 0.93, + "content": "\\mathbf { S } = \\underbrace { \\mathbf { O } _ { 1 } \\mathbf { O } _ { 1 } ^ { \\prime } } _ { \\mathbf { P } _ { 1 } } \\underbrace { \\mathbf { D } _ { 2 } \\mathbf { O } _ { 2 } } _ { \\mathbf { H } } \\underbrace { \\mathbf { O } _ { 3 } \\mathbf { O } _ { 3 } ^ { \\prime } } _ { \\mathbf { P } _ { 2 } } \\underbrace { \\mathbf { O } _ { 4 } ^ { \\prime } \\mathbf { D } _ { 4 } } _ { \\mathbf { V } } \\underbrace { \\mathbf { O } _ { 5 } \\mathbf { O } _ { 5 } ^ { \\prime } } _ { \\mathbf { P } _ { 3 } } ,", + "type": "interline_equation", + "image_path": "f4ddc3bdf875d54dacd9d6aaa9113a9957824b1947c2e065861b9fefe4c697cf.jpg" + } + ] + } + ], + "index": 23.5, + "virtual_lines": [ + { + "bbox": [ + 223, + 488, + 387, + 504.5 + ], + "spans": [], + "index": 23 + }, + { + "bbox": [ + 223, + 504.5, + 387, + 521.0 + ], + "spans": [], + "index": 24 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 527, + 506, + 567 + ], + "lines": [ + { + "bbox": [ + 105, + 527, + 507, + 543 + ], + "spans": [ + { + "bbox": [ + 105, + 527, + 145, + 543 + ], + "score": 1.0, + "content": "with each", + "type": "text" + }, + { + "bbox": [ + 145, + 527, + 178, + 541 + ], + "score": 0.92, + "content": "\\mathbf { O } _ { i } \\in \\widetilde { B }", + "type": "inline_equation" + }, + { + "bbox": [ + 178, + 527, + 200, + 543 + ], + "score": 1.0, + "content": ", each", + "type": "text" + }, + { + "bbox": [ + 200, + 527, + 238, + 542 + ], + "score": 0.93, + "content": "\\mathbf { O } _ { j } ^ { \\prime } \\in \\widetilde { B } ^ { * }", + "type": "inline_equation" + }, + { + "bbox": [ + 238, + 527, + 277, + 543 + ], + "score": 1.0, + "content": ", and each", + "type": "text" + }, + { + "bbox": [ + 277, + 529, + 292, + 541 + ], + "score": 0.89, + "content": "\\mathbf { D } _ { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 292, + 527, + 355, + 543 + ], + "score": 1.0, + "content": "diagonal. Since", + "type": "text" + }, + { + "bbox": [ + 355, + 529, + 369, + 540 + ], + "score": 0.89, + "content": "\\mathbf { P } _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 369, + 527, + 507, + 543 + ], + "score": 1.0, + "content": "is a permutation, by Corollary K.5,", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 104, + 541, + 507, + 559 + ], + "spans": [ + { + "bbox": [ + 104, + 541, + 180, + 559 + ], + "score": 1.0, + "content": "we can write it as", + "type": "text" + }, + { + "bbox": [ + 181, + 542, + 211, + 556 + ], + "score": 0.92, + "content": "\\tilde { \\bf { O } } _ { 3 } ^ { ' } \\tilde { \\bf { O } } _ { 3 }", + "type": "inline_equation" + }, + { + "bbox": [ + 211, + 541, + 250, + 559 + ], + "score": 1.0, + "content": "for some", + "type": "text" + }, + { + "bbox": [ + 251, + 542, + 292, + 556 + ], + "score": 0.93, + "content": "\\tilde { \\mathbf { O } _ { 3 } } ^ { \\prime } \\in \\widetilde { B } ^ { * }", + "type": "inline_equation" + }, + { + "bbox": [ + 292, + 541, + 311, + 559 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 311, + 542, + 345, + 556 + ], + "score": 0.93, + "content": "\\tilde { \\mathbf { O } } _ { 3 } \\in \\tilde { B }", + "type": "inline_equation" + }, + { + "bbox": [ + 345, + 541, + 441, + 559 + ], + "score": 1.0, + "content": ". 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Hence,", + "type": "text" + }, + { + "bbox": [ + 251, + 614, + 305, + 627 + ], + "score": 0.93, + "content": "\\mathbf { S } \\in ( \\mathcal { O } B B ) ^ { 4 }", + "type": "inline_equation" + }, + { + "bbox": [ + 305, + 614, + 353, + 628 + ], + "score": 1.0, + "content": ", as desired.", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 30, + "bbox_fs": [ + 106, + 614, + 353, + 628 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 640, + 506, + 675 + ], + "lines": [ + { + "bbox": [ + 106, + 641, + 505, + 654 + ], + "spans": [ + { + "bbox": [ + 106, + 641, + 505, + 654 + ], + "score": 1.0, + "content": "Just as in Appendix I, we would like to extend this orthogonal-based construction to capture matrices", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 652, + 505, + 665 + ], + "spans": [ + { + "bbox": [ + 105, + 652, + 505, + 665 + ], + "score": 1.0, + "content": "of general sparsity. To accomplish this, we introduce an addition closure lemma analogous to", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 661, + 257, + 676 + ], + "spans": [ + { + "bbox": [ + 105, + 661, + 190, + 676 + ], + "score": 1.0, + "content": "Lemma K.10 for the", + "type": "text" + }, + { + "bbox": [ + 191, + 663, + 214, + 674 + ], + "score": 0.46, + "content": "\\ O B B", + "type": "inline_equation" + }, + { + "bbox": [ + 214, + 661, + 257, + 676 + ], + "score": 1.0, + "content": "hierarchy.", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 32, + "bbox_fs": [ + 105, + 641, + 505, + 676 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 677, + 478, + 692 + ], + "lines": [ + { + "bbox": [ + 103, + 673, + 477, + 695 + ], + "spans": [ + { + "bbox": [ + 103, + 673, + 184, + 695 + ], + "score": 1.0, + "content": "Lemma K.10. Let", + "type": "text" + }, + { + "bbox": [ + 185, + 678, + 237, + 690 + ], + "score": 0.91, + "content": "\\mathbf { A } _ { 1 } , \\ldots , \\mathbf { A } _ { m }", + "type": "inline_equation" + }, + { + "bbox": [ + 237, + 673, + 250, + 695 + ], + "score": 1.0, + "content": "be", + "type": "text" + }, + { + "bbox": [ + 250, + 678, + 275, + 689 + ], + "score": 0.88, + "content": "k \\times k", + "type": "inline_equation" + }, + { + "bbox": [ + 275, + 673, + 324, + 695 + ], + "score": 1.0, + "content": "matrices in", + "type": "text" + }, + { + "bbox": [ + 324, + 678, + 360, + 691 + ], + "score": 0.78, + "content": "( O B B ) _ { e } ^ { w }", + "type": "inline_equation" + }, + { + "bbox": [ + 361, + 673, + 382, + 695 + ], + "score": 1.0, + "content": "then", + "type": "text" + }, + { + "bbox": [ + 382, + 677, + 477, + 691 + ], + "score": 0.52, + "content": "\\begin{array} { r } { \\sum _ { i = 1 } ^ { m } \\mathbf { A } _ { i } \\in ( \\mathcal { O } B \\mathcal { B } ) _ { 4 e } ^ { m w } . } \\end{array}", + "type": "inline_equation" + } + ], + "index": 34 + } + ], + "index": 34, + "bbox_fs": [ + 103, + 673, + 477, + 695 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 699, + 454, + 712 + ], + "lines": [ + { + "bbox": [ + 105, + 697, + 456, + 714 + ], + "spans": [ + { + "bbox": [ + 105, + 697, + 456, + 714 + ], + "score": 1.0, + "content": "With Lemma K.10, we arrive at the following Corollary on general orthogonal sparsity.", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 35, + "bbox_fs": [ + 105, + 697, + 456, + 714 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 715, + 430, + 733 + ], + "lines": [ + { + "bbox": [ + 104, + 713, + 430, + 735 + ], + "spans": [ + { + "bbox": [ + 104, + 713, + 226, + 735 + ], + "score": 1.0, + "content": "Corollary K.11. Let S be an", + "type": "text" + }, + { + "bbox": [ + 227, + 722, + 252, + 731 + ], + "score": 0.86, + "content": "n \\times n", + "type": "inline_equation" + }, + { + "bbox": [ + 252, + 713, + 357, + 735 + ], + "score": 1.0, + "content": "matrix with s NNZ. Then,", + "type": "text" + }, + { + "bbox": [ + 357, + 714, + 426, + 733 + ], + "score": 0.92, + "content": "\\mathbf { S } \\in \\left( \\mathcal { O } B B \\right) _ { 4 } ^ { 4 \\left\\lceil \\frac { s } { n } \\right\\rceil }", + "type": "inline_equation" + }, + { + "bbox": [ + 426, + 713, + 430, + 735 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 36, + "bbox_fs": [ + 104, + 713, + 430, + 735 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 105, + 81, + 505, + 106 + ], + "lines": [ + { + "bbox": [ + 104, + 80, + 507, + 98 + ], + "spans": [ + { + "bbox": [ + 104, + 80, + 409, + 98 + ], + "score": 1.0, + "content": "Proof. Just as in the proof of Theorem 3, we accomplish this using a sum of", + "type": "text" + }, + { + "bbox": [ + 410, + 82, + 426, + 96 + ], + "score": 0.9, + "content": "\\textstyle { \\left\\lceil { \\frac { s } { n } } \\right\\rceil }", + "type": "inline_equation" + }, + { + "bbox": [ + 426, + 80, + 507, + 98 + ], + "score": 1.0, + "content": "matrices of at most", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 94, + 505, + 106 + ], + "spans": [ + { + "bbox": [ + 106, + 96, + 114, + 104 + ], + "score": 0.45, + "content": "n", + "type": "inline_equation" + }, + { + "bbox": [ + 114, + 94, + 420, + 106 + ], + "score": 1.0, + "content": "NNZ. For handling the sum of matrices, we need to appeal to Lemma K.10.", + "type": "text" + }, + { + "bbox": [ + 495, + 95, + 505, + 104 + ], + "score": 0.999, + "content": "□", + "type": "text" + } + ], + "index": 1 + } + ], + "index": 0.5 + }, + { + "type": "text", + "bbox": [ + 106, + 116, + 354, + 129 + ], + "lines": [ + { + "bbox": [ + 106, + 117, + 354, + 130 + ], + "spans": [ + { + "bbox": [ + 106, + 117, + 354, + 130 + ], + "score": 1.0, + "content": "To conclude the argument, we give the proof of Lemma K.10.", + "type": "text" + } + ], + "index": 2 + } + ], + "index": 2 + }, + { + "type": "text", + "bbox": [ + 106, + 139, + 504, + 163 + ], + "lines": [ + { + "bbox": [ + 104, + 137, + 504, + 154 + ], + "spans": [ + { + "bbox": [ + 104, + 137, + 173, + 154 + ], + "score": 1.0, + "content": "Proof of Lemma", + "type": "text" + }, + { + "bbox": [ + 173, + 141, + 194, + 151 + ], + "score": 0.31, + "content": "K . I O", + "type": "inline_equation" + }, + { + "bbox": [ + 194, + 137, + 237, + 154 + ], + "score": 1.0, + "content": ". 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Note that", + "type": "text" + }, + { + "bbox": [ + 264, + 152, + 330, + 163 + ], + "score": 0.92, + "content": "\\mathbf { E } _ { \\mathbf { A } _ { i } } \\in ( \\mathcal { O } B B ) ^ { w }", + "type": "inline_equation" + }, + { + "bbox": [ + 331, + 150, + 461, + 164 + ], + "score": 1.0, + "content": ". 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Therefore, we can fold", + "type": "text" + }, + { + "bbox": [ + 494, + 343, + 504, + 353 + ], + "score": 0.69, + "content": "\\mathbf { O }", + "type": "inline_equation" + } + ], + "index": 13 + }, + { + "bbox": [ + 103, + 353, + 507, + 372 + ], + "spans": [ + { + "bbox": [ + 103, + 353, + 302, + 372 + ], + "score": 1.0, + "content": "into this factor (since a butterfly factor in Be (4ek)4ek", + "type": "text" + }, + { + "bbox": [ + 298, + 354, + 376, + 370 + ], + "score": 1.0, + "content": "was not yet used in", + "type": "text" + }, + { + "bbox": [ + 376, + 357, + 391, + 368 + ], + "score": 0.87, + "content": "\\mathbf { L } _ { w }", + "type": "inline_equation" + }, + { + "bbox": [ + 392, + 354, + 461, + 370 + ], + "score": 1.0, + "content": ") to conclude that", + "type": "text" + }, + { + "bbox": [ + 461, + 354, + 503, + 368 + ], + "score": 0.93, + "content": "\\mathbf { O L } _ { w } \\in \\widetilde { B }", + "type": "inline_equation" + }, + { + "bbox": [ + 504, + 354, + 507, + 370 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 103, + 366, + 507, + 387 + ], + "spans": [ + { + "bbox": [ + 103, + 366, + 303, + 387 + ], + "score": 1.0, + "content": "Similarly, since no btterfly factor from Be (4ek)2ek", + "type": "text" + }, + { + "bbox": [ + 299, + 369, + 372, + 384 + ], + "score": 1.0, + "content": "has been used in", + "type": "text" + }, + { + "bbox": [ + 373, + 371, + 387, + 383 + ], + "score": 0.88, + "content": "{ \\bf R } _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 388, + 369, + 447, + 384 + ], + "score": 1.0, + "content": ", we may fold", + "type": "text" + }, + { + "bbox": [ + 448, + 371, + 457, + 381 + ], + "score": 0.7, + "content": "\\mathbf { P }", + "type": "inline_equation" + }, + { + "bbox": [ + 458, + 369, + 478, + 384 + ], + "score": 1.0, + "content": "into", + "type": "text" + }, + { + "bbox": [ + 478, + 371, + 492, + 383 + ], + "score": 0.88, + "content": "{ \\bf R } _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 493, + 369, + 507, + 384 + ], + "score": 1.0, + "content": "to", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 383, + 506, + 397 + ], + "spans": [ + { + "bbox": [ + 106, + 384, + 162, + 397 + ], + "score": 1.0, + "content": "conclude that", + "type": "text" + }, + { + "bbox": [ + 162, + 383, + 207, + 396 + ], + "score": 0.94, + "content": "\\mathbf { R } _ { 1 } \\mathbf { P } \\in \\widetilde { B } ^ { * }", + "type": "inline_equation" + }, + { + "bbox": [ + 208, + 384, + 376, + 397 + ], + "score": 1.0, + "content": ". 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Next, we note that", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 26 + }, + { + "type": "interline_equation", + "bbox": [ + 213, + 569, + 397, + 636 + ], + "lines": [ + { + "bbox": [ + 213, + 569, + 397, + 636 + ], + "spans": [ + { + "bbox": [ + 213, + 569, + 397, + 636 + ], + "score": 0.92, + "content": "\\sum _ { i = 1 } ^ { m } \\mathbf { A } _ { i } = \\mathbf { S } \\mathbf { M } \\underbrace { \\left[ \\begin{array} { c c c c } { 0 } & { \\mathbf { I } _ { e k } } & { 0 } & { \\mathbf { I } _ { e k } } \\\\ { \\mathbf { I } _ { e k } } & { 0 } & { \\mathbf { I } _ { e k } } & { 0 } \\\\ { 0 } & { \\mathbf { I } _ { e k } } & { 0 } & { - \\mathbf { I } _ { e k } } \\\\ { \\mathbf { I } _ { e k } } & { 0 } & { - \\mathbf { I } _ { e k } } & { 0 } \\end{array} \\right] } _ { \\mathbf { Q } } \\mathbf { S } ^ { T } .", + "type": "interline_equation", + "image_path": "3b4bb35e6e3f120d2c581d877677469bda57fe8faada5eab6f72b4b12f945384.jpg" + } + ] + } + ], + "index": 28.5, + "virtual_lines": [ + { + "bbox": [ + 213, + 569, + 397, + 585.75 + ], + "spans": [], + "index": 27 + }, + { + "bbox": [ + 213, + 585.75, + 397, + 602.5 + ], + "spans": [], + "index": 28 + }, + { + "bbox": [ + 213, + 602.5, + 397, + 619.25 + ], + "spans": [], + "index": 29 + }, + { + "bbox": [ + 213, + 619.25, + 397, + 636.0 + ], + "spans": [], + "index": 30 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 641, + 506, + 664 + ], + "lines": [ + { + "bbox": [ + 106, + 641, + 505, + 654 + ], + "spans": [ + { + "bbox": [ + 106, + 641, + 261, + 654 + ], + "score": 1.0, + "content": "We would like to show that we can fold", + "type": "text" + }, + { + "bbox": [ + 262, + 642, + 272, + 653 + ], + "score": 0.8, + "content": "\\mathbf { Q }", + "type": "inline_equation" + }, + { + "bbox": [ + 272, + 641, + 344, + 654 + ], + "score": 1.0, + "content": "into the rightmost", + "type": "text" + }, + { + "bbox": [ + 344, + 641, + 367, + 652 + ], + "score": 0.39, + "content": "\\ O B B", + "type": "inline_equation" + }, + { + "bbox": [ + 368, + 641, + 505, + 654 + ], + "score": 1.0, + "content": "factor of M. The rightmost matrix", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 653, + 279, + 664 + ], + "spans": [ + { + "bbox": [ + 106, + 653, + 203, + 664 + ], + "score": 1.0, + "content": "in the decomposition of", + "type": "text" + }, + { + "bbox": [ + 204, + 653, + 216, + 662 + ], + "score": 0.32, + "content": "\\mathbf { M }", + "type": "inline_equation" + }, + { + "bbox": [ + 216, + 653, + 226, + 664 + ], + "score": 1.0, + "content": "is", + "type": "text" + }, + { + "bbox": [ + 226, + 653, + 235, + 662 + ], + "score": 0.55, + "content": "\\mathbf { P }", + "type": "inline_equation" + }, + { + "bbox": [ + 235, + 653, + 279, + 664 + ], + "score": 1.0, + "content": ". Note that", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 31.5 + }, + { + "type": "interline_equation", + "bbox": [ + 130, + 664, + 480, + 736 + ], + "lines": [ + { + "bbox": [ + 130, + 664, + 480, + 736 + ], + "spans": [ + { + "bbox": [ + 130, + 664, + 480, + 736 + ], + "score": 0.92, + "content": "\\mathbf { P Q } = \\left[ \\begin{array} { c c c c } { 0 } & { \\mathbf { I } _ { e k } } & { 0 } & { \\mathbf { I } _ { e k } } \\\\ { \\mathbf { I } _ { e k } } & { 0 } & { \\mathbf { I } _ { e k } } & { 0 } \\\\ { \\mathbf { I } _ { e k } } & { \\mathbf { I } _ { e k } } & { \\mathbf { I } _ { e k } } & { 0 } \\\\ { 0 } & { \\mathbf { I } _ { e k } } & { 0 } & { \\mathbf { - I } _ { e k } } \\end{array} \\right] = \\sqrt { 2 } \\underbrace { \\left[ \\begin{array} { c c c c } { 0 } & { \\mathbf { I } _ { e k } } & { 0 } & { 0 } \\\\ { \\mathbf { I } _ { e k } } & { 0 } & { 0 } & { 0 } \\\\ { 0 } & { 0 } & { \\mathbf { I } _ { e k } } & { 0 } \\\\ { 0 } & { 0 } & { 0 } & { \\mathbf { I } _ { e k } } \\end{array} \\right] } _ { \\widetilde { \\mathbf { B } } _ { 2 e k } ^ { ( 4 e k ) } } \\underbrace { \\left[ \\frac { 1 } { \\sqrt { 2 } } \\mathbf { I } _ { 2 e k } \\quad \\frac { 1 } { \\sqrt { 2 } } \\mathbf { I } _ { 2 e k } \\right] } _ { \\widetilde { \\mathbf { B } } _ { 4 e k } ^ { ( 4 e k ) } } .", + "type": "interline_equation", + "image_path": "8a5b6e6f1eed59213df86fb1481a447a0ed2ffb571f1e2dab42f3699c8fa5273.jpg" + } + ] + } + ], + "index": 34, + "virtual_lines": [ + { + "bbox": [ + 130, + 664, + 480, + 688.0 + ], + "spans": [], + "index": 33 + }, + { + "bbox": [ + 130, + 688.0, + 480, + 712.0 + ], + "spans": [], + "index": 34 + }, + { + "bbox": [ + 130, + 712.0, + 480, + 736.0 + ], + "spans": [], + "index": 35 + } + ] + } + ], + "page_idx": 40, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 107, + 26, + 294, + 38 + ], + "lines": [ + { + "bbox": [ + 106, + 25, + 294, + 39 + ], + "spans": [ + { + "bbox": [ + 106, + 25, + 294, + 39 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2020", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 300, + 751, + 310, + 760 + ], + "lines": [ + { + "bbox": [ + 298, + 750, + 312, + 764 + ], + "spans": [ + { + "bbox": [ + 298, + 750, + 312, + 764 + ], + "score": 1.0, + "content": "", + "type": "text", + "height": 14, + "width": 14 + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 105, + 81, + 505, + 106 + ], + "lines": [ + { + "bbox": [ + 104, + 80, + 507, + 98 + ], + "spans": [ + { + "bbox": [ + 104, + 80, + 409, + 98 + ], + "score": 1.0, + "content": "Proof. Just as in the proof of Theorem 3, we accomplish this using a sum of", + "type": "text" + }, + { + "bbox": [ + 410, + 82, + 426, + 96 + ], + "score": 0.9, + "content": "\\textstyle { \\left\\lceil { \\frac { s } { n } } \\right\\rceil }", + "type": "inline_equation" + }, + { + "bbox": [ + 426, + 80, + 507, + 98 + ], + "score": 1.0, + "content": "matrices of at most", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 94, + 505, + 106 + ], + "spans": [ + { + "bbox": [ + 106, + 96, + 114, + 104 + ], + "score": 0.45, + "content": "n", + "type": "inline_equation" + }, + { + "bbox": [ + 114, + 94, + 420, + 106 + ], + "score": 1.0, + "content": "NNZ. For handling the sum of matrices, we need to appeal to Lemma K.10.", + "type": "text" + }, + { + "bbox": [ + 495, + 95, + 505, + 104 + ], + "score": 0.999, + "content": "□", + "type": "text" + } + ], + "index": 1 + } + ], + "index": 0.5, + "bbox_fs": [ + 104, + 80, + 507, + 106 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 116, + 354, + 129 + ], + "lines": [ + { + "bbox": [ + 106, + 117, + 354, + 130 + ], + "spans": [ + { + "bbox": [ + 106, + 117, + 354, + 130 + ], + "score": 1.0, + "content": "To conclude the argument, we give the proof of Lemma K.10.", + "type": "text" + } + ], + "index": 2 + } + ], + "index": 2, + "bbox_fs": [ + 106, + 117, + 354, + 130 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 139, + 504, + 163 + ], + "lines": [ + { + "bbox": [ + 104, + 137, + 504, + 154 + ], + "spans": [ + { + "bbox": [ + 104, + 137, + 173, + 154 + ], + "score": 1.0, + "content": "Proof of Lemma", + "type": "text" + }, + { + "bbox": [ + 173, + 141, + 194, + 151 + ], + "score": 0.31, + "content": "K . I O", + "type": "inline_equation" + }, + { + "bbox": [ + 194, + 137, + 237, + 154 + ], + "score": 1.0, + "content": ". For each", + "type": "text" + }, + { + "bbox": [ + 237, + 141, + 282, + 151 + ], + "score": 0.91, + "content": "1 \\leq i \\leq m", + "type": "inline_equation" + }, + { + "bbox": [ + 282, + 137, + 298, + 154 + ], + "score": 1.0, + "content": ", let", + "type": "text" + }, + { + "bbox": [ + 299, + 140, + 358, + 152 + ], + "score": 0.92, + "content": "\\mathbf { E } _ { \\mathbf { A } _ { i } } \\in \\mathbb { F } ^ { e k \\times e k }", + "type": "inline_equation" + }, + { + "bbox": [ + 359, + 137, + 442, + 154 + ], + "score": 1.0, + "content": "be defined such that", + "type": "text" + }, + { + "bbox": [ + 442, + 140, + 504, + 152 + ], + "score": 0.91, + "content": "\\mathbf { A } _ { i } = \\mathbf { S } \\mathbf { E } _ { \\mathbf { A } _ { i } } \\mathbf { S } ^ { * }", + "type": "inline_equation" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 150, + 461, + 164 + ], + "spans": [ + { + "bbox": [ + 105, + 150, + 129, + 164 + ], + "score": 1.0, + "content": "(with", + "type": "text" + }, + { + "bbox": [ + 129, + 152, + 138, + 161 + ], + "score": 0.31, + "content": "\\mathbf { S }", + "type": "inline_equation" + }, + { + "bbox": [ + 138, + 150, + 263, + 164 + ], + "score": 1.0, + "content": "as in Definition 2.4). 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The rightmost matrix", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 653, + 279, + 664 + ], + "spans": [ + { + "bbox": [ + 106, + 653, + 203, + 664 + ], + "score": 1.0, + "content": "in the decomposition of", + "type": "text" + }, + { + "bbox": [ + 204, + 653, + 216, + 662 + ], + "score": 0.32, + "content": "\\mathbf { M }", + "type": "inline_equation" + }, + { + "bbox": [ + 216, + 653, + 226, + 664 + ], + "score": 1.0, + "content": "is", + "type": "text" + }, + { + "bbox": [ + 226, + 653, + 235, + 662 + ], + "score": 0.55, + "content": "\\mathbf { P }", + "type": "inline_equation" + }, + { + "bbox": [ + 235, + 653, + 279, + 664 + ], + "score": 1.0, + "content": ". 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Let M be an", + "type": "text" + }, + { + "bbox": [ + 232, + 189, + 257, + 198 + ], + "score": 0.88, + "content": "n \\times n", + "type": "inline_equation" + }, + { + "bbox": [ + 258, + 186, + 505, + 201 + ], + "score": 1.0, + "content": "matrix such that matrix-vector multiplication of M times an", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 197, + 505, + 212 + ], + "spans": [ + { + "bbox": [ + 105, + 197, + 406, + 212 + ], + "score": 1.0, + "content": "arbitrary vector v can be represented as a be a linear arithmetic circuit", + "type": "text" + }, + { + "bbox": [ + 407, + 199, + 416, + 208 + ], + "score": 0.78, + "content": "C", + "type": "inline_equation" + }, + { + "bbox": [ + 416, + 197, + 505, + 212 + ], + "score": 1.0, + "content": "comprised of s gates", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 104, + 208, + 360, + 227 + ], + "spans": [ + { + "bbox": [ + 104, + 208, + 251, + 226 + ], + "score": 1.0, + "content": "(including inputs) and having depth", + "type": "text" + }, + { + "bbox": [ + 251, + 212, + 258, + 222 + ], + "score": 0.34, + "content": "d .", + "type": "inline_equation" + }, + { + "bbox": [ + 258, + 208, + 286, + 226 + ], + "score": 1.0, + "content": ". Then,", + "type": "text" + }, + { + "bbox": [ + 287, + 209, + 360, + 227 + ], + "score": 0.93, + "content": "\\mathbf { M } \\in ( \\mathcal { O } B B ) _ { O ( \\frac { s } { n } ) } ^ { O ( d ) }", + "type": "inline_equation" + } + ], + "index": 9 + }, + { + "bbox": [ + 360, + 216, + 364, + 228 + ], + "spans": [ + { + "bbox": [ + 360, + 216, + 364, + 228 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 8.5 + }, + { + "type": "text", + "bbox": [ + 105, + 240, + 504, + 263 + ], + "lines": [ + { + "bbox": [ + 105, + 240, + 506, + 253 + ], + "spans": [ + { + "bbox": [ + 105, + 240, + 506, + 253 + ], + "score": 1.0, + "content": "Proof. We use the construction given in the proof of Theorem 1. Corollaries K.9 and K.4 allow us to", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 251, + 504, + 264 + ], + "spans": [ + { + "bbox": [ + 105, + 251, + 321, + 264 + ], + "score": 1.0, + "content": "recover the same width and expansion factor with the", + "type": "text" + }, + { + "bbox": [ + 322, + 252, + 345, + 262 + ], + "score": 0.26, + "content": "o s B", + "type": "inline_equation" + }, + { + "bbox": [ + 345, + 251, + 387, + 264 + ], + "score": 1.0, + "content": "hierarchy.", + "type": "text" + }, + { + "bbox": [ + 496, + 253, + 504, + 261 + ], + "score": 0.999, + "content": "□", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 11.5 + }, + { + "type": "title", + "bbox": [ + 106, + 280, + 415, + 293 + ], + "lines": [ + { + "bbox": [ + 104, + 279, + 417, + 296 + ], + "spans": [ + { + "bbox": [ + 104, + 279, + 417, + 296 + ], + "score": 1.0, + "content": "L RELU NETWORK WITH STRUCTURED WEIGHT MATRICES", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 13 + }, + { + "type": "text", + "bbox": [ + 107, + 305, + 505, + 371 + ], + "lines": [ + { + "bbox": [ + 105, + 305, + 505, + 318 + ], + "spans": [ + { + "bbox": [ + 105, + 305, + 505, + 318 + ], + "score": 1.0, + "content": "We show that for any neural network with ReLU nonlinearities and whose weight matrices have", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 317, + 505, + 329 + ], + "spans": [ + { + "bbox": [ + 106, + 317, + 505, + 329 + ], + "score": 1.0, + "content": "arithmetic circuits with few gates, its linear network counterpart (obtained by removing all the", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 327, + 506, + 340 + ], + "spans": [ + { + "bbox": [ + 105, + 327, + 506, + 340 + ], + "score": 1.0, + "content": "ReLU’s) also has an arithmetic circuit with not too many more gates. This implies that in trying to", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 338, + 505, + 351 + ], + "spans": [ + { + "bbox": [ + 105, + 338, + 505, + 351 + ], + "score": 1.0, + "content": "find the smallest arithmetic circuit augmented with ReLU gates to represent a ReLU network, one", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 349, + 505, + 361 + ], + "spans": [ + { + "bbox": [ + 106, + 349, + 505, + 361 + ], + "score": 1.0, + "content": "might as well try to find the smallest arithmetic circuits that represent the matrix-vector multiplication", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 361, + 198, + 372 + ], + "spans": [ + { + "bbox": [ + 106, + 361, + 198, + 372 + ], + "score": 1.0, + "content": "of each weight matrix.", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 16.5 + }, + { + "type": "text", + "bbox": [ + 107, + 375, + 504, + 398 + ], + "lines": [ + { + "bbox": [ + 105, + 374, + 505, + 389 + ], + "spans": [ + { + "bbox": [ + 105, + 374, + 383, + 389 + ], + "score": 1.0, + "content": "Proposition 2. Consider a neural network architecture consisting of", + "type": "text" + }, + { + "bbox": [ + 384, + 376, + 392, + 385 + ], + "score": 0.42, + "content": "L", + "type": "inline_equation" + }, + { + "bbox": [ + 392, + 374, + 505, + 389 + ], + "score": 1.0, + "content": "layers with weight matrices", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 384, + 342, + 401 + ], + "spans": [ + { + "bbox": [ + 106, + 387, + 198, + 398 + ], + "score": 0.9, + "content": "\\mathbf { W } _ { 1 } , \\dots , \\mathbf { W } _ { L } \\in \\mathbb { F } ^ { n \\times n }", + "type": "inline_equation" + }, + { + "bbox": [ + 199, + 384, + 342, + 401 + ], + "score": 1.0, + "content": "and ReLU nonlinearity in between.", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 20.5 + }, + { + "type": "text", + "bbox": [ + 107, + 403, + 505, + 450 + ], + "lines": [ + { + "bbox": [ + 106, + 404, + 506, + 416 + ], + "spans": [ + { + "bbox": [ + 106, + 404, + 291, + 416 + ], + "score": 1.0, + "content": "Suppose that matrix-vector multiplication of", + "type": "text" + }, + { + "bbox": [ + 291, + 404, + 308, + 415 + ], + "score": 0.89, + "content": "\\mathbf { W } _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 308, + 404, + 506, + 416 + ], + "score": 1.0, + "content": "times an arbitrary vector v can be represented", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 415, + 505, + 427 + ], + "spans": [ + { + "bbox": [ + 105, + 415, + 252, + 427 + ], + "score": 1.0, + "content": "as a linear arithmetic circuit with", + "type": "text" + }, + { + "bbox": [ + 252, + 416, + 262, + 426 + ], + "score": 0.82, + "content": "s _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 262, + 415, + 505, + 427 + ], + "score": 1.0, + "content": "gates (including inputs). Then there exists an arithmetic", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 101, + 421, + 509, + 445 + ], + "spans": [ + { + "bbox": [ + 101, + 421, + 509, + 445 + ], + "score": 1.0, + "content": "circuit augmented with ReLU gates with PLi=1 si + Ln total gates that computes the output", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 437, + 424, + 452 + ], + "spans": [ + { + "bbox": [ + 105, + 437, + 133, + 452 + ], + "score": 1.0, + "content": "ReLU", + "type": "text" + }, + { + "bbox": [ + 134, + 438, + 233, + 451 + ], + "score": 0.85, + "content": "( \\mathbf { W } _ { L } ( \\dots \\mathrm { R e L U } ( \\mathbf { W } _ { 1 } \\mathbf { v } ) ) )", + "type": "inline_equation" + }, + { + "bbox": [ + 233, + 437, + 424, + 452 + ], + "score": 1.0, + "content": ") of the network for an arbitrary input vector v.", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 23.5 + }, + { + "type": "text", + "bbox": [ + 106, + 455, + 505, + 500 + ], + "lines": [ + { + "bbox": [ + 106, + 455, + 505, + 467 + ], + "spans": [ + { + "bbox": [ + 106, + 455, + 505, + 467 + ], + "score": 1.0, + "content": "Conversely, if there is an arithmetic circuit augmented with ReLU gates with s total gates that", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 466, + 505, + 479 + ], + "spans": [ + { + "bbox": [ + 105, + 466, + 294, + 479 + ], + "score": 1.0, + "content": "computes all the activations of the network ReL", + "type": "text" + }, + { + "bbox": [ + 294, + 466, + 476, + 478 + ], + "score": 0.6, + "content": "\\operatorname { \\mathrm { . U } } ( \\mathbf { W } _ { 1 } \\mathbf { v } ) , \\ldots , \\operatorname { R e L U } ( \\mathbf { W } _ { L } \\ldots \\operatorname { \\mathrm { . R e L U } } ( \\mathbf { W } _ { 1 } \\mathbf { v } ) )", + "type": "inline_equation" + }, + { + "bbox": [ + 477, + 466, + 505, + 479 + ], + "score": 1.0, + "content": "for an", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 478, + 504, + 488 + ], + "spans": [ + { + "bbox": [ + 106, + 478, + 463, + 488 + ], + "score": 1.0, + "content": "arbitrary input v, then there exists an arithmetic circuit augmented with ReLU gates with", + "type": "text" + }, + { + "bbox": [ + 463, + 478, + 504, + 488 + ], + "score": 0.89, + "content": "2 s + 2 L n", + "type": "inline_equation" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 488, + 492, + 501 + ], + "spans": [ + { + "bbox": [ + 105, + 488, + 388, + 501 + ], + "score": 1.0, + "content": "total gates that computes the activations of the network without ReLU", + "type": "text" + }, + { + "bbox": [ + 388, + 488, + 488, + 500 + ], + "score": 0.87, + "content": "\\mathbf { W } _ { 1 } \\mathbf { v } , \\dots , \\mathbf { W } _ { L } \\dots \\mathbf { W } _ { 1 } \\mathbf { v }", + "type": "inline_equation" + }, + { + "bbox": [ + 489, + 488, + 492, + 501 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 27.5 + }, + { + "type": "text", + "bbox": [ + 106, + 514, + 505, + 572 + ], + "lines": [ + { + "bbox": [ + 105, + 514, + 506, + 527 + ], + "spans": [ + { + "bbox": [ + 105, + 514, + 374, + 527 + ], + "score": 1.0, + "content": "Proof of Proposition 2. To compute the output of the network Re", + "type": "text" + }, + { + "bbox": [ + 374, + 514, + 484, + 527 + ], + "score": 0.88, + "content": "\\operatorname { \\mathrm { . U } } ( \\mathbf { W } _ { L } ( \\dots \\operatorname { R e L U } ( \\mathbf { W } _ { 1 } \\mathbf { v } ) ) )", + "type": "inline_equation" + }, + { + "bbox": [ + 484, + 514, + 506, + 527 + ], + "score": 1.0, + "content": "), we", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 525, + 507, + 538 + ], + "spans": [ + { + "bbox": [ + 105, + 525, + 267, + 538 + ], + "score": 1.0, + "content": "first compute the matrix-vector product", + "type": "text" + }, + { + "bbox": [ + 268, + 525, + 291, + 536 + ], + "score": 0.9, + "content": "\\mathbf { W } _ { 1 } \\mathbf { v }", + "type": "inline_equation" + }, + { + "bbox": [ + 292, + 525, + 407, + 538 + ], + "score": 1.0, + "content": "with an arithmetic circuit of", + "type": "text" + }, + { + "bbox": [ + 408, + 527, + 418, + 536 + ], + "score": 0.85, + "content": "s _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 419, + 525, + 507, + 538 + ], + "score": 1.0, + "content": "gates by assumption,", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 536, + 506, + 549 + ], + "spans": [ + { + "bbox": [ + 105, + 536, + 139, + 549 + ], + "score": 1.0, + "content": "and use", + "type": "text" + }, + { + "bbox": [ + 140, + 538, + 147, + 546 + ], + "score": 0.72, + "content": "n", + "type": "inline_equation" + }, + { + "bbox": [ + 147, + 536, + 506, + 549 + ], + "score": 1.0, + "content": "other ReLU gates to compute the pointwise ReLU. Then we repeat the process for layer", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 547, + 506, + 559 + ], + "spans": [ + { + "bbox": [ + 106, + 547, + 151, + 559 + ], + "score": 0.91, + "content": "2 , 3 , \\ldots , L", + "type": "inline_equation" + }, + { + "bbox": [ + 151, + 547, + 277, + 559 + ], + "score": 1.0, + "content": ", using the arithmetic circuits of", + "type": "text" + }, + { + "bbox": [ + 277, + 547, + 334, + 559 + ], + "score": 0.92, + "content": "\\mathbf { W } _ { 1 } , \\dots , \\mathbf { W } _ { L }", + "type": "inline_equation" + }, + { + "bbox": [ + 335, + 547, + 352, + 559 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 352, + 547, + 366, + 558 + ], + "score": 0.79, + "content": "L n", + "type": "inline_equation" + }, + { + "bbox": [ + 367, + 547, + 506, + 559 + ], + "score": 1.0, + "content": "additional gates for ReLU. In total", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 103, + 555, + 476, + 577 + ], + "spans": [ + { + "bbox": [ + 103, + 555, + 366, + 577 + ], + "score": 1.0, + "content": "we obtain an arithmetic circuit augmented with ReLU gates with", + "type": "text" + }, + { + "bbox": [ + 367, + 558, + 426, + 573 + ], + "score": 0.92, + "content": "\\textstyle \\sum _ { i = 1 } ^ { L } s _ { i } + L n", + "type": "inline_equation" + }, + { + "bbox": [ + 427, + 555, + 476, + 577 + ], + "score": 1.0, + "content": "total gates.", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 32 + }, + { + "type": "text", + "bbox": [ + 107, + 577, + 505, + 622 + ], + "lines": [ + { + "bbox": [ + 105, + 576, + 505, + 590 + ], + "spans": [ + { + "bbox": [ + 105, + 576, + 505, + 590 + ], + "score": 1.0, + "content": "Conversely, to build an arithmetic circuit augmented with ReLU gates to compute", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 588, + 506, + 601 + ], + "spans": [ + { + "bbox": [ + 106, + 588, + 207, + 599 + ], + "score": 0.9, + "content": "\\mathbf { W } _ { 1 } \\mathbf { v } , \\dots , \\mathbf { W } _ { L } \\dots \\mathbf { W } _ { 1 } \\mathbf { v }", + "type": "inline_equation" + }, + { + "bbox": [ + 208, + 588, + 243, + 601 + ], + "score": 1.0, + "content": ", we pass", + "type": "text" + }, + { + "bbox": [ + 243, + 590, + 251, + 598 + ], + "score": 0.39, + "content": "\\mathbf { v }", + "type": "inline_equation" + }, + { + "bbox": [ + 251, + 588, + 287, + 601 + ], + "score": 1.0, + "content": "and then", + "type": "text" + }, + { + "bbox": [ + 287, + 589, + 302, + 599 + ], + "score": 0.73, + "content": "- \\mathbf { v }", + "type": "inline_equation" + }, + { + "bbox": [ + 303, + 588, + 433, + 601 + ], + "score": 1.0, + "content": "through the circuit that computes", + "type": "text" + }, + { + "bbox": [ + 433, + 588, + 489, + 599 + ], + "score": 0.58, + "content": "\\mathrm { R e L U } ( \\mathbf { W } _ { 1 } \\mathbf { x } )", + "type": "inline_equation" + }, + { + "bbox": [ + 490, + 588, + 506, + 601 + ], + "score": 1.0, + "content": "for", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 598, + 507, + 613 + ], + "spans": [ + { + "bbox": [ + 105, + 598, + 156, + 613 + ], + "score": 1.0, + "content": "an arbitrary", + "type": "text" + }, + { + "bbox": [ + 156, + 601, + 164, + 609 + ], + "score": 0.59, + "content": "\\mathbf { x }", + "type": "inline_equation" + }, + { + "bbox": [ + 164, + 598, + 217, + 613 + ], + "score": 1.0, + "content": "to get ReLU", + "type": "text" + }, + { + "bbox": [ + 217, + 599, + 247, + 611 + ], + "score": 0.63, + "content": "\\left( \\mathbf { W } _ { 1 } \\mathbf { v } \\right)", + "type": "inline_equation" + }, + { + "bbox": [ + 247, + 598, + 293, + 613 + ], + "score": 1.0, + "content": "and ReLU", + "type": "text" + }, + { + "bbox": [ + 294, + 599, + 330, + 611 + ], + "score": 0.62, + "content": "\\left( - \\mathbf { W } _ { 1 } \\mathbf { v } \\right)", + "type": "inline_equation" + }, + { + "bbox": [ + 330, + 598, + 384, + 613 + ], + "score": 1.0, + "content": ". Noting that", + "type": "text" + }, + { + "bbox": [ + 384, + 599, + 502, + 611 + ], + "score": 0.88, + "content": "x = \\mathrm { R e L U } ( x ) - \\mathrm { R e L U } ( - x )", + "type": "inline_equation" + }, + { + "bbox": [ + 502, + 598, + 507, + 613 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 610, + 463, + 623 + ], + "spans": [ + { + "bbox": [ + 105, + 610, + 152, + 623 + ], + "score": 1.0, + "content": "we can use", + "type": "text" + }, + { + "bbox": [ + 152, + 612, + 160, + 620 + ], + "score": 0.71, + "content": "n", + "type": "inline_equation" + }, + { + "bbox": [ + 160, + 610, + 273, + 623 + ], + "score": 1.0, + "content": "additional gates to compute", + "type": "text" + }, + { + "bbox": [ + 273, + 610, + 297, + 621 + ], + "score": 0.89, + "content": "\\mathbf { W } _ { 1 } \\mathbf { v }", + "type": "inline_equation" + }, + { + "bbox": [ + 298, + 610, + 347, + 623 + ], + "score": 1.0, + "content": "from ReLU", + "type": "text" + }, + { + "bbox": [ + 347, + 610, + 376, + 622 + ], + "score": 0.67, + "content": "\\left( \\mathbf { W } _ { 1 } \\mathbf { v } \\right)", + "type": "inline_equation" + }, + { + "bbox": [ + 377, + 610, + 395, + 623 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 396, + 610, + 459, + 622 + ], + "score": 0.59, + "content": "\\operatorname { R e L U } ( - \\mathbf { W } _ { 1 } \\mathbf { v } )", + "type": "inline_equation" + }, + { + "bbox": [ + 460, + 610, + 463, + 623 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 38 + } + ], + "index": 36.5 + }, + { + "type": "text", + "bbox": [ + 106, + 626, + 505, + 716 + ], + "lines": [ + { + "bbox": [ + 106, + 627, + 506, + 639 + ], + "spans": [ + { + "bbox": [ + 106, + 627, + 227, + 639 + ], + "score": 1.0, + "content": "Repeat the process for layer", + "type": "text" + }, + { + "bbox": [ + 227, + 627, + 271, + 638 + ], + "score": 0.92, + "content": "2 , 3 , \\ldots , L", + "type": "inline_equation" + }, + { + "bbox": [ + 272, + 627, + 353, + 639 + ], + "score": 1.0, + "content": "(for example, pass", + "type": "text" + }, + { + "bbox": [ + 353, + 627, + 378, + 638 + ], + "score": 0.9, + "content": "\\mathbf { W } _ { 1 } \\mathbf { v }", + "type": "inline_equation" + }, + { + "bbox": [ + 378, + 627, + 397, + 639 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 397, + 627, + 429, + 638 + ], + "score": 0.91, + "content": "- \\mathbf { W } _ { 1 } \\mathbf { v }", + "type": "inline_equation" + }, + { + "bbox": [ + 429, + 627, + 506, + 639 + ], + "score": 1.0, + "content": "to the circuit that", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 106, + 638, + 506, + 650 + ], + "spans": [ + { + "bbox": [ + 106, + 638, + 146, + 650 + ], + "score": 1.0, + "content": "computes", + "type": "text" + }, + { + "bbox": [ + 146, + 638, + 170, + 649 + ], + "score": 0.91, + "content": "\\mathbf { W } _ { \\mathrm { 2 } } \\mathbf { x }", + "type": "inline_equation" + }, + { + "bbox": [ + 171, + 638, + 231, + 650 + ], + "score": 1.0, + "content": "for an arbitrary", + "type": "text" + }, + { + "bbox": [ + 231, + 640, + 239, + 648 + ], + "score": 0.51, + "content": "\\mathbf { x }", + "type": "inline_equation" + }, + { + "bbox": [ + 240, + 638, + 506, + 650 + ], + "score": 1.0, + "content": "on layer 2). Overall we need to double the circuits that computes all", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 649, + 507, + 662 + ], + "spans": [ + { + "bbox": [ + 105, + 649, + 225, + 662 + ], + "score": 1.0, + "content": "the activations of the network", + "type": "text" + }, + { + "bbox": [ + 225, + 649, + 425, + 661 + ], + "score": 0.76, + "content": "\\operatorname { R e L U } ( \\mathbf { \\dot { W } } _ { 1 } \\mathbf { v } ) , \\dots , \\operatorname { R e L U } ( \\mathbf { W } _ { L } \\dots \\operatorname { R e L U } ( \\mathbf { W } _ { 1 } \\mathbf { v } ) )", + "type": "inline_equation" + }, + { + "bbox": [ + 426, + 649, + 469, + 662 + ], + "score": 1.0, + "content": ", requiring", + "type": "text" + }, + { + "bbox": [ + 469, + 649, + 480, + 659 + ], + "score": 0.64, + "content": "2 s", + "type": "inline_equation" + }, + { + "bbox": [ + 481, + 649, + 507, + 662 + ], + "score": 1.0, + "content": "gates.", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 104, + 658, + 507, + 675 + ], + "spans": [ + { + "bbox": [ + 104, + 658, + 164, + 675 + ], + "score": 1.0, + "content": "We also need", + "type": "text" + }, + { + "bbox": [ + 164, + 662, + 172, + 670 + ], + "score": 0.7, + "content": "n", + "type": "inline_equation" + }, + { + "bbox": [ + 172, + 658, + 507, + 675 + ], + "score": 1.0, + "content": "additional gates per layer to compute the negation of the input to that layer (e.g.", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 671, + 505, + 683 + ], + "spans": [ + { + "bbox": [ + 105, + 671, + 152, + 683 + ], + "score": 1.0, + "content": "computing", + "type": "text" + }, + { + "bbox": [ + 153, + 672, + 168, + 681 + ], + "score": 0.59, + "content": "- \\mathbf { v }", + "type": "inline_equation" + }, + { + "bbox": [ + 168, + 671, + 222, + 683 + ], + "score": 1.0, + "content": "from v), and", + "type": "text" + }, + { + "bbox": [ + 223, + 673, + 230, + 681 + ], + "score": 0.76, + "content": "n", + "type": "inline_equation" + }, + { + "bbox": [ + 230, + 671, + 505, + 683 + ], + "score": 1.0, + "content": "additional gates per layer to subtract the output of the ReLU circuit", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 104, + 681, + 506, + 695 + ], + "spans": [ + { + "bbox": [ + 104, + 681, + 174, + 695 + ], + "score": 1.0, + "content": "(e.g. computing", + "type": "text" + }, + { + "bbox": [ + 174, + 682, + 199, + 693 + ], + "score": 0.9, + "content": "\\mathbf { W } _ { 1 } \\mathbf { v }", + "type": "inline_equation" + }, + { + "bbox": [ + 199, + 681, + 223, + 695 + ], + "score": 1.0, + "content": "from", + "type": "text" + }, + { + "bbox": [ + 223, + 681, + 280, + 693 + ], + "score": 0.66, + "content": "\\mathrm { R e L U } ( \\mathbf { W } _ { 1 } \\mathbf { v } )", + "type": "inline_equation" + }, + { + "bbox": [ + 280, + 681, + 301, + 695 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 301, + 682, + 365, + 693 + ], + "score": 0.86, + "content": "\\operatorname { R e L U } ( - \\mathbf { W } _ { 1 } \\mathbf { v } )", + "type": "inline_equation" + }, + { + "bbox": [ + 366, + 681, + 506, + 695 + ], + "score": 1.0, + "content": ".) Therefore we can construct an", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 106, + 693, + 505, + 705 + ], + "spans": [ + { + "bbox": [ + 106, + 693, + 310, + 705 + ], + "score": 1.0, + "content": "arithmetic circuit augmented with ReLU gates with", + "type": "text" + }, + { + "bbox": [ + 311, + 693, + 345, + 703 + ], + "score": 0.9, + "content": "2 s + 2 L", + "type": "inline_equation" + }, + { + "bbox": [ + 345, + 693, + 505, + 705 + ], + "score": 1.0, + "content": "total gates that computes the activations", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 703, + 332, + 717 + ], + "spans": [ + { + "bbox": [ + 105, + 703, + 227, + 717 + ], + "score": 1.0, + "content": "of the network without ReLU", + "type": "text" + }, + { + "bbox": [ + 227, + 704, + 328, + 715 + ], + "score": 0.91, + "content": "\\mathbf { W } _ { 1 } \\mathbf { v } , \\dots , \\mathbf { W } _ { L } \\dots \\mathbf { W } _ { 1 } \\mathbf { v }", + "type": "inline_equation" + }, + { + "bbox": [ + 329, + 703, + 332, + 717 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 46 + } + ], + "index": 42.5 + } + ], + "page_idx": 41, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 107, + 27, + 293, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 26, + 294, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 26, + 294, + 38 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2020", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 300, + 751, + 311, + 760 + ], + "lines": [ + { + "bbox": [ + 298, + 750, + 312, + 763 + ], + "spans": [ + { + "bbox": [ + 298, + 750, + 312, + 763 + ], + "score": 1.0, + "content": "42", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 495, + 721, + 504, + 731 + ], + "lines": [ + { + "bbox": [ + 496, + 723, + 504, + 730 + ], + "spans": [ + { + "bbox": [ + 496, + 723, + 504, + 730 + ], + "score": 0.998, + "content": "□", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 81, + 505, + 127 + ], + "lines": [ + { + "bbox": [ + 104, + 81, + 506, + 95 + ], + "spans": [ + { + "bbox": [ + 104, + 81, + 222, + 95 + ], + "score": 1.0, + "content": "Just as earlier, the factor of", + "type": "text" + }, + { + "bbox": [ + 222, + 81, + 237, + 93 + ], + "score": 0.9, + "content": "\\sqrt { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 237, + 81, + 506, + 95 + ], + "score": 1.0, + "content": "can be multiplied through any diagonal matrix. Also, these two", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 94, + 505, + 106 + ], + "spans": [ + { + "bbox": [ + 106, + 94, + 388, + 106 + ], + "score": 1.0, + "content": "orthogonal butterfly factor matrices can be folded into the the rightmost", + "type": "text" + }, + { + "bbox": [ + 389, + 94, + 399, + 104 + ], + "score": 0.32, + "content": "\\mathbf { R }", + "type": "inline_equation" + }, + { + "bbox": [ + 399, + 94, + 505, + 106 + ], + "score": 1.0, + "content": "matrix (the decomposition", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 104, + 102, + 506, + 118 + ], + "spans": [ + { + "bbox": [ + 104, + 102, + 117, + 118 + ], + "score": 1.0, + "content": "of", + "type": "text" + }, + { + "bbox": [ + 118, + 105, + 128, + 115 + ], + "score": 0.33, + "content": "\\mathbf { K }", + "type": "inline_equation" + }, + { + "bbox": [ + 128, + 102, + 396, + 118 + ], + "score": 1.0, + "content": "above does not use these two, rightmost butterfly factors). Hence,", + "type": "text" + }, + { + "bbox": [ + 396, + 104, + 490, + 117 + ], + "score": 0.92, + "content": "\\begin{array} { r } { \\sum _ { i = 1 } ^ { m } \\mathbf { A } _ { i } \\in ( \\mathcal { O B } \\dot { B } ) _ { 4 e } ^ { m w } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 472, + 106, + 506, + 117 + ], + "score": 1.0, + "content": ")mw4e , as", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 115, + 505, + 126 + ], + "spans": [ + { + "bbox": [ + 105, + 115, + 140, + 126 + ], + "score": 1.0, + "content": "desired.", + "type": "text" + }, + { + "bbox": [ + 494, + 115, + 505, + 126 + ], + "score": 1.0, + "content": "□", + "type": "text" + } + ], + "index": 3 + } + ], + "index": 1.5, + "bbox_fs": [ + 104, + 81, + 506, + 126 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 141, + 243, + 153 + ], + "lines": [ + { + "bbox": [ + 106, + 140, + 243, + 154 + ], + "spans": [ + { + "bbox": [ + 106, + 140, + 243, + 154 + ], + "score": 1.0, + "content": "K.3.4 ARITHMETIC CIRCUITS", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 4 + }, + { + "type": "text", + "bbox": [ + 106, + 161, + 504, + 183 + ], + "lines": [ + { + "bbox": [ + 105, + 160, + 505, + 174 + ], + "spans": [ + { + "bbox": [ + 105, + 160, + 505, + 174 + ], + "score": 1.0, + "content": "Just as in Theorem 1, we can use the sparsity result in Lemma K.10 to place matrices with low-depth", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 172, + 432, + 185 + ], + "spans": [ + { + "bbox": [ + 106, + 172, + 366, + 185 + ], + "score": 1.0, + "content": "(linear) arithmetic circuits for matrix vector multiplication in the", + "type": "text" + }, + { + "bbox": [ + 366, + 172, + 389, + 182 + ], + "score": 0.56, + "content": "\\ O B B", + "type": "inline_equation" + }, + { + "bbox": [ + 389, + 172, + 432, + 185 + ], + "score": 1.0, + "content": "hierarchy.", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 5.5, + "bbox_fs": [ + 105, + 160, + 505, + 185 + ] + }, + { + "type": "list", + "bbox": [ + 107, + 187, + 505, + 226 + ], + "lines": [ + { + "bbox": [ + 106, + 186, + 505, + 201 + ], + "spans": [ + { + "bbox": [ + 106, + 186, + 231, + 201 + ], + "score": 1.0, + "content": "Corollary K.12. Let M be an", + "type": "text" + }, + { + "bbox": [ + 232, + 189, + 257, + 198 + ], + "score": 0.88, + "content": "n \\times n", + "type": "inline_equation" + }, + { + "bbox": [ + 258, + 186, + 505, + 201 + ], + "score": 1.0, + "content": "matrix such that matrix-vector multiplication of M times an", + "type": "text" + } + ], + "index": 7, + "is_list_start_line": true + }, + { + "bbox": [ + 105, + 197, + 505, + 212 + ], + "spans": [ + { + "bbox": [ + 105, + 197, + 406, + 212 + ], + "score": 1.0, + "content": "arbitrary vector v can be represented as a be a linear arithmetic circuit", + "type": "text" + }, + { + "bbox": [ + 407, + 199, + 416, + 208 + ], + "score": 0.78, + "content": "C", + "type": "inline_equation" + }, + { + "bbox": [ + 416, + 197, + 505, + 212 + ], + "score": 1.0, + "content": "comprised of s gates", + "type": "text" + } + ], + "index": 8, + "is_list_start_line": true + }, + { + "bbox": [ + 104, + 208, + 360, + 227 + ], + "spans": [ + { + "bbox": [ + 104, + 208, + 251, + 226 + ], + "score": 1.0, + "content": "(including inputs) and having depth", + "type": "text" + }, + { + "bbox": [ + 251, + 212, + 258, + 222 + ], + "score": 0.34, + "content": "d .", + "type": "inline_equation" + }, + { + "bbox": [ + 258, + 208, + 286, + 226 + ], + "score": 1.0, + "content": ". Then,", + "type": "text" + }, + { + "bbox": [ + 287, + 209, + 360, + 227 + ], + "score": 0.93, + "content": "\\mathbf { M } \\in ( \\mathcal { O } B B ) _ { O ( \\frac { s } { n } ) } ^ { O ( d ) }", + "type": "inline_equation" + } + ], + "index": 9, + "is_list_start_line": true, + "is_list_end_line": true + }, + { + "bbox": [ + 360, + 216, + 364, + 228 + ], + "spans": [ + { + "bbox": [ + 360, + 216, + 364, + 228 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 10, + "is_list_end_line": true + } + ], + "index": 8.5, + "bbox_fs": [ + 104, + 186, + 505, + 228 + ] + }, + { + "type": "text", + "bbox": [ + 105, + 240, + 504, + 263 + ], + "lines": [ + { + "bbox": [ + 105, + 240, + 506, + 253 + ], + "spans": [ + { + "bbox": [ + 105, + 240, + 506, + 253 + ], + "score": 1.0, + "content": "Proof. We use the construction given in the proof of Theorem 1. Corollaries K.9 and K.4 allow us to", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 251, + 504, + 264 + ], + "spans": [ + { + "bbox": [ + 105, + 251, + 321, + 264 + ], + "score": 1.0, + "content": "recover the same width and expansion factor with the", + "type": "text" + }, + { + "bbox": [ + 322, + 252, + 345, + 262 + ], + "score": 0.26, + "content": "o s B", + "type": "inline_equation" + }, + { + "bbox": [ + 345, + 251, + 387, + 264 + ], + "score": 1.0, + "content": "hierarchy.", + "type": "text" + }, + { + "bbox": [ + 496, + 253, + 504, + 261 + ], + "score": 0.999, + "content": "□", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 11.5, + "bbox_fs": [ + 105, + 240, + 506, + 264 + ] + }, + { + "type": "title", + "bbox": [ + 106, + 280, + 415, + 293 + ], + "lines": [ + { + "bbox": [ + 104, + 279, + 417, + 296 + ], + "spans": [ + { + "bbox": [ + 104, + 279, + 417, + 296 + ], + "score": 1.0, + "content": "L RELU NETWORK WITH STRUCTURED WEIGHT MATRICES", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 13 + }, + { + "type": "text", + "bbox": [ + 107, + 305, + 505, + 371 + ], + "lines": [ + { + "bbox": [ + 105, + 305, + 505, + 318 + ], + "spans": [ + { + "bbox": [ + 105, + 305, + 505, + 318 + ], + "score": 1.0, + "content": "We show that for any neural network with ReLU nonlinearities and whose weight matrices have", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 317, + 505, + 329 + ], + "spans": [ + { + "bbox": [ + 106, + 317, + 505, + 329 + ], + "score": 1.0, + "content": "arithmetic circuits with few gates, its linear network counterpart (obtained by removing all the", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 327, + 506, + 340 + ], + "spans": [ + { + "bbox": [ + 105, + 327, + 506, + 340 + ], + "score": 1.0, + "content": "ReLU’s) also has an arithmetic circuit with not too many more gates. This implies that in trying to", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 338, + 505, + 351 + ], + "spans": [ + { + "bbox": [ + 105, + 338, + 505, + 351 + ], + "score": 1.0, + "content": "find the smallest arithmetic circuit augmented with ReLU gates to represent a ReLU network, one", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 349, + 505, + 361 + ], + "spans": [ + { + "bbox": [ + 106, + 349, + 505, + 361 + ], + "score": 1.0, + "content": "might as well try to find the smallest arithmetic circuits that represent the matrix-vector multiplication", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 361, + 198, + 372 + ], + "spans": [ + { + "bbox": [ + 106, + 361, + 198, + 372 + ], + "score": 1.0, + "content": "of each weight matrix.", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 16.5, + "bbox_fs": [ + 105, + 305, + 506, + 372 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 375, + 504, + 398 + ], + "lines": [ + { + "bbox": [ + 105, + 374, + 505, + 389 + ], + "spans": [ + { + "bbox": [ + 105, + 374, + 383, + 389 + ], + "score": 1.0, + "content": "Proposition 2. Consider a neural network architecture consisting of", + "type": "text" + }, + { + "bbox": [ + 384, + 376, + 392, + 385 + ], + "score": 0.42, + "content": "L", + "type": "inline_equation" + }, + { + "bbox": [ + 392, + 374, + 505, + 389 + ], + "score": 1.0, + "content": "layers with weight matrices", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 384, + 342, + 401 + ], + "spans": [ + { + "bbox": [ + 106, + 387, + 198, + 398 + ], + "score": 0.9, + "content": "\\mathbf { W } _ { 1 } , \\dots , \\mathbf { W } _ { L } \\in \\mathbb { F } ^ { n \\times n }", + "type": "inline_equation" + }, + { + "bbox": [ + 199, + 384, + 342, + 401 + ], + "score": 1.0, + "content": "and ReLU nonlinearity in between.", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 20.5, + "bbox_fs": [ + 105, + 374, + 505, + 401 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 403, + 505, + 450 + ], + "lines": [ + { + "bbox": [ + 106, + 404, + 506, + 416 + ], + "spans": [ + { + "bbox": [ + 106, + 404, + 291, + 416 + ], + "score": 1.0, + "content": "Suppose that matrix-vector multiplication of", + "type": "text" + }, + { + "bbox": [ + 291, + 404, + 308, + 415 + ], + "score": 0.89, + "content": "\\mathbf { W } _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 308, + 404, + 506, + 416 + ], + "score": 1.0, + "content": "times an arbitrary vector v can be represented", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 415, + 505, + 427 + ], + "spans": [ + { + "bbox": [ + 105, + 415, + 252, + 427 + ], + "score": 1.0, + "content": "as a linear arithmetic circuit with", + "type": "text" + }, + { + "bbox": [ + 252, + 416, + 262, + 426 + ], + "score": 0.82, + "content": "s _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 262, + 415, + 505, + 427 + ], + "score": 1.0, + "content": "gates (including inputs). Then there exists an arithmetic", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 101, + 421, + 509, + 445 + ], + "spans": [ + { + "bbox": [ + 101, + 421, + 509, + 445 + ], + "score": 1.0, + "content": "circuit augmented with ReLU gates with PLi=1 si + Ln total gates that computes the output", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 437, + 424, + 452 + ], + "spans": [ + { + "bbox": [ + 105, + 437, + 133, + 452 + ], + "score": 1.0, + "content": "ReLU", + "type": "text" + }, + { + "bbox": [ + 134, + 438, + 233, + 451 + ], + "score": 0.85, + "content": "( \\mathbf { W } _ { L } ( \\dots \\mathrm { R e L U } ( \\mathbf { W } _ { 1 } \\mathbf { v } ) ) )", + "type": "inline_equation" + }, + { + "bbox": [ + 233, + 437, + 424, + 452 + ], + "score": 1.0, + "content": ") of the network for an arbitrary input vector v.", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 23.5, + "bbox_fs": [ + 101, + 404, + 509, + 452 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 455, + 505, + 500 + ], + "lines": [ + { + "bbox": [ + 106, + 455, + 505, + 467 + ], + "spans": [ + { + "bbox": [ + 106, + 455, + 505, + 467 + ], + "score": 1.0, + "content": "Conversely, if there is an arithmetic circuit augmented with ReLU gates with s total gates that", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 466, + 505, + 479 + ], + "spans": [ + { + "bbox": [ + 105, + 466, + 294, + 479 + ], + "score": 1.0, + "content": "computes all the activations of the network ReL", + "type": "text" + }, + { + "bbox": [ + 294, + 466, + 476, + 478 + ], + "score": 0.6, + "content": "\\operatorname { \\mathrm { . U } } ( \\mathbf { W } _ { 1 } \\mathbf { v } ) , \\ldots , \\operatorname { R e L U } ( \\mathbf { W } _ { L } \\ldots \\operatorname { \\mathrm { . R e L U } } ( \\mathbf { W } _ { 1 } \\mathbf { v } ) )", + "type": "inline_equation" + }, + { + "bbox": [ + 477, + 466, + 505, + 479 + ], + "score": 1.0, + "content": "for an", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 478, + 504, + 488 + ], + "spans": [ + { + "bbox": [ + 106, + 478, + 463, + 488 + ], + "score": 1.0, + "content": "arbitrary input v, then there exists an arithmetic circuit augmented with ReLU gates with", + "type": "text" + }, + { + "bbox": [ + 463, + 478, + 504, + 488 + ], + "score": 0.89, + "content": "2 s + 2 L n", + "type": "inline_equation" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 488, + 492, + 501 + ], + "spans": [ + { + "bbox": [ + 105, + 488, + 388, + 501 + ], + "score": 1.0, + "content": "total gates that computes the activations of the network without ReLU", + "type": "text" + }, + { + "bbox": [ + 388, + 488, + 488, + 500 + ], + "score": 0.87, + "content": "\\mathbf { W } _ { 1 } \\mathbf { v } , \\dots , \\mathbf { W } _ { L } \\dots \\mathbf { W } _ { 1 } \\mathbf { v }", + "type": "inline_equation" + }, + { + "bbox": [ + 489, + 488, + 492, + 501 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 27.5, + "bbox_fs": [ + 105, + 455, + 505, + 501 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 514, + 505, + 572 + ], + "lines": [ + { + "bbox": [ + 105, + 514, + 506, + 527 + ], + "spans": [ + { + "bbox": [ + 105, + 514, + 374, + 527 + ], + "score": 1.0, + "content": "Proof of Proposition 2. To compute the output of the network Re", + "type": "text" + }, + { + "bbox": [ + 374, + 514, + 484, + 527 + ], + "score": 0.88, + "content": "\\operatorname { \\mathrm { . U } } ( \\mathbf { W } _ { L } ( \\dots \\operatorname { R e L U } ( \\mathbf { W } _ { 1 } \\mathbf { v } ) ) )", + "type": "inline_equation" + }, + { + "bbox": [ + 484, + 514, + 506, + 527 + ], + "score": 1.0, + "content": "), we", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 525, + 507, + 538 + ], + "spans": [ + { + "bbox": [ + 105, + 525, + 267, + 538 + ], + "score": 1.0, + "content": "first compute the matrix-vector product", + "type": "text" + }, + { + "bbox": [ + 268, + 525, + 291, + 536 + ], + "score": 0.9, + "content": "\\mathbf { W } _ { 1 } \\mathbf { v }", + "type": "inline_equation" + }, + { + "bbox": [ + 292, + 525, + 407, + 538 + ], + "score": 1.0, + "content": "with an arithmetic circuit of", + "type": "text" + }, + { + "bbox": [ + 408, + 527, + 418, + 536 + ], + "score": 0.85, + "content": "s _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 419, + 525, + 507, + 538 + ], + "score": 1.0, + "content": "gates by assumption,", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 536, + 506, + 549 + ], + "spans": [ + { + "bbox": [ + 105, + 536, + 139, + 549 + ], + "score": 1.0, + "content": "and use", + "type": "text" + }, + { + "bbox": [ + 140, + 538, + 147, + 546 + ], + "score": 0.72, + "content": "n", + "type": "inline_equation" + }, + { + "bbox": [ + 147, + 536, + 506, + 549 + ], + "score": 1.0, + "content": "other ReLU gates to compute the pointwise ReLU. Then we repeat the process for layer", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 547, + 506, + 559 + ], + "spans": [ + { + "bbox": [ + 106, + 547, + 151, + 559 + ], + "score": 0.91, + "content": "2 , 3 , \\ldots , L", + "type": "inline_equation" + }, + { + "bbox": [ + 151, + 547, + 277, + 559 + ], + "score": 1.0, + "content": ", using the arithmetic circuits of", + "type": "text" + }, + { + "bbox": [ + 277, + 547, + 334, + 559 + ], + "score": 0.92, + "content": "\\mathbf { W } _ { 1 } , \\dots , \\mathbf { W } _ { L }", + "type": "inline_equation" + }, + { + "bbox": [ + 335, + 547, + 352, + 559 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 352, + 547, + 366, + 558 + ], + "score": 0.79, + "content": "L n", + "type": "inline_equation" + }, + { + "bbox": [ + 367, + 547, + 506, + 559 + ], + "score": 1.0, + "content": "additional gates for ReLU. In total", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 103, + 555, + 476, + 577 + ], + "spans": [ + { + "bbox": [ + 103, + 555, + 366, + 577 + ], + "score": 1.0, + "content": "we obtain an arithmetic circuit augmented with ReLU gates with", + "type": "text" + }, + { + "bbox": [ + 367, + 558, + 426, + 573 + ], + "score": 0.92, + "content": "\\textstyle \\sum _ { i = 1 } ^ { L } s _ { i } + L n", + "type": "inline_equation" + }, + { + "bbox": [ + 427, + 555, + 476, + 577 + ], + "score": 1.0, + "content": "total gates.", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 32, + "bbox_fs": [ + 103, + 514, + 507, + 577 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 577, + 505, + 622 + ], + "lines": [ + { + "bbox": [ + 105, + 576, + 505, + 590 + ], + "spans": [ + { + "bbox": [ + 105, + 576, + 505, + 590 + ], + "score": 1.0, + "content": "Conversely, to build an arithmetic circuit augmented with ReLU gates to compute", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 588, + 506, + 601 + ], + "spans": [ + { + "bbox": [ + 106, + 588, + 207, + 599 + ], + "score": 0.9, + "content": "\\mathbf { W } _ { 1 } \\mathbf { v } , \\dots , \\mathbf { W } _ { L } \\dots \\mathbf { W } _ { 1 } \\mathbf { v }", + "type": "inline_equation" + }, + { + "bbox": [ + 208, + 588, + 243, + 601 + ], + "score": 1.0, + "content": ", we pass", + "type": "text" + }, + { + "bbox": [ + 243, + 590, + 251, + 598 + ], + "score": 0.39, + "content": "\\mathbf { v }", + "type": "inline_equation" + }, + { + "bbox": [ + 251, + 588, + 287, + 601 + ], + "score": 1.0, + "content": "and then", + "type": "text" + }, + { + "bbox": [ + 287, + 589, + 302, + 599 + ], + "score": 0.73, + "content": "- \\mathbf { v }", + "type": "inline_equation" + }, + { + "bbox": [ + 303, + 588, + 433, + 601 + ], + "score": 1.0, + "content": "through the circuit that computes", + "type": "text" + }, + { + "bbox": [ + 433, + 588, + 489, + 599 + ], + "score": 0.58, + "content": "\\mathrm { R e L U } ( \\mathbf { W } _ { 1 } \\mathbf { x } )", + "type": "inline_equation" + }, + { + "bbox": [ + 490, + 588, + 506, + 601 + ], + "score": 1.0, + "content": "for", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 598, + 507, + 613 + ], + "spans": [ + { + "bbox": [ + 105, + 598, + 156, + 613 + ], + "score": 1.0, + "content": "an arbitrary", + "type": "text" + }, + { + "bbox": [ + 156, + 601, + 164, + 609 + ], + "score": 0.59, + "content": "\\mathbf { x }", + "type": "inline_equation" + }, + { + "bbox": [ + 164, + 598, + 217, + 613 + ], + "score": 1.0, + "content": "to get ReLU", + "type": "text" + }, + { + "bbox": [ + 217, + 599, + 247, + 611 + ], + "score": 0.63, + "content": "\\left( \\mathbf { W } _ { 1 } \\mathbf { v } \\right)", + "type": "inline_equation" + }, + { + "bbox": [ + 247, + 598, + 293, + 613 + ], + "score": 1.0, + "content": "and ReLU", + "type": "text" + }, + { + "bbox": [ + 294, + 599, + 330, + 611 + ], + "score": 0.62, + "content": "\\left( - \\mathbf { W } _ { 1 } \\mathbf { v } \\right)", + "type": "inline_equation" + }, + { + "bbox": [ + 330, + 598, + 384, + 613 + ], + "score": 1.0, + "content": ". Noting that", + "type": "text" + }, + { + "bbox": [ + 384, + 599, + 502, + 611 + ], + "score": 0.88, + "content": "x = \\mathrm { R e L U } ( x ) - \\mathrm { R e L U } ( - x )", + "type": "inline_equation" + }, + { + "bbox": [ + 502, + 598, + 507, + 613 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 610, + 463, + 623 + ], + "spans": [ + { + "bbox": [ + 105, + 610, + 152, + 623 + ], + "score": 1.0, + "content": "we can use", + "type": "text" + }, + { + "bbox": [ + 152, + 612, + 160, + 620 + ], + "score": 0.71, + "content": "n", + "type": "inline_equation" + }, + { + "bbox": [ + 160, + 610, + 273, + 623 + ], + "score": 1.0, + "content": "additional gates to compute", + "type": "text" + }, + { + "bbox": [ + 273, + 610, + 297, + 621 + ], + "score": 0.89, + "content": "\\mathbf { W } _ { 1 } \\mathbf { v }", + "type": "inline_equation" + }, + { + "bbox": [ + 298, + 610, + 347, + 623 + ], + "score": 1.0, + "content": "from ReLU", + "type": "text" + }, + { + "bbox": [ + 347, + 610, + 376, + 622 + ], + "score": 0.67, + "content": "\\left( \\mathbf { W } _ { 1 } \\mathbf { v } \\right)", + "type": "inline_equation" + }, + { + "bbox": [ + 377, + 610, + 395, + 623 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 396, + 610, + 459, + 622 + ], + "score": 0.59, + "content": "\\operatorname { R e L U } ( - \\mathbf { W } _ { 1 } \\mathbf { v } )", + "type": "inline_equation" + }, + { + "bbox": [ + 460, + 610, + 463, + 623 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 38 + } + ], + "index": 36.5, + "bbox_fs": [ + 105, + 576, + 507, + 623 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 626, + 505, + 716 + ], + "lines": [ + { + "bbox": [ + 106, + 627, + 506, + 639 + ], + "spans": [ + { + "bbox": [ + 106, + 627, + 227, + 639 + ], + "score": 1.0, + "content": "Repeat the process for layer", + "type": "text" + }, + { + "bbox": [ + 227, + 627, + 271, + 638 + ], + "score": 0.92, + "content": "2 , 3 , \\ldots , L", + "type": "inline_equation" + }, + { + "bbox": [ + 272, + 627, + 353, + 639 + ], + "score": 1.0, + "content": "(for example, pass", + "type": "text" + }, + { + "bbox": [ + 353, + 627, + 378, + 638 + ], + "score": 0.9, + "content": "\\mathbf { W } _ { 1 } \\mathbf { v }", + "type": "inline_equation" + }, + { + "bbox": [ + 378, + 627, + 397, + 639 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 397, + 627, + 429, + 638 + ], + "score": 0.91, + "content": "- \\mathbf { W } _ { 1 } \\mathbf { v }", + "type": "inline_equation" + }, + { + "bbox": [ + 429, + 627, + 506, + 639 + ], + "score": 1.0, + "content": "to the circuit that", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 106, + 638, + 506, + 650 + ], + "spans": [ + { + "bbox": [ + 106, + 638, + 146, + 650 + ], + "score": 1.0, + "content": "computes", + "type": "text" + }, + { + "bbox": [ + 146, + 638, + 170, + 649 + ], + "score": 0.91, + "content": "\\mathbf { W } _ { \\mathrm { 2 } } \\mathbf { x }", + "type": "inline_equation" + }, + { + "bbox": [ + 171, + 638, + 231, + 650 + ], + "score": 1.0, + "content": "for an arbitrary", + "type": "text" + }, + { + "bbox": [ + 231, + 640, + 239, + 648 + ], + "score": 0.51, + "content": "\\mathbf { x }", + "type": "inline_equation" + }, + { + "bbox": [ + 240, + 638, + 506, + 650 + ], + "score": 1.0, + "content": "on layer 2). Overall we need to double the circuits that computes all", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 649, + 507, + 662 + ], + "spans": [ + { + "bbox": [ + 105, + 649, + 225, + 662 + ], + "score": 1.0, + "content": "the activations of the network", + "type": "text" + }, + { + "bbox": [ + 225, + 649, + 425, + 661 + ], + "score": 0.76, + "content": "\\operatorname { R e L U } ( \\mathbf { \\dot { W } } _ { 1 } \\mathbf { v } ) , \\dots , \\operatorname { R e L U } ( \\mathbf { W } _ { L } \\dots \\operatorname { R e L U } ( \\mathbf { W } _ { 1 } \\mathbf { v } ) )", + "type": "inline_equation" + }, + { + "bbox": [ + 426, + 649, + 469, + 662 + ], + "score": 1.0, + "content": ", requiring", + "type": "text" + }, + { + "bbox": [ + 469, + 649, + 480, + 659 + ], + "score": 0.64, + "content": "2 s", + "type": "inline_equation" + }, + { + "bbox": [ + 481, + 649, + 507, + 662 + ], + "score": 1.0, + "content": "gates.", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 104, + 658, + 507, + 675 + ], + "spans": [ + { + "bbox": [ + 104, + 658, + 164, + 675 + ], + "score": 1.0, + "content": "We also need", + "type": "text" + }, + { + "bbox": [ + 164, + 662, + 172, + 670 + ], + "score": 0.7, + "content": "n", + "type": "inline_equation" + }, + { + "bbox": [ + 172, + 658, + 507, + 675 + ], + "score": 1.0, + "content": "additional gates per layer to compute the negation of the input to that layer (e.g.", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 671, + 505, + 683 + ], + "spans": [ + { + "bbox": [ + 105, + 671, + 152, + 683 + ], + "score": 1.0, + "content": "computing", + "type": "text" + }, + { + "bbox": [ + 153, + 672, + 168, + 681 + ], + "score": 0.59, + "content": "- \\mathbf { v }", + "type": "inline_equation" + }, + { + "bbox": [ + 168, + 671, + 222, + 683 + ], + "score": 1.0, + "content": "from v), and", + "type": "text" + }, + { + "bbox": [ + 223, + 673, + 230, + 681 + ], + "score": 0.76, + "content": "n", + "type": "inline_equation" + }, + { + "bbox": [ + 230, + 671, + 505, + 683 + ], + "score": 1.0, + "content": "additional gates per layer to subtract the output of the ReLU circuit", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 104, + 681, + 506, + 695 + ], + "spans": [ + { + "bbox": [ + 104, + 681, + 174, + 695 + ], + "score": 1.0, + "content": "(e.g. computing", + "type": "text" + }, + { + "bbox": [ + 174, + 682, + 199, + 693 + ], + "score": 0.9, + "content": "\\mathbf { W } _ { 1 } \\mathbf { v }", + "type": "inline_equation" + }, + { + "bbox": [ + 199, + 681, + 223, + 695 + ], + "score": 1.0, + "content": "from", + "type": "text" + }, + { + "bbox": [ + 223, + 681, + 280, + 693 + ], + "score": 0.66, + "content": "\\mathrm { R e L U } ( \\mathbf { W } _ { 1 } \\mathbf { v } )", + "type": "inline_equation" + }, + { + "bbox": [ + 280, + 681, + 301, + 695 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 301, + 682, + 365, + 693 + ], + "score": 0.86, + "content": "\\operatorname { R e L U } ( - \\mathbf { W } _ { 1 } \\mathbf { v } )", + "type": "inline_equation" + }, + { + "bbox": [ + 366, + 681, + 506, + 695 + ], + "score": 1.0, + "content": ".) Therefore we can construct an", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 106, + 693, + 505, + 705 + ], + "spans": [ + { + "bbox": [ + 106, + 693, + 310, + 705 + ], + "score": 1.0, + "content": "arithmetic circuit augmented with ReLU gates with", + "type": "text" + }, + { + "bbox": [ + 311, + 693, + 345, + 703 + ], + "score": 0.9, + "content": "2 s + 2 L", + "type": "inline_equation" + }, + { + "bbox": [ + 345, + 693, + 505, + 705 + ], + "score": 1.0, + "content": "total gates that computes the activations", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 703, + 332, + 717 + ], + "spans": [ + { + "bbox": [ + 105, + 703, + 227, + 717 + ], + "score": 1.0, + "content": "of the network without ReLU", + "type": "text" + }, + { + "bbox": [ + 227, + 704, + 328, + 715 + ], + "score": 0.91, + "content": "\\mathbf { W } _ { 1 } \\mathbf { v } , \\dots , \\mathbf { W } _ { L } \\dots \\mathbf { W } _ { 1 } \\mathbf { v }", + "type": "inline_equation" + }, + { + "bbox": [ + 329, + 703, + 332, + 717 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 46 + } + ], + "index": 42.5, + "bbox_fs": [ + 104, + 627, + 507, + 717 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 105, + 82, + 504, + 105 + ], + "lines": [ + { + "bbox": [ + 105, + 82, + 505, + 95 + ], + "spans": [ + { + "bbox": [ + 105, + 82, + 505, + 95 + ], + "score": 1.0, + "content": "We now prove an asymptotic bound on the VC dimension of a ReLU network whose weight matrices", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 93, + 356, + 106 + ], + "spans": [ + { + "bbox": [ + 105, + 93, + 356, + 106 + ], + "score": 1.0, + "content": "are kaleidoscope matrices with bounded width and expansion.", + "type": "text" + } + ], + "index": 1 + } + ], + "index": 0.5 + }, + { + "type": "text", + "bbox": [ + 106, + 106, + 505, + 151 + ], + "lines": [ + { + "bbox": [ + 105, + 106, + 506, + 119 + ], + "spans": [ + { + "bbox": [ + 105, + 106, + 188, + 119 + ], + "score": 1.0, + "content": "Proposition 3. Let", + "type": "text" + }, + { + "bbox": [ + 188, + 107, + 198, + 117 + ], + "score": 0.73, + "content": "\\mathcal { F }", + "type": "inline_equation" + }, + { + "bbox": [ + 199, + 106, + 415, + 119 + ], + "score": 1.0, + "content": "be the class of ReLU neural networks consisting of", + "type": "text" + }, + { + "bbox": [ + 415, + 107, + 423, + 117 + ], + "score": 0.7, + "content": "L", + "type": "inline_equation" + }, + { + "bbox": [ + 424, + 106, + 506, + 119 + ], + "score": 1.0, + "content": "layers, where each", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 106, + 118, + 505, + 129 + ], + "spans": [ + { + "bbox": [ + 106, + 118, + 150, + 129 + ], + "score": 1.0, + "content": "layer is a", + "type": "text" + }, + { + "bbox": [ + 150, + 118, + 159, + 128 + ], + "score": 0.61, + "content": "K", + "type": "inline_equation" + }, + { + "bbox": [ + 159, + 118, + 416, + 129 + ], + "score": 1.0, + "content": "-matrix with width and expansion bounded by some constant", + "type": "text" + }, + { + "bbox": [ + 416, + 118, + 425, + 128 + ], + "score": 0.77, + "content": "C", + "type": "inline_equation" + }, + { + "bbox": [ + 425, + 118, + 505, + 129 + ], + "score": 1.0, + "content": ". Suppose that the", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 129, + 506, + 141 + ], + "spans": [ + { + "bbox": [ + 105, + 129, + 160, + 141 + ], + "score": 1.0, + "content": "network has", + "type": "text" + }, + { + "bbox": [ + 160, + 129, + 172, + 138 + ], + "score": 0.65, + "content": "W", + "type": "inline_equation" + }, + { + "bbox": [ + 173, + 129, + 286, + 141 + ], + "score": 1.0, + "content": "total parameters. Let sign", + "type": "text" + }, + { + "bbox": [ + 286, + 129, + 296, + 139 + ], + "score": 0.82, + "content": "\\mathcal { F }", + "type": "inline_equation" + }, + { + "bbox": [ + 297, + 129, + 506, + 141 + ], + "score": 1.0, + "content": "denote the corresponding classification functions:", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 107, + 140, + 357, + 153 + ], + "spans": [ + { + "bbox": [ + 107, + 140, + 211, + 152 + ], + "score": 0.92, + "content": "\\{ x \\mapsto \\operatorname { s i g n } f ( x ) : f \\in { \\mathcal { F } } \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 212, + 140, + 357, + 153 + ], + "score": 1.0, + "content": ". Then this class has VC dimension:", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 3.5 + }, + { + "type": "interline_equation", + "bbox": [ + 233, + 153, + 377, + 166 + ], + "lines": [ + { + "bbox": [ + 233, + 153, + 377, + 166 + ], + "spans": [ + { + "bbox": [ + 233, + 153, + 377, + 166 + ], + "score": 0.9, + "content": "\\operatorname { V C d i m } ( \\operatorname { s i g n } { \\mathcal { F } } ) = O ( L W \\log W ) .", + "type": "interline_equation", + "image_path": "b07fcddd39796f289207532c245eb6a0f0438676278f9347655be1f0eef21dc2.jpg" + } + ] + } + ], + "index": 6, + "virtual_lines": [ + { + "bbox": [ + 233, + 153, + 377, + 166 + ], + "spans": [], + "index": 6 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 172, + 505, + 206 + ], + "lines": [ + { + "bbox": [ + 105, + 172, + 506, + 186 + ], + "spans": [ + { + "bbox": [ + 105, + 172, + 506, + 186 + ], + "score": 1.0, + "content": "We leverage the result from Thomas et al. (2018) for the case where the entries of the weight matrices", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 184, + 506, + 196 + ], + "spans": [ + { + "bbox": [ + 105, + 184, + 506, + 196 + ], + "score": 1.0, + "content": "interact multiplicatively, but with polynomially bounded degrees. This proof is similar to the VC", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 195, + 469, + 207 + ], + "spans": [ + { + "bbox": [ + 106, + 195, + 469, + 207 + ], + "score": 1.0, + "content": "bound for ReLU networks whose weight matrices are butterfly matrices (Dao et al., 2019).", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 8 + }, + { + "type": "text", + "bbox": [ + 106, + 217, + 505, + 295 + ], + "lines": [ + { + "bbox": [ + 106, + 218, + 506, + 231 + ], + "spans": [ + { + "bbox": [ + 106, + 218, + 506, + 231 + ], + "score": 1.0, + "content": "Proof. To use Theorem 3 of Thomas et al. (2018), we simply need to check that the entries of the", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 229, + 506, + 243 + ], + "spans": [ + { + "bbox": [ + 105, + 229, + 366, + 243 + ], + "score": 1.0, + "content": "linear layer, as polynomials of the parameters, has degree at most", + "type": "text" + }, + { + "bbox": [ + 366, + 230, + 393, + 242 + ], + "score": 0.92, + "content": "c _ { 1 } m _ { l } ^ { c _ { 2 } }", + "type": "inline_equation" + }, + { + "bbox": [ + 393, + 229, + 506, + 243 + ], + "score": 1.0, + "content": "for some universal constant", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 240, + 506, + 253 + ], + "spans": [ + { + "bbox": [ + 106, + 241, + 151, + 252 + ], + "score": 0.9, + "content": "c _ { 1 } , c _ { 2 } > 0", + "type": "inline_equation" + }, + { + "bbox": [ + 152, + 240, + 184, + 253 + ], + "score": 1.0, + "content": ", where", + "type": "text" + }, + { + "bbox": [ + 184, + 242, + 197, + 251 + ], + "score": 0.84, + "content": "m _ { l }", + "type": "inline_equation" + }, + { + "bbox": [ + 198, + 240, + 313, + 253 + ], + "score": 1.0, + "content": "is the size of output of the", + "type": "text" + }, + { + "bbox": [ + 313, + 241, + 317, + 250 + ], + "score": 0.61, + "content": "l", + "type": "inline_equation" + }, + { + "bbox": [ + 318, + 240, + 506, + 253 + ], + "score": 1.0, + "content": "-th layer. If the network weight matrices are", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 250, + 504, + 265 + ], + "spans": [ + { + "bbox": [ + 105, + 250, + 465, + 265 + ], + "score": 1.0, + "content": "K-matrices with bounded width and expansion, each weight matrix is a product of at most", + "type": "text" + }, + { + "bbox": [ + 466, + 252, + 504, + 263 + ], + "score": 0.87, + "content": "c _ { 3 } \\log m _ { l }", + "type": "inline_equation" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 262, + 505, + 275 + ], + "spans": [ + { + "bbox": [ + 105, + 262, + 285, + 275 + ], + "score": 1.0, + "content": "sparse factors, for some universal constant", + "type": "text" + }, + { + "bbox": [ + 286, + 263, + 317, + 273 + ], + "score": 0.91, + "content": "c _ { 3 } > 0", + "type": "inline_equation" + }, + { + "bbox": [ + 317, + 262, + 505, + 275 + ], + "score": 1.0, + "content": ". This means that the degree is polynomially", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 274, + 505, + 285 + ], + "spans": [ + { + "bbox": [ + 106, + 274, + 505, + 285 + ], + "score": 1.0, + "content": "bounded, which satisfies the condition of the theorem. Therefore the VC dimension is bounded to be", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 284, + 278, + 296 + ], + "spans": [ + { + "bbox": [ + 106, + 284, + 278, + 296 + ], + "score": 1.0, + "content": "almost linear in the number of parameters:", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 13 + }, + { + "type": "interline_equation", + "bbox": [ + 233, + 297, + 378, + 310 + ], + "lines": [ + { + "bbox": [ + 233, + 297, + 378, + 310 + ], + "spans": [ + { + "bbox": [ + 233, + 297, + 378, + 310 + ], + "score": 0.91, + "content": "\\operatorname { V C d i m } ( \\operatorname { s i g n } { \\mathcal { F } } ) = O ( L W \\log W ) .", + "type": "interline_equation", + "image_path": "f76055338f318f1bde28f1efe8b39133be9d88513589e943c274bfb13b326720.jpg" + } + ] + } + ], + "index": 17, + "virtual_lines": [ + { + "bbox": [ + 233, + 297, + 378, + 310 + ], + "spans": [], + "index": 17 + } + ] + }, + { + "type": "title", + "bbox": [ + 108, + 338, + 288, + 351 + ], + "lines": [ + { + "bbox": [ + 105, + 337, + 290, + 352 + ], + "spans": [ + { + "bbox": [ + 105, + 337, + 290, + 352 + ], + "score": 1.0, + "content": "M ARITHMETIC CIRCUIT PRIMER", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 18 + }, + { + "type": "text", + "bbox": [ + 106, + 362, + 505, + 418 + ], + "lines": [ + { + "bbox": [ + 106, + 362, + 506, + 375 + ], + "spans": [ + { + "bbox": [ + 106, + 362, + 506, + 375 + ], + "score": 1.0, + "content": "We give a quick overview of arithmetic circuits. This is a model of computation that has been", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 372, + 505, + 387 + ], + "spans": [ + { + "bbox": [ + 106, + 372, + 505, + 387 + ], + "score": 1.0, + "content": "studied for numerous computational problems (and is the basic model for algebraic complexity", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 384, + 506, + 397 + ], + "spans": [ + { + "bbox": [ + 106, + 384, + 506, + 397 + ], + "score": 1.0, + "content": "theory). For our purposes, we will exclusively focus on arithmetic circuits for the matrix-vector", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 395, + 506, + 408 + ], + "spans": [ + { + "bbox": [ + 105, + 395, + 506, + 408 + ], + "score": 1.0, + "content": "multiplication problem. For a more detailed exposition, the reader is referred to the standard book on", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 406, + 242, + 420 + ], + "spans": [ + { + "bbox": [ + 106, + 406, + 242, + 420 + ], + "score": 1.0, + "content": "this topic (Bürgisser et al., 2013).", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 21 + }, + { + "type": "text", + "bbox": [ + 106, + 419, + 505, + 486 + ], + "lines": [ + { + "bbox": [ + 104, + 417, + 507, + 432 + ], + "spans": [ + { + "bbox": [ + 104, + 417, + 400, + 432 + ], + "score": 1.0, + "content": "Definition M.1 (Arithmetic Circuits). An arithmetic circuit that computes", + "type": "text" + }, + { + "bbox": [ + 400, + 420, + 437, + 431 + ], + "score": 0.89, + "content": "\\mathbf { y } = \\mathbf { A x }", + "type": "inline_equation" + }, + { + "bbox": [ + 437, + 417, + 454, + 432 + ], + "score": 1.0, + "content": "(for", + "type": "text" + }, + { + "bbox": [ + 455, + 419, + 502, + 430 + ], + "score": 0.86, + "content": "\\mathbf { A } \\in \\mathbb { F } ^ { m \\times n } ,", + "type": "inline_equation" + }, + { + "bbox": [ + 502, + 417, + 507, + 432 + ], + "score": 1.0, + "content": ")", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 430, + 506, + 443 + ], + "spans": [ + { + "bbox": [ + 105, + 430, + 124, + 443 + ], + "score": 1.0, + "content": "has", + "type": "text" + }, + { + "bbox": [ + 124, + 433, + 132, + 441 + ], + "score": 0.65, + "content": "n", + "type": "inline_equation" + }, + { + "bbox": [ + 132, + 430, + 261, + 443 + ], + "score": 1.0, + "content": "input gates (corresponding to", + "type": "text" + }, + { + "bbox": [ + 261, + 430, + 340, + 443 + ], + "score": 0.9, + "content": "\\mathbf { x } [ 0 ] , \\ldots , \\mathbf { x } [ n - 1 ] )", + "type": "inline_equation" + }, + { + "bbox": [ + 340, + 430, + 362, + 443 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 362, + 432, + 372, + 441 + ], + "score": 0.67, + "content": "m", + "type": "inline_equation" + }, + { + "bbox": [ + 372, + 430, + 506, + 443 + ], + "score": 1.0, + "content": "output gates (corresponding to", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 442, + 506, + 455 + ], + "spans": [ + { + "bbox": [ + 106, + 442, + 186, + 453 + ], + "score": 0.9, + "content": "\\mathbf { y } [ 0 ] , \\ldots , \\mathbf { y } [ m - 1 ] )", + "type": "inline_equation" + }, + { + "bbox": [ + 187, + 442, + 506, + 455 + ], + "score": 1.0, + "content": ". All the internal gates correspond to addition, subtraction, multiplication and", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 104, + 450, + 507, + 466 + ], + "spans": [ + { + "bbox": [ + 104, + 450, + 507, + 466 + ], + "score": 1.0, + "content": "division13 over the underlying field F. The circuit is also allowed to use constants from F for ‘free.’", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 463, + 505, + 476 + ], + "spans": [ + { + "bbox": [ + 105, + 463, + 505, + 476 + ], + "score": 1.0, + "content": "The definition of the internal gates can depend on A (as well as x of course). In other words, one", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 475, + 309, + 486 + ], + "spans": [ + { + "bbox": [ + 106, + 475, + 309, + 486 + ], + "score": 1.0, + "content": "can ‘bake’ the knowledge about A into the circuit.", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 26.5 + }, + { + "type": "text", + "bbox": [ + 107, + 491, + 505, + 525 + ], + "lines": [ + { + "bbox": [ + 105, + 491, + 505, + 504 + ], + "spans": [ + { + "bbox": [ + 105, + 491, + 505, + 504 + ], + "score": 1.0, + "content": "The size s of a circuit is n plus the number of addition, multiplication, subtraction and division gates", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 502, + 505, + 514 + ], + "spans": [ + { + "bbox": [ + 105, + 502, + 505, + 514 + ], + "score": 1.0, + "content": "used in the circuit. The depth d of a circuit is the minimum number of layers such that all gates in a", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 104, + 511, + 338, + 528 + ], + "spans": [ + { + "bbox": [ + 104, + 511, + 338, + 528 + ], + "score": 1.0, + "content": "given layer take as its input gates from previous layers.14", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 31 + }, + { + "type": "text", + "bbox": [ + 107, + 532, + 505, + 587 + ], + "lines": [ + { + "bbox": [ + 106, + 532, + 505, + 544 + ], + "spans": [ + { + "bbox": [ + 106, + 532, + 382, + 544 + ], + "score": 1.0, + "content": "One drawback of arithmetic circuits (especially for infinite fields e.g.", + "type": "text" + }, + { + "bbox": [ + 383, + 533, + 411, + 543 + ], + "score": 0.9, + "content": "\\mathbb { F } = \\mathbb { R }", + "type": "inline_equation" + }, + { + "bbox": [ + 411, + 532, + 505, + 544 + ], + "score": 1.0, + "content": ", which is our preferred", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 543, + 506, + 556 + ], + "spans": [ + { + "bbox": [ + 105, + 543, + 329, + 556 + ], + "score": 1.0, + "content": "choice in this work) is that they assume operations over", + "type": "text" + }, + { + "bbox": [ + 329, + 544, + 338, + 553 + ], + "score": 0.76, + "content": "\\mathbb { F }", + "type": "inline_equation" + }, + { + "bbox": [ + 338, + 543, + 506, + 556 + ], + "score": 1.0, + "content": "can be performed exactly. In particular, it", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 553, + 506, + 568 + ], + "spans": [ + { + "bbox": [ + 105, + 553, + 506, + 568 + ], + "score": 1.0, + "content": "ignores precision issues involved with real arithmetic. Nonetheless, this model turns out to be a very", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 564, + 505, + 578 + ], + "spans": [ + { + "bbox": [ + 105, + 564, + 505, + 578 + ], + "score": 1.0, + "content": "useful model in reasoning about the complexity of doing matrix-vector multiplication for any family", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 575, + 156, + 589 + ], + "spans": [ + { + "bbox": [ + 105, + 575, + 156, + 589 + ], + "score": 1.0, + "content": "of matrices.", + "type": "text" + } + ], + "index": 37 + } + ], + "index": 35 + }, + { + "type": "text", + "bbox": [ + 106, + 593, + 506, + 663 + ], + "lines": [ + { + "bbox": [ + 105, + 593, + 506, + 606 + ], + "spans": [ + { + "bbox": [ + 105, + 593, + 506, + 606 + ], + "score": 1.0, + "content": "Perhaps the strongest argument in support of arithmetic circuits is that a large (if not an overwhelming)", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 604, + 505, + 617 + ], + "spans": [ + { + "bbox": [ + 105, + 604, + 505, + 617 + ], + "score": 1.0, + "content": "majority of matrix-vector multiplication algorithm also imply an arithmetic circuit of size comparable", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 614, + 506, + 628 + ], + "spans": [ + { + "bbox": [ + 105, + 614, + 506, + 628 + ], + "score": 1.0, + "content": "to the runtime of the algorithm (and the depth of the circuit roughly correponds to the time taken to", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 102, + 626, + 510, + 658 + ], + "spans": [ + { + "bbox": [ + 102, + 631, + 142, + 658 + ], + "score": 1.0, + "content": "for each", + "type": "text" + }, + { + "bbox": [ + 105, + 626, + 471, + 639 + ], + "score": 1.0, + "content": "compute it by a parallel algorithm). For example consider the obvious algorithm to compute", + "type": "text" + }, + { + "bbox": [ + 142, + 639, + 174, + 651 + ], + "score": 0.92, + "content": "i \\in [ m ]", + "type": "inline_equation" + }, + { + "bbox": [ + 175, + 631, + 216, + 658 + ], + "score": 1.0, + "content": ", compute", + "type": "text" + }, + { + "bbox": [ + 217, + 638, + 233, + 651 + ], + "score": 0.85, + "content": "\\mathbf { y } [ i ]", + "type": "inline_equation" + }, + { + "bbox": [ + 234, + 631, + 281, + 658 + ], + "score": 1.0, + "content": "as the sum", + "type": "text" + }, + { + "bbox": [ + 281, + 637, + 356, + 651 + ], + "score": 0.92, + "content": "\\begin{array} { r } { \\bar { \\sum } _ { i = 0 } ^ { n - 1 } \\mathbf { A } [ i , j ] \\mathbf { x } [ j ] ) } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 357, + 631, + 510, + 658 + ], + "score": 1.0, + "content": ". It is easy to see that this algorithm", + "type": "text" + }, + { + "bbox": [ + 472, + 626, + 488, + 637 + ], + "score": 0.27, + "content": "\\mathbf { A x }", + "type": "inline_equation" + }, + { + "bbox": [ + 489, + 626, + 506, + 639 + ], + "score": 1.0, + "content": "(i.e.", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 104, + 649, + 374, + 664 + ], + "spans": [ + { + "bbox": [ + 104, + 649, + 250, + 664 + ], + "score": 1.0, + "content": "implies an arithmetic circuit of size", + "type": "text" + }, + { + "bbox": [ + 250, + 651, + 281, + 663 + ], + "score": 0.91, + "content": "O ( n m )", + "type": "inline_equation" + }, + { + "bbox": [ + 281, + 649, + 324, + 664 + ], + "score": 1.0, + "content": "and depth", + "type": "text" + }, + { + "bbox": [ + 324, + 651, + 361, + 663 + ], + "score": 0.91, + "content": "O ( \\log n )", + "type": "inline_equation" + }, + { + "bbox": [ + 361, + 649, + 374, + 664 + ], + "score": 1.0, + "content": ".15", + "type": "text" + } + ], + "index": 42 + } + ], + "index": 40 + } + ], + "page_idx": 42, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 106, + 669, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 116, + 667, + 290, + 682 + ], + "spans": [ + { + "bbox": [ + 116, + 667, + 290, + 682 + ], + "score": 1.0, + "content": "13Here we assume all the gates have two inputs.", + "type": "text" + } + ] + }, + { + "bbox": [ + 114, + 677, + 416, + 694 + ], + "spans": [ + { + "bbox": [ + 114, + 677, + 416, + 694 + ], + "score": 1.0, + "content": "14The input layer corresponding to the input gates does not contriubte to the depth.", + "type": "text" + } + ] + }, + { + "bbox": [ + 113, + 686, + 508, + 708 + ], + "spans": [ + { + "bbox": [ + 113, + 686, + 363, + 708 + ], + "score": 1.0, + "content": "15The claim on the depth follow from the fact that each of the sums", + "type": "text" + }, + { + "bbox": [ + 363, + 690, + 430, + 703 + ], + "score": 0.91, + "content": "\\begin{array} { r } { \\sum _ { i = 0 } ^ { n - 1 } \\mathbf { A } [ i ] [ j ] \\mathbf { x } [ j ] } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 431, + 686, + 508, + 708 + ], + "score": 1.0, + "content": "can be computed in", + "type": "text" + } + ] + }, + { + "bbox": [ + 105, + 701, + 505, + 713 + ], + "spans": [ + { + "bbox": [ + 105, + 701, + 235, + 713 + ], + "score": 1.0, + "content": "parallel. Further, the sum for each", + "type": "text" + }, + { + "bbox": [ + 235, + 702, + 265, + 712 + ], + "score": 0.9, + "content": "i \\in [ m ]", + "type": "inline_equation" + }, + { + "bbox": [ + 266, + 701, + 322, + 713 + ], + "score": 1.0, + "content": "can be done in", + "type": "text" + }, + { + "bbox": [ + 323, + 702, + 350, + 712 + ], + "score": 0.89, + "content": "\\log _ { 2 } m", + "type": "inline_equation" + }, + { + "bbox": [ + 350, + 701, + 505, + 713 + ], + "score": 1.0, + "content": "depth by first computing the partial sums", + "type": "text" + } + ] + }, + { + "bbox": [ + 106, + 711, + 506, + 723 + ], + "spans": [ + { + "bbox": [ + 106, + 711, + 261, + 722 + ], + "score": 0.91, + "content": "\\mathbf { \\hat { A } } [ i ] [ 2 j ^ { \\prime } ] \\mathbf { x } [ 2 j ^ { \\prime } ] + \\mathbf { A } [ i ] [ 2 j ^ { \\prime } + 1 ] \\mathbf { x } [ 2 j ^ { \\prime } + \\mathrm { \\hat { 1 } } ]", + "type": "inline_equation" + }, + { + "bbox": [ + 262, + 711, + 288, + 723 + ], + "score": 1.0, + "content": "for all", + "type": "text" + }, + { + "bbox": [ + 288, + 711, + 328, + 722 + ], + "score": 0.92, + "content": "j ^ { \\prime } \\in [ n / 2 ]", + "type": "inline_equation" + }, + { + "bbox": [ + 328, + 711, + 506, + 723 + ], + "score": 1.0, + "content": "in parallel and recursively computing pair-wise", + "type": "text" + } + ] + }, + { + "bbox": [ + 105, + 720, + 188, + 732 + ], + "spans": [ + { + "bbox": [ + 105, + 720, + 188, + 732 + ], + "score": 1.0, + "content": "sums till we are done.", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 107, + 27, + 293, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 26, + 294, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 26, + 294, + 38 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2020", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 300, + 751, + 311, + 760 + ], + "lines": [ + { + "bbox": [ + 298, + 750, + 312, + 763 + ], + "spans": [ + { + "bbox": [ + 298, + 750, + 312, + 763 + ], + "score": 1.0, + "content": "43", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 495, + 312, + 504, + 322 + ], + "lines": [ + { + "bbox": [ + 496, + 313, + 504, + 321 + ], + "spans": [ + { + "bbox": [ + 496, + 313, + 504, + 321 + ], + "score": 0.999, + "content": "□", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 105, + 82, + 504, + 105 + ], + "lines": [ + { + "bbox": [ + 105, + 82, + 505, + 95 + ], + "spans": [ + { + "bbox": [ + 105, + 82, + 505, + 95 + ], + "score": 1.0, + "content": "We now prove an asymptotic bound on the VC dimension of a ReLU network whose weight matrices", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 93, + 356, + 106 + ], + "spans": [ + { + "bbox": [ + 105, + 93, + 356, + 106 + ], + "score": 1.0, + "content": "are kaleidoscope matrices with bounded width and expansion.", + "type": "text" + } + ], + "index": 1 + } + ], + "index": 0.5, + "bbox_fs": [ + 105, + 82, + 505, + 106 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 106, + 505, + 151 + ], + "lines": [ + { + "bbox": [ + 105, + 106, + 506, + 119 + ], + "spans": [ + { + "bbox": [ + 105, + 106, + 188, + 119 + ], + "score": 1.0, + "content": "Proposition 3. Let", + "type": "text" + }, + { + "bbox": [ + 188, + 107, + 198, + 117 + ], + "score": 0.73, + "content": "\\mathcal { F }", + "type": "inline_equation" + }, + { + "bbox": [ + 199, + 106, + 415, + 119 + ], + "score": 1.0, + "content": "be the class of ReLU neural networks consisting of", + "type": "text" + }, + { + "bbox": [ + 415, + 107, + 423, + 117 + ], + "score": 0.7, + "content": "L", + "type": "inline_equation" + }, + { + "bbox": [ + 424, + 106, + 506, + 119 + ], + "score": 1.0, + "content": "layers, where each", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 106, + 118, + 505, + 129 + ], + "spans": [ + { + "bbox": [ + 106, + 118, + 150, + 129 + ], + "score": 1.0, + "content": "layer is a", + "type": "text" + }, + { + "bbox": [ + 150, + 118, + 159, + 128 + ], + "score": 0.61, + "content": "K", + "type": "inline_equation" + }, + { + "bbox": [ + 159, + 118, + 416, + 129 + ], + "score": 1.0, + "content": "-matrix with width and expansion bounded by some constant", + "type": "text" + }, + { + "bbox": [ + 416, + 118, + 425, + 128 + ], + "score": 0.77, + "content": "C", + "type": "inline_equation" + }, + { + "bbox": [ + 425, + 118, + 505, + 129 + ], + "score": 1.0, + "content": ". Suppose that the", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 129, + 506, + 141 + ], + "spans": [ + { + "bbox": [ + 105, + 129, + 160, + 141 + ], + "score": 1.0, + "content": "network has", + "type": "text" + }, + { + "bbox": [ + 160, + 129, + 172, + 138 + ], + "score": 0.65, + "content": "W", + "type": "inline_equation" + }, + { + "bbox": [ + 173, + 129, + 286, + 141 + ], + "score": 1.0, + "content": "total parameters. Let sign", + "type": "text" + }, + { + "bbox": [ + 286, + 129, + 296, + 139 + ], + "score": 0.82, + "content": "\\mathcal { F }", + "type": "inline_equation" + }, + { + "bbox": [ + 297, + 129, + 506, + 141 + ], + "score": 1.0, + "content": "denote the corresponding classification functions:", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 107, + 140, + 357, + 153 + ], + "spans": [ + { + "bbox": [ + 107, + 140, + 211, + 152 + ], + "score": 0.92, + "content": "\\{ x \\mapsto \\operatorname { s i g n } f ( x ) : f \\in { \\mathcal { F } } \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 212, + 140, + 357, + 153 + ], + "score": 1.0, + "content": ". Then this class has VC dimension:", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 3.5, + "bbox_fs": [ + 105, + 106, + 506, + 153 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 233, + 153, + 377, + 166 + ], + "lines": [ + { + "bbox": [ + 233, + 153, + 377, + 166 + ], + "spans": [ + { + "bbox": [ + 233, + 153, + 377, + 166 + ], + "score": 0.9, + "content": "\\operatorname { V C d i m } ( \\operatorname { s i g n } { \\mathcal { F } } ) = O ( L W \\log W ) .", + "type": "interline_equation", + "image_path": "b07fcddd39796f289207532c245eb6a0f0438676278f9347655be1f0eef21dc2.jpg" + } + ] + } + ], + "index": 6, + "virtual_lines": [ + { + "bbox": [ + 233, + 153, + 377, + 166 + ], + "spans": [], + "index": 6 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 172, + 505, + 206 + ], + "lines": [ + { + "bbox": [ + 105, + 172, + 506, + 186 + ], + "spans": [ + { + "bbox": [ + 105, + 172, + 506, + 186 + ], + "score": 1.0, + "content": "We leverage the result from Thomas et al. (2018) for the case where the entries of the weight matrices", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 184, + 506, + 196 + ], + "spans": [ + { + "bbox": [ + 105, + 184, + 506, + 196 + ], + "score": 1.0, + "content": "interact multiplicatively, but with polynomially bounded degrees. This proof is similar to the VC", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 195, + 469, + 207 + ], + "spans": [ + { + "bbox": [ + 106, + 195, + 469, + 207 + ], + "score": 1.0, + "content": "bound for ReLU networks whose weight matrices are butterfly matrices (Dao et al., 2019).", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 8, + "bbox_fs": [ + 105, + 172, + 506, + 207 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 217, + 505, + 295 + ], + "lines": [ + { + "bbox": [ + 106, + 218, + 506, + 231 + ], + "spans": [ + { + "bbox": [ + 106, + 218, + 506, + 231 + ], + "score": 1.0, + "content": "Proof. To use Theorem 3 of Thomas et al. (2018), we simply need to check that the entries of the", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 229, + 506, + 243 + ], + "spans": [ + { + "bbox": [ + 105, + 229, + 366, + 243 + ], + "score": 1.0, + "content": "linear layer, as polynomials of the parameters, has degree at most", + "type": "text" + }, + { + "bbox": [ + 366, + 230, + 393, + 242 + ], + "score": 0.92, + "content": "c _ { 1 } m _ { l } ^ { c _ { 2 } }", + "type": "inline_equation" + }, + { + "bbox": [ + 393, + 229, + 506, + 243 + ], + "score": 1.0, + "content": "for some universal constant", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 240, + 506, + 253 + ], + "spans": [ + { + "bbox": [ + 106, + 241, + 151, + 252 + ], + "score": 0.9, + "content": "c _ { 1 } , c _ { 2 } > 0", + "type": "inline_equation" + }, + { + "bbox": [ + 152, + 240, + 184, + 253 + ], + "score": 1.0, + "content": ", where", + "type": "text" + }, + { + "bbox": [ + 184, + 242, + 197, + 251 + ], + "score": 0.84, + "content": "m _ { l }", + "type": "inline_equation" + }, + { + "bbox": [ + 198, + 240, + 313, + 253 + ], + "score": 1.0, + "content": "is the size of output of the", + "type": "text" + }, + { + "bbox": [ + 313, + 241, + 317, + 250 + ], + "score": 0.61, + "content": "l", + "type": "inline_equation" + }, + { + "bbox": [ + 318, + 240, + 506, + 253 + ], + "score": 1.0, + "content": "-th layer. If the network weight matrices are", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 250, + 504, + 265 + ], + "spans": [ + { + "bbox": [ + 105, + 250, + 465, + 265 + ], + "score": 1.0, + "content": "K-matrices with bounded width and expansion, each weight matrix is a product of at most", + "type": "text" + }, + { + "bbox": [ + 466, + 252, + 504, + 263 + ], + "score": 0.87, + "content": "c _ { 3 } \\log m _ { l }", + "type": "inline_equation" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 262, + 505, + 275 + ], + "spans": [ + { + "bbox": [ + 105, + 262, + 285, + 275 + ], + "score": 1.0, + "content": "sparse factors, for some universal constant", + "type": "text" + }, + { + "bbox": [ + 286, + 263, + 317, + 273 + ], + "score": 0.91, + "content": "c _ { 3 } > 0", + "type": "inline_equation" + }, + { + "bbox": [ + 317, + 262, + 505, + 275 + ], + "score": 1.0, + "content": ". This means that the degree is polynomially", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 274, + 505, + 285 + ], + "spans": [ + { + "bbox": [ + 106, + 274, + 505, + 285 + ], + "score": 1.0, + "content": "bounded, which satisfies the condition of the theorem. Therefore the VC dimension is bounded to be", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 284, + 278, + 296 + ], + "spans": [ + { + "bbox": [ + 106, + 284, + 278, + 296 + ], + "score": 1.0, + "content": "almost linear in the number of parameters:", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 13, + "bbox_fs": [ + 105, + 218, + 506, + 296 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 233, + 297, + 378, + 310 + ], + "lines": [ + { + "bbox": [ + 233, + 297, + 378, + 310 + ], + "spans": [ + { + "bbox": [ + 233, + 297, + 378, + 310 + ], + "score": 0.91, + "content": "\\operatorname { V C d i m } ( \\operatorname { s i g n } { \\mathcal { F } } ) = O ( L W \\log W ) .", + "type": "interline_equation", + "image_path": "f76055338f318f1bde28f1efe8b39133be9d88513589e943c274bfb13b326720.jpg" + } + ] + } + ], + "index": 17, + "virtual_lines": [ + { + "bbox": [ + 233, + 297, + 378, + 310 + ], + "spans": [], + "index": 17 + } + ] + }, + { + "type": "title", + "bbox": [ + 108, + 338, + 288, + 351 + ], + "lines": [ + { + "bbox": [ + 105, + 337, + 290, + 352 + ], + "spans": [ + { + "bbox": [ + 105, + 337, + 290, + 352 + ], + "score": 1.0, + "content": "M ARITHMETIC CIRCUIT PRIMER", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 18 + }, + { + "type": "text", + "bbox": [ + 106, + 362, + 505, + 418 + ], + "lines": [ + { + "bbox": [ + 106, + 362, + 506, + 375 + ], + "spans": [ + { + "bbox": [ + 106, + 362, + 506, + 375 + ], + "score": 1.0, + "content": "We give a quick overview of arithmetic circuits. This is a model of computation that has been", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 372, + 505, + 387 + ], + "spans": [ + { + "bbox": [ + 106, + 372, + 505, + 387 + ], + "score": 1.0, + "content": "studied for numerous computational problems (and is the basic model for algebraic complexity", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 384, + 506, + 397 + ], + "spans": [ + { + "bbox": [ + 106, + 384, + 506, + 397 + ], + "score": 1.0, + "content": "theory). For our purposes, we will exclusively focus on arithmetic circuits for the matrix-vector", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 395, + 506, + 408 + ], + "spans": [ + { + "bbox": [ + 105, + 395, + 506, + 408 + ], + "score": 1.0, + "content": "multiplication problem. For a more detailed exposition, the reader is referred to the standard book on", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 406, + 242, + 420 + ], + "spans": [ + { + "bbox": [ + 106, + 406, + 242, + 420 + ], + "score": 1.0, + "content": "this topic (Bürgisser et al., 2013).", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 21, + "bbox_fs": [ + 105, + 362, + 506, + 420 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 419, + 505, + 486 + ], + "lines": [ + { + "bbox": [ + 104, + 417, + 507, + 432 + ], + "spans": [ + { + "bbox": [ + 104, + 417, + 400, + 432 + ], + "score": 1.0, + "content": "Definition M.1 (Arithmetic Circuits). An arithmetic circuit that computes", + "type": "text" + }, + { + "bbox": [ + 400, + 420, + 437, + 431 + ], + "score": 0.89, + "content": "\\mathbf { y } = \\mathbf { A x }", + "type": "inline_equation" + }, + { + "bbox": [ + 437, + 417, + 454, + 432 + ], + "score": 1.0, + "content": "(for", + "type": "text" + }, + { + "bbox": [ + 455, + 419, + 502, + 430 + ], + "score": 0.86, + "content": "\\mathbf { A } \\in \\mathbb { F } ^ { m \\times n } ,", + "type": "inline_equation" + }, + { + "bbox": [ + 502, + 417, + 507, + 432 + ], + "score": 1.0, + "content": ")", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 430, + 506, + 443 + ], + "spans": [ + { + "bbox": [ + 105, + 430, + 124, + 443 + ], + "score": 1.0, + "content": "has", + "type": "text" + }, + { + "bbox": [ + 124, + 433, + 132, + 441 + ], + "score": 0.65, + "content": "n", + "type": "inline_equation" + }, + { + "bbox": [ + 132, + 430, + 261, + 443 + ], + "score": 1.0, + "content": "input gates (corresponding to", + "type": "text" + }, + { + "bbox": [ + 261, + 430, + 340, + 443 + ], + "score": 0.9, + "content": "\\mathbf { x } [ 0 ] , \\ldots , \\mathbf { x } [ n - 1 ] )", + "type": "inline_equation" + }, + { + "bbox": [ + 340, + 430, + 362, + 443 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 362, + 432, + 372, + 441 + ], + "score": 0.67, + "content": "m", + "type": "inline_equation" + }, + { + "bbox": [ + 372, + 430, + 506, + 443 + ], + "score": 1.0, + "content": "output gates (corresponding to", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 442, + 506, + 455 + ], + "spans": [ + { + "bbox": [ + 106, + 442, + 186, + 453 + ], + "score": 0.9, + "content": "\\mathbf { y } [ 0 ] , \\ldots , \\mathbf { y } [ m - 1 ] )", + "type": "inline_equation" + }, + { + "bbox": [ + 187, + 442, + 506, + 455 + ], + "score": 1.0, + "content": ". All the internal gates correspond to addition, subtraction, multiplication and", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 104, + 450, + 507, + 466 + ], + "spans": [ + { + "bbox": [ + 104, + 450, + 507, + 466 + ], + "score": 1.0, + "content": "division13 over the underlying field F. The circuit is also allowed to use constants from F for ‘free.’", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 463, + 505, + 476 + ], + "spans": [ + { + "bbox": [ + 105, + 463, + 505, + 476 + ], + "score": 1.0, + "content": "The definition of the internal gates can depend on A (as well as x of course). In other words, one", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 475, + 309, + 486 + ], + "spans": [ + { + "bbox": [ + 106, + 475, + 309, + 486 + ], + "score": 1.0, + "content": "can ‘bake’ the knowledge about A into the circuit.", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 26.5, + "bbox_fs": [ + 104, + 417, + 507, + 486 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 491, + 505, + 525 + ], + "lines": [ + { + "bbox": [ + 105, + 491, + 505, + 504 + ], + "spans": [ + { + "bbox": [ + 105, + 491, + 505, + 504 + ], + "score": 1.0, + "content": "The size s of a circuit is n plus the number of addition, multiplication, subtraction and division gates", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 502, + 505, + 514 + ], + "spans": [ + { + "bbox": [ + 105, + 502, + 505, + 514 + ], + "score": 1.0, + "content": "used in the circuit. The depth d of a circuit is the minimum number of layers such that all gates in a", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 104, + 511, + 338, + 528 + ], + "spans": [ + { + "bbox": [ + 104, + 511, + 338, + 528 + ], + "score": 1.0, + "content": "given layer take as its input gates from previous layers.14", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 31, + "bbox_fs": [ + 104, + 491, + 505, + 528 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 532, + 505, + 587 + ], + "lines": [ + { + "bbox": [ + 106, + 532, + 505, + 544 + ], + "spans": [ + { + "bbox": [ + 106, + 532, + 382, + 544 + ], + "score": 1.0, + "content": "One drawback of arithmetic circuits (especially for infinite fields e.g.", + "type": "text" + }, + { + "bbox": [ + 383, + 533, + 411, + 543 + ], + "score": 0.9, + "content": "\\mathbb { F } = \\mathbb { R }", + "type": "inline_equation" + }, + { + "bbox": [ + 411, + 532, + 505, + 544 + ], + "score": 1.0, + "content": ", which is our preferred", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 543, + 506, + 556 + ], + "spans": [ + { + "bbox": [ + 105, + 543, + 329, + 556 + ], + "score": 1.0, + "content": "choice in this work) is that they assume operations over", + "type": "text" + }, + { + "bbox": [ + 329, + 544, + 338, + 553 + ], + "score": 0.76, + "content": "\\mathbb { F }", + "type": "inline_equation" + }, + { + "bbox": [ + 338, + 543, + 506, + 556 + ], + "score": 1.0, + "content": "can be performed exactly. In particular, it", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 553, + 506, + 568 + ], + "spans": [ + { + "bbox": [ + 105, + 553, + 506, + 568 + ], + "score": 1.0, + "content": "ignores precision issues involved with real arithmetic. Nonetheless, this model turns out to be a very", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 564, + 505, + 578 + ], + "spans": [ + { + "bbox": [ + 105, + 564, + 505, + 578 + ], + "score": 1.0, + "content": "useful model in reasoning about the complexity of doing matrix-vector multiplication for any family", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 575, + 156, + 589 + ], + "spans": [ + { + "bbox": [ + 105, + 575, + 156, + 589 + ], + "score": 1.0, + "content": "of matrices.", + "type": "text" + } + ], + "index": 37 + } + ], + "index": 35, + "bbox_fs": [ + 105, + 532, + 506, + 589 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 593, + 506, + 663 + ], + "lines": [ + { + "bbox": [ + 105, + 593, + 506, + 606 + ], + "spans": [ + { + "bbox": [ + 105, + 593, + 506, + 606 + ], + "score": 1.0, + "content": "Perhaps the strongest argument in support of arithmetic circuits is that a large (if not an overwhelming)", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 604, + 505, + 617 + ], + "spans": [ + { + "bbox": [ + 105, + 604, + 505, + 617 + ], + "score": 1.0, + "content": "majority of matrix-vector multiplication algorithm also imply an arithmetic circuit of size comparable", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 614, + 506, + 628 + ], + "spans": [ + { + "bbox": [ + 105, + 614, + 506, + 628 + ], + "score": 1.0, + "content": "to the runtime of the algorithm (and the depth of the circuit roughly correponds to the time taken to", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 102, + 626, + 510, + 658 + ], + "spans": [ + { + "bbox": [ + 102, + 631, + 142, + 658 + ], + "score": 1.0, + "content": "for each", + "type": "text" + }, + { + "bbox": [ + 105, + 626, + 471, + 639 + ], + "score": 1.0, + "content": "compute it by a parallel algorithm). For example consider the obvious algorithm to compute", + "type": "text" + }, + { + "bbox": [ + 142, + 639, + 174, + 651 + ], + "score": 0.92, + "content": "i \\in [ m ]", + "type": "inline_equation" + }, + { + "bbox": [ + 175, + 631, + 216, + 658 + ], + "score": 1.0, + "content": ", compute", + "type": "text" + }, + { + "bbox": [ + 217, + 638, + 233, + 651 + ], + "score": 0.85, + "content": "\\mathbf { y } [ i ]", + "type": "inline_equation" + }, + { + "bbox": [ + 234, + 631, + 281, + 658 + ], + "score": 1.0, + "content": "as the sum", + "type": "text" + }, + { + "bbox": [ + 281, + 637, + 356, + 651 + ], + "score": 0.92, + "content": "\\begin{array} { r } { \\bar { \\sum } _ { i = 0 } ^ { n - 1 } \\mathbf { A } [ i , j ] \\mathbf { x } [ j ] ) } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 357, + 631, + 510, + 658 + ], + "score": 1.0, + "content": ". It is easy to see that this algorithm", + "type": "text" + }, + { + "bbox": [ + 472, + 626, + 488, + 637 + ], + "score": 0.27, + "content": "\\mathbf { A x }", + "type": "inline_equation" + }, + { + "bbox": [ + 489, + 626, + 506, + 639 + ], + "score": 1.0, + "content": "(i.e.", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 104, + 649, + 374, + 664 + ], + "spans": [ + { + "bbox": [ + 104, + 649, + 250, + 664 + ], + "score": 1.0, + "content": "implies an arithmetic circuit of size", + "type": "text" + }, + { + "bbox": [ + 250, + 651, + 281, + 663 + ], + "score": 0.91, + "content": "O ( n m )", + "type": "inline_equation" + }, + { + "bbox": [ + 281, + 649, + 324, + 664 + ], + "score": 1.0, + "content": "and depth", + "type": "text" + }, + { + "bbox": [ + 324, + 651, + 361, + 663 + ], + "score": 0.91, + "content": "O ( \\log n )", + "type": "inline_equation" + }, + { + "bbox": [ + 361, + 649, + 374, + 664 + ], + "score": 1.0, + "content": ".15", + "type": "text" + } + ], + "index": 42 + } + ], + "index": 40, + "bbox_fs": [ + 102, + 593, + 510, + 664 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 82, + 504, + 116 + ], + "lines": [ + { + "bbox": [ + 106, + 82, + 505, + 95 + ], + "spans": [ + { + "bbox": [ + 106, + 82, + 505, + 95 + ], + "score": 1.0, + "content": "One thing to note about the arithmetic circuit above is that all the multiplications involve at least one", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 93, + 505, + 105 + ], + "spans": [ + { + "bbox": [ + 105, + 93, + 219, + 105 + ], + "score": 1.0, + "content": "input that is a constant from", + "type": "text" + }, + { + "bbox": [ + 219, + 94, + 228, + 104 + ], + "score": 0.5, + "content": "\\mathbb { F }", + "type": "inline_equation" + }, + { + "bbox": [ + 228, + 93, + 505, + 105 + ], + "score": 1.0, + "content": "(recall that we can assume that the entries of A are constants that can", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 106, + 105, + 498, + 116 + ], + "spans": [ + { + "bbox": [ + 106, + 105, + 498, + 116 + ], + "score": 1.0, + "content": "be used to build the circuit). This leads to the following important sub-class of arithmetic circuits:", + "type": "text" + } + ], + "index": 2 + } + ], + "index": 1 + }, + { + "type": "text", + "bbox": [ + 106, + 130, + 505, + 175 + ], + "lines": [ + { + "bbox": [ + 106, + 131, + 505, + 142 + ], + "spans": [ + { + "bbox": [ + 106, + 131, + 505, + 142 + ], + "score": 1.0, + "content": "Definition M.2 (Linear Arithmetic Circuits). An arithmetic circuit is called a linear arithmetic", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 106, + 142, + 505, + 154 + ], + "spans": [ + { + "bbox": [ + 106, + 142, + 505, + 154 + ], + "score": 1.0, + "content": "circuit if it only uses addition, subtraction and multiplication. Further, every multiplcation has a", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 153, + 505, + 165 + ], + "spans": [ + { + "bbox": [ + 105, + 153, + 187, + 165 + ], + "score": 1.0, + "content": "fixed constant from", + "type": "text" + }, + { + "bbox": [ + 187, + 153, + 195, + 163 + ], + "score": 0.61, + "content": "\\mathbb { F }", + "type": "inline_equation" + }, + { + "bbox": [ + 196, + 153, + 505, + 165 + ], + "score": 1.0, + "content": "as at least one of its two inputs. In other words, all gates in the circuit are", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 163, + 446, + 176 + ], + "spans": [ + { + "bbox": [ + 105, + 163, + 295, + 176 + ], + "score": 1.0, + "content": "linear functions of their inputs (i.e. of the form", + "type": "text" + }, + { + "bbox": [ + 295, + 164, + 319, + 174 + ], + "score": 0.37, + "content": "a x +", + "type": "inline_equation" + }, + { + "bbox": [ + 320, + 163, + 405, + 176 + ], + "score": 1.0, + "content": "by for fixed constants", + "type": "text" + }, + { + "bbox": [ + 406, + 164, + 439, + 175 + ], + "score": 0.9, + "content": "a , b \\in \\mathbb { F }", + "type": "inline_equation" + }, + { + "bbox": [ + 440, + 163, + 446, + 176 + ], + "score": 1.0, + "content": ").", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 4.5 + }, + { + "type": "text", + "bbox": [ + 107, + 196, + 505, + 229 + ], + "lines": [ + { + "bbox": [ + 106, + 196, + 505, + 208 + ], + "spans": [ + { + "bbox": [ + 106, + 196, + 505, + 208 + ], + "score": 1.0, + "content": "Intuitively for the matrix-vector multiplication, it makes sense to consider linear arithmetic circuits", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 207, + 505, + 219 + ], + "spans": [ + { + "bbox": [ + 105, + 207, + 281, + 219 + ], + "score": 1.0, + "content": "since the final function we want to compute", + "type": "text" + }, + { + "bbox": [ + 282, + 207, + 298, + 217 + ], + "score": 0.26, + "content": "\\mathbf { A x }", + "type": "inline_equation" + }, + { + "bbox": [ + 298, + 207, + 505, + 219 + ], + "score": 1.0, + "content": "is indeed a linear function of its inputs. For inifinite", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 217, + 466, + 231 + ], + "spans": [ + { + "bbox": [ + 105, + 217, + 151, + 231 + ], + "score": 1.0, + "content": "fields (e.g.", + "type": "text" + }, + { + "bbox": [ + 151, + 218, + 179, + 228 + ], + "score": 0.9, + "content": "\\mathbb { F } = \\mathbb { R }", + "type": "inline_equation" + }, + { + "bbox": [ + 180, + 217, + 191, + 231 + ], + "score": 1.0, + "content": "or", + "type": "text" + }, + { + "bbox": [ + 191, + 218, + 219, + 228 + ], + "score": 0.89, + "content": "\\mathbb { F } = \\mathbb { C }", + "type": "inline_equation" + }, + { + "bbox": [ + 220, + 217, + 466, + 231 + ], + "score": 1.0, + "content": "), it turns out that this is essentially without loss of generality:", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 8 + }, + { + "type": "text", + "bbox": [ + 107, + 243, + 505, + 277 + ], + "lines": [ + { + "bbox": [ + 105, + 243, + 505, + 256 + ], + "spans": [ + { + "bbox": [ + 105, + 243, + 505, + 256 + ], + "score": 1.0, + "content": "Theorem 4 ((Bürgisser et al., 2013)). Let F be an infinite field. Any (general) arithmetic circuit to", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 254, + 505, + 267 + ], + "spans": [ + { + "bbox": [ + 105, + 254, + 181, + 267 + ], + "score": 1.0, + "content": "compute Ax over", + "type": "text" + }, + { + "bbox": [ + 182, + 255, + 190, + 265 + ], + "score": 0.67, + "content": "\\mathbb { F }", + "type": "inline_equation" + }, + { + "bbox": [ + 190, + 254, + 505, + 267 + ], + "score": 1.0, + "content": "of size s and depth d can be converted into a linear arithmetic circuit of size", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 107, + 264, + 198, + 279 + ], + "spans": [ + { + "bbox": [ + 107, + 266, + 128, + 278 + ], + "score": 0.9, + "content": "O ( s )", + "type": "inline_equation" + }, + { + "bbox": [ + 128, + 264, + 171, + 279 + ], + "score": 1.0, + "content": "and depth", + "type": "text" + }, + { + "bbox": [ + 172, + 266, + 193, + 278 + ], + "score": 0.91, + "content": "O ( d )", + "type": "inline_equation" + }, + { + "bbox": [ + 193, + 264, + 198, + 279 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 11 + }, + { + "type": "text", + "bbox": [ + 107, + 298, + 504, + 320 + ], + "lines": [ + { + "bbox": [ + 106, + 298, + 505, + 310 + ], + "spans": [ + { + "bbox": [ + 106, + 298, + 505, + 310 + ], + "score": 1.0, + "content": "The above result implies that for asymptotic considerations, linear arithmetic circuits for matrix-vector", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 308, + 351, + 321 + ], + "spans": [ + { + "bbox": [ + 105, + 308, + 351, + 321 + ], + "score": 1.0, + "content": "multiplication are equivalent to general arithmetic circuits.16", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 13.5 + }, + { + "type": "text", + "bbox": [ + 107, + 325, + 505, + 358 + ], + "lines": [ + { + "bbox": [ + 106, + 325, + 506, + 338 + ], + "spans": [ + { + "bbox": [ + 106, + 325, + 349, + 338 + ], + "score": 1.0, + "content": "One important property of linear arithmetic circuits of depth", + "type": "text" + }, + { + "bbox": [ + 349, + 326, + 356, + 336 + ], + "score": 0.71, + "content": "d", + "type": "inline_equation" + }, + { + "bbox": [ + 356, + 325, + 506, + 338 + ], + "score": 1.0, + "content": ", which we will use in our arguments,", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 336, + 506, + 349 + ], + "spans": [ + { + "bbox": [ + 105, + 336, + 367, + 349 + ], + "score": 1.0, + "content": "is that such a circuit can be equivalently represented as product of", + "type": "text" + }, + { + "bbox": [ + 368, + 338, + 374, + 347 + ], + "score": 0.82, + "content": "d", + "type": "inline_equation" + }, + { + "bbox": [ + 375, + 336, + 506, + 349 + ], + "score": 1.0, + "content": "sparse matrices (see the proof of", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 347, + 267, + 359 + ], + "spans": [ + { + "bbox": [ + 105, + 347, + 267, + 359 + ], + "score": 1.0, + "content": "Theorem 1 for the precise derivation17).", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 16 + }, + { + "type": "text", + "bbox": [ + 107, + 364, + 505, + 431 + ], + "lines": [ + { + "bbox": [ + 105, + 364, + 506, + 377 + ], + "spans": [ + { + "bbox": [ + 105, + 364, + 506, + 377 + ], + "score": 1.0, + "content": "As mentioned earlier, a vast majority of efficient matrix vector multiplication algorithms are equivalent", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 375, + 506, + 388 + ], + "spans": [ + { + "bbox": [ + 105, + 375, + 506, + 388 + ], + "score": 1.0, + "content": "to small (both in size and depth) linear arithmetic circuit. For example the FFT can be thought of as an", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 386, + 505, + 399 + ], + "spans": [ + { + "bbox": [ + 105, + 386, + 505, + 399 + ], + "score": 1.0, + "content": "efficient arithmetic circuit to compute the Discrete Fourier Transform (indeed when one converts the", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 397, + 505, + 410 + ], + "spans": [ + { + "bbox": [ + 105, + 397, + 505, + 410 + ], + "score": 1.0, + "content": "linear arithmetic circuit for FFT into a matrix decomposition,18 then each matrix in the decomposition", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 408, + 505, + 421 + ], + "spans": [ + { + "bbox": [ + 105, + 408, + 505, + 421 + ], + "score": 1.0, + "content": "is a butterfly factor, with each block matrix in each factor being the same). For an illustration of this", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 419, + 333, + 432 + ], + "spans": [ + { + "bbox": [ + 105, + 419, + 199, + 432 + ], + "score": 1.0, + "content": "consider the DFT with", + "type": "text" + }, + { + "bbox": [ + 199, + 420, + 225, + 430 + ], + "score": 0.89, + "content": "n = 4", + "type": "inline_equation" + }, + { + "bbox": [ + 225, + 419, + 333, + 432 + ], + "score": 1.0, + "content": "as illustrated in Figure 11.", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 20.5 + }, + { + "type": "image", + "bbox": [ + 266, + 468, + 344, + 546 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 266, + 468, + 344, + 546 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 266, + 468, + 344, + 546 + ], + "spans": [ + { + "bbox": [ + 266, + 468, + 344, + 546 + ], + "score": 0.941, + "type": "image", + "image_path": "a3cbb522b99a7e6b2265f68e5eb9f6e0cce9cdc88c2032108159d5ac677d7182.jpg" + } + ] + } + ], + "index": 24.5, + "virtual_lines": [ + { + "bbox": [ + 266, + 468, + 344, + 507.0 + ], + "spans": [], + "index": 24 + }, + { + "bbox": [ + 266, + 507.0, + 344, + 546.0 + ], + "spans": [], + "index": 25 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 251, + 560, + 360, + 572 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 249, + 559, + 361, + 572 + ], + "spans": [ + { + "bbox": [ + 249, + 559, + 361, + 572 + ], + "score": 1.0, + "content": "Figure 11: DFT of order 4.", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 26 + } + ], + "index": 25.25 + }, + { + "type": "text", + "bbox": [ + 106, + 614, + 409, + 626 + ], + "lines": [ + { + "bbox": [ + 105, + 613, + 407, + 628 + ], + "spans": [ + { + "bbox": [ + 105, + 613, + 381, + 628 + ], + "score": 1.0, + "content": "Figure 12 represent the arithmetic circuit corresponding to FFT with", + "type": "text" + }, + { + "bbox": [ + 382, + 615, + 407, + 624 + ], + "score": 0.88, + "content": "n = 4", + "type": "inline_equation" + } + ], + "index": 27 + } + ], + "index": 27 + } + ], + "page_idx": 43, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 108, + 27, + 293, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 25, + 294, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 25, + 294, + 38 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2020", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 300, + 752, + 311, + 759 + ], + "lines": [ + { + "bbox": [ + 298, + 750, + 312, + 763 + ], + "spans": [ + { + "bbox": [ + 298, + 750, + 312, + 763 + ], + "score": 1.0, + "content": "", + "type": "text", + "height": 13, + "width": 14 + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 105, + 679, + 506, + 732 + ], + "lines": [ + { + "bbox": [ + 114, + 675, + 507, + 694 + ], + "spans": [ + { + "bbox": [ + 114, + 675, + 507, + 694 + ], + "score": 1.0, + "content": "16This follows from the fact that by definition any linear arithmetic circuit is also an arithmetic circuit; the", + "type": "text" + } + ] + }, + { + "bbox": [ + 106, + 689, + 253, + 700 + ], + "spans": [ + { + "bbox": [ + 106, + 689, + 253, + 700 + ], + "score": 1.0, + "content": "other direction follows from Theorem 4.", + "type": "text" + } + ] + }, + { + "bbox": [ + 116, + 696, + 506, + 715 + ], + "spans": [ + { + "bbox": [ + 116, + 699, + 134, + 710 + ], + "score": 0.26, + "content": "^ { 1 7 } \\mathrm { T o }", + "type": "inline_equation" + }, + { + "bbox": [ + 134, + 696, + 506, + 715 + ], + "score": 1.0, + "content": "the best of our knowledge, this connection was explicitly made by De Sa et al. (2018) though the", + "type": "text" + } + ] + }, + { + "bbox": [ + 106, + 710, + 225, + 721 + ], + "spans": [ + { + "bbox": [ + 106, + 710, + 225, + 721 + ], + "score": 1.0, + "content": "connection seems to be folklore.", + "type": "text" + } + ] + }, + { + "bbox": [ + 114, + 718, + 328, + 735 + ], + "spans": [ + { + "bbox": [ + 114, + 718, + 328, + 735 + ], + "score": 1.0, + "content": "18Using the conversion mentioned in the paragraph above.", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 82, + 504, + 116 + ], + "lines": [ + { + "bbox": [ + 106, + 82, + 505, + 95 + ], + "spans": [ + { + "bbox": [ + 106, + 82, + 505, + 95 + ], + "score": 1.0, + "content": "One thing to note about the arithmetic circuit above is that all the multiplications involve at least one", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 93, + 505, + 105 + ], + "spans": [ + { + "bbox": [ + 105, + 93, + 219, + 105 + ], + "score": 1.0, + "content": "input that is a constant from", + "type": "text" + }, + { + "bbox": [ + 219, + 94, + 228, + 104 + ], + "score": 0.5, + "content": "\\mathbb { F }", + "type": "inline_equation" + }, + { + "bbox": [ + 228, + 93, + 505, + 105 + ], + "score": 1.0, + "content": "(recall that we can assume that the entries of A are constants that can", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 106, + 105, + 498, + 116 + ], + "spans": [ + { + "bbox": [ + 106, + 105, + 498, + 116 + ], + "score": 1.0, + "content": "be used to build the circuit). This leads to the following important sub-class of arithmetic circuits:", + "type": "text" + } + ], + "index": 2 + } + ], + "index": 1, + "bbox_fs": [ + 105, + 82, + 505, + 116 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 130, + 505, + 175 + ], + "lines": [ + { + "bbox": [ + 106, + 131, + 505, + 142 + ], + "spans": [ + { + "bbox": [ + 106, + 131, + 505, + 142 + ], + "score": 1.0, + "content": "Definition M.2 (Linear Arithmetic Circuits). An arithmetic circuit is called a linear arithmetic", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 106, + 142, + 505, + 154 + ], + "spans": [ + { + "bbox": [ + 106, + 142, + 505, + 154 + ], + "score": 1.0, + "content": "circuit if it only uses addition, subtraction and multiplication. Further, every multiplcation has a", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 153, + 505, + 165 + ], + "spans": [ + { + "bbox": [ + 105, + 153, + 187, + 165 + ], + "score": 1.0, + "content": "fixed constant from", + "type": "text" + }, + { + "bbox": [ + 187, + 153, + 195, + 163 + ], + "score": 0.61, + "content": "\\mathbb { F }", + "type": "inline_equation" + }, + { + "bbox": [ + 196, + 153, + 505, + 165 + ], + "score": 1.0, + "content": "as at least one of its two inputs. In other words, all gates in the circuit are", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 163, + 446, + 176 + ], + "spans": [ + { + "bbox": [ + 105, + 163, + 295, + 176 + ], + "score": 1.0, + "content": "linear functions of their inputs (i.e. of the form", + "type": "text" + }, + { + "bbox": [ + 295, + 164, + 319, + 174 + ], + "score": 0.37, + "content": "a x +", + "type": "inline_equation" + }, + { + "bbox": [ + 320, + 163, + 405, + 176 + ], + "score": 1.0, + "content": "by for fixed constants", + "type": "text" + }, + { + "bbox": [ + 406, + 164, + 439, + 175 + ], + "score": 0.9, + "content": "a , b \\in \\mathbb { F }", + "type": "inline_equation" + }, + { + "bbox": [ + 440, + 163, + 446, + 176 + ], + "score": 1.0, + "content": ").", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 4.5, + "bbox_fs": [ + 105, + 131, + 505, + 176 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 196, + 505, + 229 + ], + "lines": [ + { + "bbox": [ + 106, + 196, + 505, + 208 + ], + "spans": [ + { + "bbox": [ + 106, + 196, + 505, + 208 + ], + "score": 1.0, + "content": "Intuitively for the matrix-vector multiplication, it makes sense to consider linear arithmetic circuits", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 207, + 505, + 219 + ], + "spans": [ + { + "bbox": [ + 105, + 207, + 281, + 219 + ], + "score": 1.0, + "content": "since the final function we want to compute", + "type": "text" + }, + { + "bbox": [ + 282, + 207, + 298, + 217 + ], + "score": 0.26, + "content": "\\mathbf { A x }", + "type": "inline_equation" + }, + { + "bbox": [ + 298, + 207, + 505, + 219 + ], + "score": 1.0, + "content": "is indeed a linear function of its inputs. For inifinite", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 217, + 466, + 231 + ], + "spans": [ + { + "bbox": [ + 105, + 217, + 151, + 231 + ], + "score": 1.0, + "content": "fields (e.g.", + "type": "text" + }, + { + "bbox": [ + 151, + 218, + 179, + 228 + ], + "score": 0.9, + "content": "\\mathbb { F } = \\mathbb { R }", + "type": "inline_equation" + }, + { + "bbox": [ + 180, + 217, + 191, + 231 + ], + "score": 1.0, + "content": "or", + "type": "text" + }, + { + "bbox": [ + 191, + 218, + 219, + 228 + ], + "score": 0.89, + "content": "\\mathbb { F } = \\mathbb { C }", + "type": "inline_equation" + }, + { + "bbox": [ + 220, + 217, + 466, + 231 + ], + "score": 1.0, + "content": "), it turns out that this is essentially without loss of generality:", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 8, + "bbox_fs": [ + 105, + 196, + 505, + 231 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 243, + 505, + 277 + ], + "lines": [ + { + "bbox": [ + 105, + 243, + 505, + 256 + ], + "spans": [ + { + "bbox": [ + 105, + 243, + 505, + 256 + ], + "score": 1.0, + "content": "Theorem 4 ((Bürgisser et al., 2013)). Let F be an infinite field. Any (general) arithmetic circuit to", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 254, + 505, + 267 + ], + "spans": [ + { + "bbox": [ + 105, + 254, + 181, + 267 + ], + "score": 1.0, + "content": "compute Ax over", + "type": "text" + }, + { + "bbox": [ + 182, + 255, + 190, + 265 + ], + "score": 0.67, + "content": "\\mathbb { F }", + "type": "inline_equation" + }, + { + "bbox": [ + 190, + 254, + 505, + 267 + ], + "score": 1.0, + "content": "of size s and depth d can be converted into a linear arithmetic circuit of size", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 107, + 264, + 198, + 279 + ], + "spans": [ + { + "bbox": [ + 107, + 266, + 128, + 278 + ], + "score": 0.9, + "content": "O ( s )", + "type": "inline_equation" + }, + { + "bbox": [ + 128, + 264, + 171, + 279 + ], + "score": 1.0, + "content": "and depth", + "type": "text" + }, + { + "bbox": [ + 172, + 266, + 193, + 278 + ], + "score": 0.91, + "content": "O ( d )", + "type": "inline_equation" + }, + { + "bbox": [ + 193, + 264, + 198, + 279 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 11, + "bbox_fs": [ + 105, + 243, + 505, + 279 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 298, + 504, + 320 + ], + "lines": [ + { + "bbox": [ + 106, + 298, + 505, + 310 + ], + "spans": [ + { + "bbox": [ + 106, + 298, + 505, + 310 + ], + "score": 1.0, + "content": "The above result implies that for asymptotic considerations, linear arithmetic circuits for matrix-vector", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 308, + 351, + 321 + ], + "spans": [ + { + "bbox": [ + 105, + 308, + 351, + 321 + ], + "score": 1.0, + "content": "multiplication are equivalent to general arithmetic circuits.16", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 13.5, + "bbox_fs": [ + 105, + 298, + 505, + 321 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 325, + 505, + 358 + ], + "lines": [ + { + "bbox": [ + 106, + 325, + 506, + 338 + ], + "spans": [ + { + "bbox": [ + 106, + 325, + 349, + 338 + ], + "score": 1.0, + "content": "One important property of linear arithmetic circuits of depth", + "type": "text" + }, + { + "bbox": [ + 349, + 326, + 356, + 336 + ], + "score": 0.71, + "content": "d", + "type": "inline_equation" + }, + { + "bbox": [ + 356, + 325, + 506, + 338 + ], + "score": 1.0, + "content": ", which we will use in our arguments,", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 336, + 506, + 349 + ], + "spans": [ + { + "bbox": [ + 105, + 336, + 367, + 349 + ], + "score": 1.0, + "content": "is that such a circuit can be equivalently represented as product of", + "type": "text" + }, + { + "bbox": [ + 368, + 338, + 374, + 347 + ], + "score": 0.82, + "content": "d", + "type": "inline_equation" + }, + { + "bbox": [ + 375, + 336, + 506, + 349 + ], + "score": 1.0, + "content": "sparse matrices (see the proof of", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 347, + 267, + 359 + ], + "spans": [ + { + "bbox": [ + 105, + 347, + 267, + 359 + ], + "score": 1.0, + "content": "Theorem 1 for the precise derivation17).", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 16, + "bbox_fs": [ + 105, + 325, + 506, + 359 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 364, + 505, + 431 + ], + "lines": [ + { + "bbox": [ + 105, + 364, + 506, + 377 + ], + "spans": [ + { + "bbox": [ + 105, + 364, + 506, + 377 + ], + "score": 1.0, + "content": "As mentioned earlier, a vast majority of efficient matrix vector multiplication algorithms are equivalent", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 375, + 506, + 388 + ], + "spans": [ + { + "bbox": [ + 105, + 375, + 506, + 388 + ], + "score": 1.0, + "content": "to small (both in size and depth) linear arithmetic circuit. For example the FFT can be thought of as an", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 386, + 505, + 399 + ], + "spans": [ + { + "bbox": [ + 105, + 386, + 505, + 399 + ], + "score": 1.0, + "content": "efficient arithmetic circuit to compute the Discrete Fourier Transform (indeed when one converts the", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 397, + 505, + 410 + ], + "spans": [ + { + "bbox": [ + 105, + 397, + 505, + 410 + ], + "score": 1.0, + "content": "linear arithmetic circuit for FFT into a matrix decomposition,18 then each matrix in the decomposition", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 408, + 505, + 421 + ], + "spans": [ + { + "bbox": [ + 105, + 408, + 505, + 421 + ], + "score": 1.0, + "content": "is a butterfly factor, with each block matrix in each factor being the same). For an illustration of this", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 419, + 333, + 432 + ], + "spans": [ + { + "bbox": [ + 105, + 419, + 199, + 432 + ], + "score": 1.0, + "content": "consider the DFT with", + "type": "text" + }, + { + "bbox": [ + 199, + 420, + 225, + 430 + ], + "score": 0.89, + "content": "n = 4", + "type": "inline_equation" + }, + { + "bbox": [ + 225, + 419, + 333, + 432 + ], + "score": 1.0, + "content": "as illustrated in Figure 11.", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 20.5, + "bbox_fs": [ + 105, + 364, + 506, + 432 + ] + }, + { + "type": "image", + "bbox": [ + 266, + 468, + 344, + 546 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 266, + 468, + 344, + 546 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 266, + 468, + 344, + 546 + ], + "spans": [ + { + "bbox": [ + 266, + 468, + 344, + 546 + ], + "score": 0.941, + "type": "image", + "image_path": "a3cbb522b99a7e6b2265f68e5eb9f6e0cce9cdc88c2032108159d5ac677d7182.jpg" + } + ] + } + ], + "index": 24.5, + "virtual_lines": [ + { + "bbox": [ + 266, + 468, + 344, + 507.0 + ], + "spans": [], + "index": 24 + }, + { + "bbox": [ + 266, + 507.0, + 344, + 546.0 + ], + "spans": [], + "index": 25 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 251, + 560, + 360, + 572 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 249, + 559, + 361, + 572 + ], + "spans": [ + { + "bbox": [ + 249, + 559, + 361, + 572 + ], + "score": 1.0, + "content": "Figure 11: DFT of order 4.", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 26 + } + ], + "index": 25.25 + }, + { + "type": "text", + "bbox": [ + 106, + 614, + 409, + 626 + ], + "lines": [ + { + "bbox": [ + 105, + 613, + 407, + 628 + ], + "spans": [ + { + "bbox": [ + 105, + 613, + 381, + 628 + ], + "score": 1.0, + "content": "Figure 12 represent the arithmetic circuit corresponding to FFT with", + "type": "text" + }, + { + "bbox": [ + 382, + 615, + 407, + 624 + ], + "score": 0.88, + "content": "n = 4", + "type": "inline_equation" + } + ], + "index": 27 + } + ], + "index": 27, + "bbox_fs": [ + 105, + 613, + 407, + 628 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "image", + "bbox": [ + 106, + 80, + 504, + 248 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 106, + 80, + 504, + 248 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 106, + 80, + 504, + 248 + ], + "spans": [ + { + "bbox": [ + 106, + 80, + 504, + 248 + ], + "score": 0.968, + "type": "image", + "image_path": "ed59117b4e076a126cd83314d57abe6b306ececc3fd8efefc8ff242d70ff5fe8.jpg" + } + ] + } + ], + "index": 1, + "virtual_lines": [ + { + "bbox": [ + 106, + 80, + 504, + 136.0 + ], + "spans": [], + "index": 0 + }, + { + "bbox": [ + 106, + 136.0, + 504, + 192.0 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 106, + 192.0, + 504, + 248.0 + ], + "spans": [], + "index": 2 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 191, + 259, + 418, + 272 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 192, + 259, + 419, + 273 + ], + "spans": [ + { + "bbox": [ + 192, + 259, + 419, + 273 + ], + "score": 1.0, + "content": "Figure 12: Arithmetic circuit for 4-DFT from Figure 11.", + "type": "text" + } + ], + "index": 3 + } + ], + "index": 3 + } + ], + "index": 2.0 + }, + { + "type": "text", + "bbox": [ + 107, + 291, + 505, + 325 + ], + "lines": [ + { + "bbox": [ + 105, + 291, + 505, + 305 + ], + "spans": [ + { + "bbox": [ + 105, + 291, + 505, + 305 + ], + "score": 1.0, + "content": "Finally, Figure 13 is representation of the arithmetic circuit of Figure 12 as a product of a butterfly", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 302, + 505, + 315 + ], + "spans": [ + { + "bbox": [ + 105, + 302, + 505, + 315 + ], + "score": 1.0, + "content": "matrix and (the bit-reversal) permutation. 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Each of these pieces are", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 540, + 506, + 552 + ], + "spans": [ + { + "bbox": [ + 106, + 540, + 506, + 552 + ], + "score": 1.0, + "content": "well known to have small (depth and size) linear arithmetic circuits (since FFT has these properties).", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 551, + 505, + 565 + ], + "spans": [ + { + "bbox": [ + 105, + 551, + 505, + 565 + ], + "score": 1.0, + "content": "Finally, the divide and conquer structure of the algorithms leads to the circuit being of low depth. See", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 563, + 426, + 575 + ], + "spans": [ + { + "bbox": [ + 105, + 563, + 426, + 575 + ], + "score": 1.0, + "content": "the book of Pan (Pan, 2001) for a more elaborate description of this connection.", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 20 + }, + { + "type": "text", + "bbox": [ + 107, + 579, + 505, + 668 + ], + "lines": [ + { + "bbox": [ + 106, + 580, + 504, + 591 + ], + "spans": [ + { + "bbox": [ + 106, + 580, + 504, + 591 + ], + "score": 1.0, + "content": "In fact, the recent work of De Sa et al. (De Sa et al., 2018) makes this fact explicit and presents the", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 591, + 505, + 603 + ], + "spans": [ + { + "bbox": [ + 105, + 591, + 505, + 603 + ], + "score": 1.0, + "content": "most general known structure on matrices that imply near-linear size linear arithmetic circuits for the", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 602, + 505, + 614 + ], + "spans": [ + { + "bbox": [ + 105, + 602, + 505, + 614 + ], + "score": 1.0, + "content": "corresponding matrix vector multiplication. 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We note", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 646, + 505, + 658 + ], + "spans": [ + { + "bbox": [ + 105, + 646, + 505, + 658 + ], + "score": 1.0, + "content": "that structured matrices with low displacement rank have been used to replace fully connected layers", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 657, + 428, + 669 + ], + "spans": [ + { + "bbox": [ + 105, + 657, + 428, + 669 + ], + "score": 1.0, + "content": "in some neural network architectures (Sainath et al., 2013; Thomas et al., 2018).", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 27.5 + } + ], + "page_idx": 44, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 107, + 27, + 293, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 26, + 293, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 26, + 293, + 38 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2020", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 300, + 751, + 311, + 760 + ], + "lines": [ + { + "bbox": [ + 298, + 750, + 312, + 763 + ], + "spans": [ + { + "bbox": [ + 298, + 750, + 312, + 763 + ], + "score": 1.0, + "content": "45", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "image", + "bbox": [ + 106, + 80, + 504, + 248 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 106, + 80, + 504, + 248 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 106, + 80, + 504, + 248 + ], + "spans": [ + { + "bbox": [ + 106, + 80, + 504, + 248 + ], + "score": 0.968, + "type": "image", + "image_path": "ed59117b4e076a126cd83314d57abe6b306ececc3fd8efefc8ff242d70ff5fe8.jpg" + } + ] + } + ], + "index": 1, + "virtual_lines": [ + { + "bbox": [ + 106, + 80, + 504, + 136.0 + ], + "spans": [], + "index": 0 + }, + { + "bbox": [ + 106, + 136.0, + 504, + 192.0 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 106, + 192.0, + 504, + 248.0 + ], + "spans": [], + "index": 2 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 191, + 259, + 418, + 272 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 192, + 259, + 419, + 273 + ], + "spans": [ + { + "bbox": [ + 192, + 259, + 419, + 273 + ], + "score": 1.0, + "content": "Figure 12: Arithmetic circuit for 4-DFT from Figure 11.", + "type": "text" + } + ], + "index": 3 + } + ], + "index": 3 + } + ], + "index": 2.0 + }, + { + "type": "text", + "bbox": [ + 107, + 291, + 505, + 325 + ], + "lines": [ + { + "bbox": [ + 105, + 291, + 505, + 305 + ], + "spans": [ + { + "bbox": [ + 105, + 291, + 505, + 305 + ], + "score": 1.0, + "content": "Finally, Figure 13 is representation of the arithmetic circuit of Figure 12 as a product of a butterfly", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 302, + 505, + 315 + ], + "spans": [ + { + "bbox": [ + 105, + 302, + 505, + 315 + ], + "score": 1.0, + "content": "matrix and (the bit-reversal) permutation. 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Each of these pieces are", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 540, + 506, + 552 + ], + "spans": [ + { + "bbox": [ + 106, + 540, + 506, + 552 + ], + "score": 1.0, + "content": "well known to have small (depth and size) linear arithmetic circuits (since FFT has these properties).", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 551, + 505, + 565 + ], + "spans": [ + { + "bbox": [ + 105, + 551, + 505, + 565 + ], + "score": 1.0, + "content": "Finally, the divide and conquer structure of the algorithms leads to the circuit being of low depth. See", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 563, + 426, + 575 + ], + "spans": [ + { + "bbox": [ + 105, + 563, + 426, + 575 + ], + "score": 1.0, + "content": "the book of Pan (Pan, 2001) for a more elaborate description of this connection.", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 20, + "bbox_fs": [ + 105, + 497, + 506, + 575 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 579, + 505, + 668 + ], + "lines": [ + { + "bbox": [ + 106, + 580, + 504, + 591 + ], + "spans": [ + { + "bbox": [ + 106, + 580, + 504, + 591 + ], + "score": 1.0, + "content": "In fact, the recent work of De Sa et al. (De Sa et al., 2018) makes this fact explicit and presents the", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 591, + 505, + 603 + ], + "spans": [ + { + "bbox": [ + 105, + 591, + 505, + 603 + ], + "score": 1.0, + "content": "most general known structure on matrices that imply near-linear size linear arithmetic circuits for the", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 602, + 505, + 614 + ], + "spans": [ + { + "bbox": [ + 105, + 602, + 505, + 614 + ], + "score": 1.0, + "content": "corresponding matrix vector multiplication. Their work combines two separate classes of structures", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 613, + 505, + 625 + ], + "spans": [ + { + "bbox": [ + 105, + 613, + 505, + 625 + ], + "score": 1.0, + "content": "matrices– orthogonal polynomial transforms (Driscoll et al., 1997; Szegö, 1967) as well as matrices", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 622, + 506, + 637 + ], + "spans": [ + { + "bbox": [ + 105, + 622, + 506, + 637 + ], + "score": 1.0, + "content": "with low displacement rank (Kailath et al., 1979; Olshevsky & Shokrollahi, 2000)– and presents a", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 634, + 505, + 646 + ], + "spans": [ + { + "bbox": [ + 105, + 634, + 505, + 646 + ], + "score": 1.0, + "content": "linear class of linear arithmetic circuits to solve their matrix vector multiplication problem. 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SincNet (Ravanelli et al., 2019)17.2
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ShuffleHadamardKaleidoscope (K.)K. vs. Shuffle
0.25 ShuffleNet g844.1% (0.46M)43.9% (0.46M)49.2% (0.51M)+5.0% (+0.05M)
0.5 ShuffleNet g857.1% (1.0M)56.2% (1.0M)59.5% (1.1M)+2.4% (+0.1M)
1.0 ShuffleNet g865.3% (2.5M)65.0% (2.5M)66.5% (2.8M)+1.2% (+0.2M)
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ModelFCRNNCNNDense +CNNK +CNN Unpermuted
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Model# paramsBLEUSentences/secTokens/sec
Transformer (Vaswani et al., 2017)43M34.43.066.4
DynamicConv Transformer (Wu et al., 2019)39M35.23.680.2
DynamicConv Transformer w/ K-matrices (ours)30M34.24.9103.4
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ModelTest set PER%#Parameters
Low rank + LSTM23.6 ± 0.915.5M
Sparse +LSTM21.7 ± 0.915.5M
Circulant +LSTM23.9 ± 0.915.4M
Dense +LSTM15.4 ± 0.615.9M
FFT+LSTM15.7 ± 0.115.4M
Identity +LSTM20.7 ± 0.315.4M
Kaleidoscope + LSTM14.6 ± 0.315.4M
MFSC features + LSTM14.2 ± 0.214.3M
SincNet (Ravanelli et al., 2019)17.210.0M
LiGRU (Ravanelli et al., 2018)13.812.3M
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Target MethodKaleidoscopeLow-rankSparseConvolutionFastfoodRandom
Global Min.Kaleidoscope0.00.00.00.00.0
Low-rank14.90.010.814.611.615.5
Sparse11.712.20.013.17.114.1
With SGDKaleidoscope0.00.018.00.05.114.5
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at end of file diff --git a/parse/train/H1ersoRqtm/H1ersoRqtm.md b/parse/train/H1ersoRqtm/H1ersoRqtm.md new file mode 100644 index 0000000000000000000000000000000000000000..c791e2dc65e5a7dfe1d7a9cd6adf1ad12d6fd0ac --- /dev/null +++ b/parse/train/H1ersoRqtm/H1ersoRqtm.md @@ -0,0 +1,350 @@ +# STRUCTURED NEURAL SUMMARIZATION + +Patrick Fernandes, Miltiadis Allamanis & Marc Brockschmidt + +Microsoft Research +Cambridge, United Kingdom +{t-pafern,miallama,mabrocks}@microsoft.com + +# ABSTRACT + +Summarization of long sequences into a concise statement is a core problem in natural language processing, requiring non-trivial understanding of the input. Based on the promising results of graph neural networks on highly structured data, we develop a framework to extend existing sequence encoders with a graph component that can reason about long-distance relationships in weakly structured data such as text. In an extensive evaluation, we show that the resulting hybrid sequence-graph models outperform both pure sequence models as well as pure graph models on a range of summarization tasks. + +# 1 INTRODUCTION + +Summarization, the task of condensing a large and complex input into a smaller representation that retains the core semantics of the input, is a classical task for natural language processing systems. Automatic summarization requires a machine learning component to identify important entities and relationships between them, while ignoring redundancies and common concepts. + +Current approaches to summarization are based on the sequence-to-sequence paradigm over the words of some text, with a sequence encoder — typically a recurrent neural network, but sometimes a 1D-CNN (Narayan et al., 2018) or using self-attention (McCann et al., 2018) — processing the input and a sequence decoder generating the output. Recent successful implementations of this paradigm have substantially improved performance by focusing on the decoder, extending it with an attention mechanism over the input sequence and copying facilities (See et al., 2017; McCann et al., 2018). However, while standard encoders (e.g. bidirectional LSTMs) theoretically have the ability to handle arbitrary long-distance relationships, in practice they often fail to correctly handle long texts and are easily distracted by simple noise (Jia & Liang, 2017). + +In this work, we focus on an improvement of sequence encoders that is compatible with a wide range of decoder choices. To mitigate the long-distance relationship problem, we draw inspiration from recent work on highly-structured objects (Li et al., 2015; Kipf & Welling, 2017; Gilmer et al., 2017; Allamanis et al., 2018; Cvitkovic et al., 2018). In this line of work, highly-structured data such as entity relationships, molecules and programs is modelled using graphs. Graph neural networks are then successfully applied to directly learn from these graph representations. Here, we propose to extend this idea to weakly-structured data such as natural language. Using existing tools, we can annotate (accepting some noise) such data with additional relationships (e.g. co-references) to obtain a graph. However, the sequential aspect of the input data is still rich in meaning, and thus we propose a hybrid model in which a standard sequence encoder generates rich input for a graph neural network. In our experiments, the resulting combination outperforms baselines that use pure sequence or pure graph-based representations. + +Briefly, the contributions of our work are: 1. A framework that extends standard sequence encoder models with a graph component that leverages additional structure in sequence data. 2. Application of this extension to a range of existing sequence models and an extensive evaluation on three summarization tasks from the literature. 3. We release all used code and data at https://github.com/CoderPat/structured-neural-summarization. + +public void Add(string name, object value $=$ null, DbType? dbType $=$ null, ParameterDirection? direction $=$ null, int? size $=$ null, byte? precision $=$ null, byte? scale $=$ null) { parameters[Clean(name) $] =$ new ParamInfo{ Name $=$ name, Value $=$ value, ParameterDirection $=$ direction ?? ParameterDirection.Input, DbType $=$ dbType, Size $=$ size, Precision $=$ precision, Scale $=$ scale };} + +Ground truth: BILSTM LSTM: $\mathbf { B I L S T M + G N N } \to \mathbf { L S T M } .$ : $\mathbf { B I L S T M + G N N } \to \mathbf { L S T M + P O I N T E R }$ : + +add a parameter to this dynamic parameter list adds a new parameter to the specified parameter creates a new instance of the dynamic type specified add a parameter to a list of parameters + +Figure 1: An example from the dataset for the METHODDOC source code summarization task along with the outputs of a baseline and our models. In the METHODNAMING dataset, this method appears as a sample requiring to predict the name Add as a subtoken sequence of length 1. + +# 2 STRUCTURED SUMMARIZATION TASKS + +In this work, we consider three summarization tasks with different properties. All tasks follow the common pattern of translating a long (structured) sequence into a shorter sequence while trying to preserve as much meaning as possible. The first two tasks are related to the summarization of source code (Figure 1), which is highly structured and thus can profit most from models that can take advantage of this structure; the final task is a classical natural language task illustrating that hybrid sequence-graph models are applicable for less structured inputs as well. + +METHODNAMING The aim of this task is to infer the name of a function (or method in objectoriented languages, such as Java, Python and C#) given its source code (Allamanis et al., 2016). Although method names are a single token, they are usually composed of one or more subtokens (split using snake case or camelCase) and thus, the method naming task can be cast as predicting a sequence of subtokens. Consequently, method names represent an “extreme” summary of the functionality of a given function (on average, the names in the Java dataset have only 2.9 subtokens). Notably, the vocabulary of tokens used in names is very large (due to abbreviations and domainspecific jargon), but this is mitigated by the fact that $33 \%$ of subtokens in names can be copied directly from subtokens in the method’s source code. Finally, source code is highly structured input data with known semantics, which can be exploited to support name prediction. + +METHODDOC Similar to the first task, the aim of this task is to predict a succinct description of the functionality of a method given its source code (Barone & Sennrich, 2017). Such descriptions usually appear as documentation of methods (e.g. “docstrings” in Python or “JavaDocs” in Java). While the task shares many characteristics with the METHODNAMING task, the target sequence is substantially longer (on average 19.1 tokens in our C# dataset) and only $1 9 . 4 \%$ of tokens in the documentation can be copied from the code. While method documentation is nearer to standard natural language than method names, it mixes project-specific jargon, code segments and often describes non-functional aspects of the code, such as performance characteristics and design considerations. + +NLSUMMARIZATION Finally, we consider the classic summarization of natural language as widely studied in NLP research. Specifically, we are interested in abstractive summarization, where given some text input (e.g. a news article) a machine learning model produces a novel natural language summary. Traditionally, NLP summarization methods treat text as a sequence of sentences and each one of them as a sequence of words (tokens). The input data has less explicitly defined structure than our first two tasks. However, we recast the task as a structured summarization problem by considering additional linguistic structure, including named entities and entity coreferences as inferred by existing NLP tools. + +# 3 MODEL + +As discussed above, standard neural approaches to summarization follow the sequence-to-sequence framework. In this setting, most decoders only require a representation $^ { h }$ of the complete input sequence (e.g. the final state of an RNN) and per-token representations $h _ { t _ { i } }$ for each input token $t _ { i }$ These token representations are then used as the “memories” of an attention mechanism (Bahdanau et al., 2014; Luong et al., 2015) or a pointer network (Vinyals et al., 2015a). + +In this work, we propose an extension of sequence encoders that allows us to leverage known (or inferred) relationships among elements in the input data. To achieve that, we combine sequence encoders with graph neural networks (GNNs) (Li et al., 2015; Gilmer et al., 2017; Kipf & Welling, 2017). For this, we first use a standard sequential encoder (e.g. bidirectional RNNs) to obtain a pertoken representation $h _ { t _ { i } }$ , which we then feed into a GNN as the initial node representations. The resulting per-node (i.e. per-token) representations $\boldsymbol { h } _ { t _ { i } } ^ { \prime }$ can then be used by an unmodified decoder. Experimentally, we found this to surpass models that use either only the sequential structure or only the graph structure (see Sect. 4). We now discuss the different parts of our model in detail. + +Gated Graph Neural Networks To process graphs, we follow Li et al. (2015) and briefly summarize the core concepts of GGNNs here. A graph $\mathcal { G } = ( \nu , \pmb { \varepsilon } , \pmb { X } )$ is composed of a set of nodes $\nu$ , node features $\boldsymbol { X }$ , and a list of directed edge sets ${ \pmb { \mathcal { E } } } = ( { \mathcal { E } } _ { 1 } , \dots , { \mathcal { E } } _ { K } )$ where $K$ is the number of edge types. Each $v \in \nu$ is associated with a real-valued vector $\scriptstyle { \mathbf { { \mathit { x } } } } _ { \mathit { v } }$ representing the features of the node (e.g., the embedding of a string label of that node), which is used for the initial state ${ h } _ { v } ^ { ( 0 ) }$ of a node. + +Information is propagated through the graph using neural message passing (Gilmer et al., 2017). For this, every node $v$ sends messages to its neighbors by transforming its current representation $\boldsymbol { h } _ { v } ^ { ( i ) }$ using an edge-type dependent function $f _ { k }$ . Here, $f _ { k }$ can be an arbitrary function; we use a simple linear layer. By computing all messages at the same time, all states can be updated simultaneously. In particular, a new state for a node $v$ is computed by aggregating all incoming messages as $m _ { v } ^ { ( i ) } = \bar { g } ( \{ f _ { k } ( h _ { u } ^ { ( i ) } ) \} )$ there is an edge of type $k$ from $u$ to $v \}$ ). $g$ is an aggregation function; we use elementwise summation for $g$ . Given the aggregated message $\mathbf { \it { m } } _ { v } ^ { ( i ) }$ and the current state vector $\boldsymbol { h } _ { v } ^ { ( i ) }$ of node $v$ , we can compute the new state 1) = GRU(m(i)v , h(i)v ), where GRU is the recurrent cell function of a gated recurrent unit. These dynamics are rolled out for a fixed number of timesteps $T$ , and the state vectors resulting from the final step are used as output node representations, i.e., $\mathrm { G N N } ( ( \mathcal { V } , \pmb { \mathcal { E } } , X ) ) = \{ \pmb { h } _ { v } ^ { ( T ) } \} _ { v \in \mathcal { V } }$ . + +Sequence GNNs We now explain our novel combination of GGNNs and standard sequence encoders. As input, we take a sequence $\boldsymbol { S } = \left[ s _ { 1 } \ldots s _ { N } \right]$ and $K$ binary relationships $R _ { 1 } \ldots R _ { K } \in S \times S$ between elements of the sequence. For example, $R _ { = }$ could be the equality relationship $\left\{ \left( s _ { i } , s _ { j } \right) \right. \mid$ $s _ { i } = s _ { j } \}$ . The choice and construction of relationships is dataset-dependent, and will be discussed in detail in Sect. 4. Given any sequence encoder $\mathcal { S E }$ that maps $S$ to per-element representations $\left[ \mathbf { e } _ { 1 } \ldots \mathbf { e } _ { N } \right]$ and a sequence representation e (e.g. a bidirectional RNN), we can construct the sequence GNN $\mathcal { S } \mathcal { E } _ { G N N }$ by simply computing $\mathbf { \bar { e } } _ { 1 } ^ { \prime } \cdot \cdot \cdot \mathbf { e } _ { N } ^ { \prime } ] = \mathbf { G N N } ( ( S , [ R _ { 1 } \dots R _ { K } ] , [ \mathbf { e } _ { 1 } \dots \mathbf { e } _ { N } ] ) )$ . To obtain a graph-level representation, we use the weighted averaging mechanism from Gilmer et al. (2017). Concretely, for each node $v$ in the graph, we compute a weight $\sigma ( w ( \pmb { h } _ { v } ^ { ( T ) } ) ) \in [ 0 , 1 ]$ using a learnable function $w$ and the logistic sigmoid $\sigma$ and compute a graph-level representation as $\begin{array} { r } { \hat { \mathbf { e } } \equiv \sum _ { 1 \leq i \leq N } \sigma ( w ( \mathbf { e } _ { i } ^ { \prime } ) ) \cdot \mathbb { N } ( \mathbf { e } _ { i } ^ { \prime } ) } \end{array}$ , where $\aleph$ is another learnable projection function. We found that best results were achieved by computing the final $\mathbf { e ^ { \prime } }$ as $W \cdot ( \mathbf { e } \hat { \mathbf { e } } )$ for some learnable matrix $W$ . + +This method can easily be extended to support additional nodes not present in the original sequence $S$ after running $\mathcal { S E }$ (e.g., to accommodate meta-nodes representing sentences, or non-terminal nodes from a syntax tree). The initial node representation for these additional nodes can come from other sources, such as a simple embedding of their label. + +Implementation Details. Processing large graphs of different shapes efficiently requires to overcome some engineering challenges. For example, the CNN/DM corpus has (on average) about 900 nodes per graph. To allow efficient computation, we use the trick of Allamanis et al. (2018) where all graphs in a minibatch are “flattened” into a single graph with multiple disconnected components. The varying graph sizes also represent a problem for the attention and copying mechanisms in the decoder, as they require to compute a softmax over a variable-sized list of memories. To handle this efficiently without padding, we associate each node in the (flattened) “batch” graph with the index of the sample in the minibatch from which the node originated. Then, using TensorFlow’s unsorted segment $\mathbf { \nabla } _ { \cdot } \star$ operations, we can perform an efficient and numerically stable softmax over the variable number of representations of the nodes of each graph. + +# 4 EVALUATION + +# 4.1 QUANTITATIVE EVALUATION + +We evaluate Sequence GNNs on our three tasks by comparing them to models that use only sequence or graph information, as well as by comparing them to task-specific baselines. We discuss the three tasks, their respective baselines and how we present the data to the models (including the relationships considered in the graph component) next before analyzing the results. + +# 4.1.1 SETUP FOR METHODNAMING + +Datasets, Metrics, and Models. We consider two datasets for the METHODNAMING task. First, we consider the “Java (small)” dataset of Alon et al. (2018a), re-using the train-validation-test splits they have picked. We additionally generated a new dataset from 23 open-source C# projects mined from GitHub (see below for the reasons for this second dataset), removing any duplicates. More information about these datasets can be found in Appendix C. We follow earlier work on METHODNAMING (Allamanis et al., 2016; Alon et al., 2018a) and measure performance using the F1 score over the generated subtokens. However, since the task can be viewed as a form of (extreme) summarization, we also report ROUGE-2 and ROUGE-L scores (Lin, 2004), which we believe to be additional useful indicators for the quality of results. ROUGE-1 is omitted since it is equivalent to F1 score. We note that there is no widely accepted metric for this task and further work identifying the most appropriate metric is required. + +We compare to the current state of the art (Alon et al., 2018a), as well as a sequence-to-sequence implementation from the OpenNMT project (Klein et al.). Concretely, we combine two encoders (a bidirectional LSTM encoder with 1 layer and 256 hidden units, and its sequence GNN extension with 128 hidden units unrolled over 8 timesteps) with two decoders (an LSTM decoder with 1 layer and 256 hidden units with attention over the input sequence, and an extension using a pointer network-style copying mechanism (Vinyals et al., 2015a)). Additionally, we consider self-attention as an alternative to RNN-based sequence encoding architectures. For this, we use the Transformer (Vaswani et al., 2017) implementation in OpenNMT (i.e., using self-attention both for the decoder and the encoder) as a baseline and compare it to a version whose encoder is extended with a GNN component. + +Data Representation Following the work of Allamanis et al. (2016); Alon et al. (2018a), we break up all identifier tokens (i.e. variables, methods, classes, etc.) in the source code into subtokens by splitting them according to camelCase and pascal case heuristics. This allows the models to extract information from the information-rich subtoken structure, and ensures that a copying mechanism in the decoder can directly copy relevant subtokens, something that we found to be very effective for this task. All models are provided with all (sub)tokens belonging to the source code of a method, including its declaration, with the actual method name replaced by a placeholder symbol. + +To construct a graph from the (sub)tokens, we implement a simplified form of the work of Allamanis et al. (2018). First, we introduce additional nodes for each (full) identifier token, and connect the constituent subtokens appearing in the input sequence using a INTOKEN edge; we additionally connect these nodes using a NEXTTOKEN edge. We also add nodes for the parse tree and use edges to indicate that one node is a CHILD of another. Finally, we add LASTLEXICALUSE edges to connect identifiers to their most (lexically) recent use in the source code. + +# 4.1.2 SETUP FOR METHODDOC + +Datasets, Metrics, and Models. We tried to evaluate on the Python dataset of Barone & Sennrich (2017) that contains pairs of method declarations and their documentation (“docstring”). However, following the work of Lopes et al. (2017), we found extensive duplication between different folds of the dataset and were only able to reach comparable results by substantially overfitting to the training data that overlapped with the test set. We have documented details in subsection C.3 and in Allamanis (2018), and decided to instead evaluate on our new dataset of 23 open-source C# projects from above, again removing duplicates and methods without documentation. Following Barone & Sennrich (2017), we measure the BLEU score for all models. However, we also report F1, ROUGE-2 and ROUGE-L scores, which should better reflect the summarization aspect of the task. We consider the same models as for the METHODNAMING task, using the same configuration, and use the same data representation. + +![](images/0f24ece1ecd933d082df2284a4d3190ac4a48d4c10138fc5c01fc0d240b7a43f.jpg) +Figure 2: (Partial) graph of an example input from the CNN/DM corpus. + +# 4.1.3 SETUP FOR NLSUMMARIZATION + +Datasets, Metrics, and Models. We use the CNN/DM dataset (Hermann et al., 2015) using the exact data and split provided by See et al. (2017). The data is constructed from CNN and Daily Mail news articles along with a few sentences that summarize each article. To measure performance, we use the standard ROUGE metrics. We compare our model with the near-to-state-of-the-art work of See et al. (2017), who use a sequence-to-sequence model with attention and copying as basis, but have additionally substantially improved the decoder component. As our contribution is entirely on the encoder side and our model uses a standard sequence decoder, we are not expecting to outperform more recent models that introduce substantial novelty in the structure or training objective of the decoder (Chen & Bansal, 2018; Narayan et al., 2018). Again, we evaluate our contribution using an OpenNMT-based encoder/decoder combination. Concretely, we use a bidirectional LSTM encoder with 1 layer and 256 hidden units, and its sequence GNN extension with 128 hidden units unrolled over 8 timesteps. As decoder, we use an LSTM with 1 layer and 256 hidden units with attention over the input sequence, and an extension using a pointer network-style copying mechanism. + +Data Representation We use Stanford CoreNLP (Manning et al., 2014) (version 3.9.1) to tokenize the text and provide the resulting tokens to the encoder. For the graph construction (Figure 2), we extract the named entities and run coreference resolution using CoreNLP. We connect tokens using a NEXT edge and introduce additional super-nodes for each sentence, connecting each token to the corresponding sentence-node using a IN edge. We also connect subsequent sentence-nodes using a NEXT edge. Then, for each multi-token named entity we create a new node, labeling it with the type of the entity and connecting it with all tokens referring to that entity using an IN edge. Finally, coreferences of entities are connected with a special REF edge. Figure 2 shows a partial graph for an article in the CNN/DM dataset. The goal of this graph construction process is to explicitly annotate important relationships that can be useful for summarization. We note that (a) in early efforts we experimented with adding dependency parse edges, but found that they do not provide significant benefits and (b) that since we retrieve the annotations from CoreNLP, they can contain errors and thus, the performance of the our method is influenced by the accuracy of the upstream annotators of named entities and coreferences. + +Table 1: Evaluation results for all models and tasks. + +
METHODNAMINGF1ROUGE-2ROUGE-L
Java
Alon et al. (2018a)43.0一1
SELFATT →SELFATT24.98.327.4
SELFATT+GNN →SELFATT44.520.943.4
BILSTM →LSTM35.817.939.7
BILSTM+GNN →LSTM44.721.143.1
BILSTM →LSTM+POINTER42.522.445.6
GNN →LSTM+POINTER50.524.848.9
BILSTM+GNN→LSTM+POINTER51.425.050.0
C#
SELFATT →SELFATT41.325.243.2
SELFATT+GNN→→SELFATT62.131.061.1
BILSTM →LSTM48.832.851.8
BILSTM+GNN →LSTM62.631.061.3
BILSTM →LSTM+POINTER57.229.760.4
GNN→LSTM+POINTER63.031.561.3
BILSTM+GNN→LSTM+POINTER63.431.962.4
METHODDOCF1ROUGE-2ROUGE-LBLEU
C#
SELFATT →SELFATT40.027.841.113.9
SELFATT+GNN→SELFATT37.625.637.921.4
BILSTM →LSTM35.215.330.810.0
BILSTM+GNN →LSTM41.128.941.022.5
BILSTM →LSTM+POINTER35.220.836.714.7
GNN →LSTM+POINTER38.925.637.117.7
BILSTM+GNN →→LSTM+POINTER (average pooling)43.229.041.021.3
BILSTM+GNN→LSTM+POINTER45.428.341.122.2
NLSUMMARIZATIONROUGE-1ROUGE-2ROUGE-L
CNN/DM
BILSTM →LSTM33.611.427.9
BILSTM+GNN→LSTM33.013.328.3
See et al. (2017) (+ Pointer)36.415.733.4
BILSTM →LSTM+POINTER35.913.930.3
BILSTM+GNN →LSTM+POINTER38.116.133.2
See et al. (2017) (+ Pointer + Coverage)39.517.336.4
+ +# 4.1.4 RESULTS & ANALYSIS + +We show all results in Tab. 1. Results for models from the literature are taken from the respective papers and repeated here. Across all tasks, the results show the advantage of our hybrid sequence GNN encoders over pure sequence encoders. + +On METHODNAMING, we can see that all GNN-augmented models are able to outperform the current specialized state of the art, requiring only simple graph structure that can easily be obtained using existing parsers for a programming language. The results in performance between the different encoder and decoder configurations nicely show that their effects are largely orthogonal. + +On METHODDOC, the unmodified SELFATT $ s$ ELFATT model already performs quite well, and the augmentation with graph data only improves the BLEU score and worsens the results on ROUGE. Inspection of the results shows that this is due to the length of predictions. Whereas the ground truth data has on average 19 tokens in each result, SELFATT $ s$ ELFATT predicts on average 11 tokens, and SELFATT+GNN SELFATT 16 tokens. Additionally, we experimented with an ablation in which a model is only using graph information, e.g., a setting comparable to a simplification of the architecture of Allamanis et al. (2018). For this, we configured the GNN to use 128-dimensional representations and unrolled it for 10 timesteps, keeping the decoder configuration as for the other models. The results indicate that this configuration performs less well than a pure sequenced model. We speculate that this is mainly due to the fact that 10 timesteps are insufficient to propagate inforpublic static bool TryFormat(float value, Span destination, out int bytesWritten, StandardFormat format $=$ default) { return TryFormatFloatingPoint(value, destination, out bytesWritten, format); } + +Table 2: Ablations on CNN/DM Corpus + +
NLSUMMARIZATION (CNN/DM)ROUGE-1ROUGE-2ROUGE-L
See et al.(2017) (base)31.311.828.8
See et al. (2017) (+ Pointer)36.415.733.4
See et al.(2017) (+ Pointer + Coverage)39.517.336.4
BILSTM →LSTM33.611.427.9
BILSTM →LSTM+POINTER35.913.930.3
BILSTM →LSTM+PoINTER (+ coref/entity annotations)36.214.230.5
BILSTM+GNN →LSTM33.013.328.3
BILSTM+GNN→LSTM+POINTER (only sentence nodes)36.015.229.6
BILSTM+GNN →LSTM+POINTER (sentence nodes + eq edges)36.115.430.3
BILSTM+GNN→LSTM+POINTER38.116.133.2
+ +Ground truth BILSTM LSTM BILST $\mathbf { M } \mathbf { + } \mathbf { G N N } \to \mathbf { L S T M }$ B $\mathbf { \mathbf { \mathbf { I L S T M + G N N } } } \mathbf { \mathbf { \Phi } } \to \mathbf { \mathbf { \mathbf { L S T M + P O I N T E R } } }$ formats a single as a utf8 string formats a number of bytes in a utf8 string formats a timespan as a utf8 string formats a float as a utf8 string + +Figure 3: An example from the dataset for the METHODDOC source code summarization task along with the outputs of a baseline and our models. + +mation across the whole graph, especially in combination with summation as aggregation function for messages in graph information propagation. + +Finally, on NLSUMMARIZATION, our experiments show that the same model suitable for tasks on highly structured code is competitive with specialized models for natural language tasks. While there is still a gap to the best configuration of See et al. (2017) (and an even larger one to more recent work in the area), we believe that this is entirely due to our simplistic decoder and training objective, and that our contribution can be combined with these advances. + +In Table 2 we show some ablations for NLSUMMARIZATION. As we use the same hyperparameters across all datasets and tasks, we additionally perform an experiment with the model of See et al. (2017) (as implemented in OpenNMT) but using our settings. The results achieved by these baselines trend to be a bit worse than the results reported in the original paper, which we believe is due to a lack of hyperparameter optimization for this task. We then evaluated how much the additional linguistic structure provided by CoreNLP helps. First, we add the coreference and entity annotations to the baseline $\mathrm { B I L S T M } \to \mathrm { L S T M } + \mathrm { P O I N T E R }$ model (by extending the embedding of tokens with an embedding of the entity information, and inserting fresh $\mathrm { ~ \omega ~ } _ { \mathrm { i R E F 1 } } \mathrm { ~ \omega ~ }$ ”, . . . tokens at the sources/targets of co-references) and observe only minimal improvements. This suggests that our graph-based encoder is better-suited to exploit additional structured information compared to a biLSTM encoder. We then drop all linguistic structure information from our model, keeping only the sentence edges/nodes. This still improves on the baseline $\mathrm { B I L S T M } \to \mathrm { L S T M } + \mathrm { P O I I }$ NTER model (in the ROUGE-2 score), suggesting that the GNN still yields improvements in the absence of linguistic structure. Finally, we add long-range dependency edges by connecting tokens with equivalent string representations of their stems and observe further minor improvements, indicating that even using only purely syntactical information, without a semantic parse, can already provide gains. + +# 4.2 QUALITATIVE EVALUATION + +We look at a few sample suggestions in our dataset across the tasks. Here we highlight some observations we make that point out interesting aspects and failure cases of our model. + +Figure 4: Sample natural language translations from the CNN-DM dataset. + +
Input: Arsenal,Newcastle United and Southampton have checked on Caen midfelder N'golo Kante .Paris-born Kante is a defensive minded player who has impressed for Caen this season and they are wiling to sellfor around f 5milion.Marseille have been in constant contact with Caen over signing the 24-year-old who has similarities with Lassna Diarra and Claude Makelele in terms of stature and style .N'Golo Kante is attractinginterest froma host of PremierLeague clubs including Arsenal.Caen would be wiling to sellKante for aroundf5million.
Reference:n'golo kante is wanted by arsenal,newcastle and southampton.marseille are also keen on thef5mrated midfielder.kante has been compared to lassana diarra and claude makelele.click here for the latestpremier league news .
See et al.(2O17) (+ Pointer): arsenal,newcastle united and southampton have checked on caen midfieldern'golo kante .paris-born kante is attracting interest from a host of premier league clubs including arsenal .paris-born kante is attracting interest from a host of premier league clubs including arsenal
See et al. (2O17)(+ Pointer + Coverage): arsenal, newcastle united and southampton have checked on caenmidfielder n'golo kante.paris-born kante is a defensive minded player who has impressed for caen this season.marseille have been in constant contact with caen over signing the 24-year-old .
BILSTM+GNN →LSTM: marseille have been linked with caen midfielder %UNK% %UNK% . marseillehave been interested from a host of premier league clubs including arsenal .caen have been interested from ahost of premier league clubs including arsenal .
BILSTM+GNN →LSTM+PoINTER n'golo kante is attracting interest from a host of premier league clubs .marseilehave been in constant contact with caen over signing the24-year-old.the 24-year-old has similaritieswith lassana diarra and claude makelele in terms of stature .
+ +METHODDOC Figures 1 and 3 illustrate typical results of baselines and our model on the METHODDOC task (see Appendix A for more examples). The hardness of the task stems from the large number of distractors and the need to identify the most relevant parts of the input. In Figure 1, the token “parameter” and variations appears many times, and identifying the correct relationship is non-trivial, but is evidently eased by graph edges explicitly denoting these relationships. Similarly, in Figure 3, many variables are passed around, and the semantics of the method require understanding how information flows between them. + +NLSUMMARIZATION Figure 4 shows one sample summarization. More samples for this task can be found in Appendix B. First, we notice that the model produces natural-looking summaries with no noticeable negative impact on the fluency of the language over existing methods. Furthermore, the GNN-based model seems to capture the central named entity in the article and creates a summary centered around that entity. We hypothesize that the GNN component that links long-distance relationships helps capture and maintain a better “global” view of the article, allowing for better identification of central entities. Our model still suffers from repetition of information (see Appendix B), and so we believe that our model would also profit from advances such as taking coverage into account (See et al., 2017) or optimizing for ROUGE-L scores directly via reinforcement learning (Chen & Bansal, 2018; Narayan et al., 2018). + +# 5 RELATED WORK + +Natural language processing research has studied summarization for a long time. Most related is work on abstractive summarization, in which the core content of a given text (usually a news article) is summarized in a novel and concise sentence. Chopra et al. (2016) and Nallapati et al. (2016) use deep learning models with attention on the input text to guide a decoder that generates a summary. See et al. (2017) and McCann et al. (2018) extend this idea with pointer networks (Vinyals et al., 2015a) to allow for copying tokens from the input text to the output summary. These approaches treat text as a simple token sequences, not explicitly exposing additional structure. In principle, deep sequence networks are known to be able to learn the inherent structure of natural language (e.g. in parsing (Vinyals et al., 2015b) and entity recognition (Lample et al., 2016)), but our experiments indicate that explicitly exposing this structure by separating concerns improves performance. + +Recent work in summarization has proposed improved training objectives for summarization, such as tracking coverage of the input document (See et al., 2017) or using reinforcement learning to directly identify actions in the decoder that improve target measures such as ROUGE-L (Chen & Bansal, 2018; Narayan et al., 2018). These objectives are orthogonal to the graph-augmented encoder discussed in this work, and we are interested in combining these efforts in future work. + +Exposing more language structure explicitly has been studied over the last years, with a focus on tree-based models (Tai et al., 2015). Very recently, first uses of graphs in natural language processing have been explored. Marcheggiani & Titov (2017) use graph convolutional networks to encode single sentences and assist machine translation. De Cao et al. (2018) create a graph over named entities over a set of documents to assist question answering. Closer to our work is the work of Liu et al. (2018), who use abstract meaning representation (AMR), in which the source document is first parsed into AMR graphs, before a summary graph is created, which is finally rendered in natural language. In contrast to that work we do not use AMRs but directly encode relatively simple relationships directly on the tokenized text, and do not treat summarization as a graph rewrite problem. Combining our encoder with AMRs to use richer graph structures may be a promising future direction. + +Finally, summarization in source code has also been studied in the forms of method naming, comment and documentation prediction. Method naming has been tackled with a series of models. For example, Allamanis et al. (2015) use a log-bilinear network to predict method names from features, and later extend this idea to use a convolutional attention network over the tokens of a method to predict the subtokens of names (Allamanis et al., 2016). Raychev et al. (2015) and Bichsel et al. (2016) use CRFs for a range of tasks on source code, including the inference of names for variables and methods. Recently, Alon et al. (2018b;a) extract and encode paths from the syntax tree of a program, setting the state of the art in accuracy on method naming. + +Linking text to code can have useful applications, such as code search (Gu et al., 2018), traceability (Guo et al., 2017), and detection of redundant method comments (Louis et al., 2018). Most approaches on source code either treat it as natural language (i.e., a token sequence), or use a language parser to explicitly expose its tree structure. For example, Barone & Sennrich (2017) use a simple sequenceto-sequence baseline, whereas Hu et al. (2017) summarize source code by linearizing the abstract syntax tree of the code and using a sequence-to-sequence model. Wan et al. (2018) instead directly operate on the tree structure using tree recurrent neural networks (Tai et al., 2015). The use of additional structure on related tasks on source code has been studied recently, for example in models that are conditioned on learned traversals of the syntax tree (Bielik et al., 2016) and in graph-based approaches (Allamanis et al., 2018; Cvitkovic et al., 2018). However, as noted by Liao et al. (2018), GNN-based approaches suffer from a tension between the ability to propagate information across large distances in a graph and the computational expense of the propagation function, which is linear in the number of graph edges per propagation step. + +# 6 DISCUSSION & CONCLUSIONS + +We presented a framework for extending sequence encoders with a graph component that can leverage rich additional structure. In an evaluation on three different summarization tasks, we have shown that this augmentation improves the performance of a range of different sequence models across all tasks. We are excited about this initial progress and look forward to deeper integration of mixed sequence-graph modeling in a wide range of tasks across both formal and natural languages. The key insight, which we believe to be widely applicable, is that inductive biases induced by explicit relationship modeling are a simple way to boost the practical performance of existing deep learning systems. + +# REFERENCES + +Miltiadis Allamanis. The adverse effects of code duplication in machine learning models of code. arXiv preprint arXiv:1812.06469, 2018. + +Miltiadis Allamanis, Earl T Barr, Christian Bird, and Charles Sutton. Suggesting accurate method and class names. In Proceedings of the 2015 10th Joint Meeting on Foundations of Software Engineering, pp. 38–49. ACM, 2015. + +Miltiadis Allamanis, Hao Peng, and Charles Sutton. A convolutional attention network for extreme summarization of source code. In International Conference on Machine Learning, pp. 2091–2100, 2016. + +Miltiadis Allamanis, Marc Brockschmidt, and Mahmoud Khademi. Learning to represent programs with graphs. In International Conference on Learning Representations, 2018. + +Uri Alon, Omer Levy, and Eran Yahav. code2seq: Generating sequences from structured representations of code. arXiv preprint arXiv:1808.01400, 2018a. + +Uri Alon, Meital Zilberstein, Omer Levy, and Eran Yahav. A general path-based representation for predicting program properties. In Proceedings of the 39th ACM SIGPLAN Conference on Programming Language Design and Implementation, pp. 404–419. ACM, 2018b. + +Dzmitry Bahdanau, Kyunghyun Cho, and Yoshua Bengio. Neural machine translation by jointly learning to align and translate. arXiv preprint arXiv:1409.0473, 2014. + +Antonio Valerio Miceli Barone and Rico Sennrich. A parallel corpus of Python functions and documentation strings for automated code documentation and code generation. In Proceedings of the Eighth International Joint Conference on Natural Language Processing (Volume 2: Short Papers), volume 2, pp. 314–319, 2017. + +Benjamin Bichsel, Veselin Raychev, Petar Tsankov, and Martin Vechev. Statistical deobfuscation of android applications. In Proceedings of the 2016 ACM SIGSAC Conference on Computer and Communications Security, pp. 343–355. ACM, 2016. + +Pavol Bielik, Veselin Raychev, and Martin Vechev. PHOG: probabilistic model for code. In International Conference on Machine Learning (ICML), 2016. + +Yen-Chun Chen and Mohit Bansal. Fast abstractive summarization with reinforce-selected sentence rewriting. arXiv preprint arXiv:1805.11080, 2018. + +Sumit Chopra, Michael Auli, and Alexander M Rush. Abstractive sentence summarization with attentive recurrent neural networks. In Proceedings of the 2016 Conference of the North American Chapter of the Association for Computational Linguistics: Human Language Technologies, pp. 93–98, 2016. + +Milan Cvitkovic, Badal Singh, and Anima Anandkumar. Deep learning on code with an unbounded vocabulary. In Machine Learning 4 Programming, 2018. + +Nicola De Cao, Wilker Aziz, and Ivan Titov. Question answering by reasoning across documents with graph convolutional networks. arXiv preprint arXiv:1808.09920, 2018. + +Justin Gilmer, Samuel S Schoenholz, Patrick F Riley, Oriol Vinyals, and George E Dahl. Neural message passing for quantum chemistry. In International Conference on Machine Learning, pp. 1263–1272, 2017. + +Xiaodong Gu, Hongyu Zhang, and Sunghun Kim. Deep code search. In Proceedings of the 40th International Conference on Software Engineering, pp. 933–944. ACM, 2018. + +Jin Guo, Jinghui Cheng, and Jane Cleland-Huang. Semantically enhanced software traceability using deep learning techniques. In Software Engineering (ICSE), 2017 IEEE/ACM 39th International Conference on, pp. 3–14. IEEE, 2017. + +Karl Moritz Hermann, Tomas Kocisky, Edward Grefenstette, Lasse Espeholt, Will Kay, Mustafa Suleyman, and Phil Blunsom. Teaching machines to read and comprehend. In Advances in Neural Information Processing Systems, pp. 1693–1701, 2015. + +Xing Hu, Yuhan Wei, Ge Li, and Zhi Jin. Codesum: Translate program language to natural language. arXiv preprint arXiv:1708.01837, 2017. + +Robin Jia and Percy Liang. Adversarial examples for evaluating reading comprehension systems. In Empirical Methods in Natural Language Processing (EMNLP), 2017. + +Thomas N. Kipf and Max Welling. Semi-supervised classification with graph convolutional networks. In International Conference on Learning Representations, 2017. + +G. Klein, Y. Kim, Y. Deng, J. Senellart, and A. M. Rush. OpenNMT: Open-Source Toolkit for Neural Machine Translation. ArXiv e-prints. + +Guillaume Lample, Miguel Ballesteros, Sandeep Subramanian, Kazuya Kawakami, and Chris Dyer. Neural architectures for named entity recognition. In Proceedings the North American Chapter of the Association for Computational Linguistics: Human Language Technologies, 2016. + +Yujia Li, Daniel Tarlow, Marc Brockschmidt, and Richard Zemel. Gated graph sequence neural networks. arXiv preprint arXiv:1511.05493, 2015. + +Renjie Liao, Marc Brockschmidt, Daniel Tarlow, Alexander Gaunt, Raquel Urtasun, and Richard S. Zemel. Graph partition neural networks for semi-supervised classification. In International Conference on Learning Representations (ICLR) [Workshop Track], 2018. + +Chin-Yew Lin. ROUGE: A package for automatic evaluation of summaries. Text Summarization Branches Out, 2004. + +Fei Liu, Jeffrey Flanigan, Sam Thomson, Norman Sadeh, and Noah A Smith. Toward abstractive summarization using semantic representations. arXiv preprint arXiv:1805.10399, 2018. + +Cristina V Lopes, Petr Maj, Pedro Martins, Vaibhav Saini, Di Yang, Jakub Zitny, Hitesh Sajnani, and Jan Vitek. Dej´ avu: a map of code duplicates on github. \` Proceedings of the ACM on Programming Languages, 1(OOPSLA):84, 2017. + +Annie Louis, Santanu Kumar Dash, Earl T Barr, and Charles Sutton. Deep learning to detect redundant method comments. arXiv preprint arXiv:1806.04616, 2018. + +Minh-Thang Luong, Hieu Pham, and Christopher D Manning. Effective approaches to attentionbased neural machine translation. arXiv preprint arXiv:1508.04025, 2015. + +Christopher Manning, Mihai Surdeanu, John Bauer, Jenny Finkel, Steven Bethard, and David McClosky. The stanford corenlp natural language processing toolkit. In Proceedings of 52nd annual meeting of the association for computational linguistics: system demonstrations, pp. 55–60, 2014. + +Diego Marcheggiani and Ivan Titov. Encoding sentences with graph convolutional networks for semantic role labeling. In Proceedings of the 2017 Conference on Empirical Methods in Natural Language Processing, pp. 1506–1515, 2017. + +Bryan McCann, Nitish Shirish Keskar, Caiming Xiong, and Richard Socher. The natural language decathlon: Multitask learning as question answering. arXiv preprint arXiv:1806.08730, 2018. + +Ramesh Nallapati, Bowen Zhou, Cicero dos Santos, Caglar Gulcehre, and Bing Xiang. Abstractive text summarization using sequence-to-sequence rnns and beyond. In Proceedings of The 20th SIGNLL Conference on Computational Natural Language Learning, pp. 280–290, 2016. + +Shashi Narayan, Shay B Cohen, and Mirella Lapata. Ranking sentences for extractive summarization with reinforcement learning. arXiv preprint arXiv:1802.08636, 2018. + +Veselin Raychev, Martin Vechev, and Andreas Krause. Predicting program properties from Big Code. In Principles of Programming Languages (POPL), 2015. + +Abigail See, Peter J Liu, and Christopher D Manning. Get to the point: Summarization with pointergenerator networks. In Proceedings of the 55th Annual Meeting of the Association for Computational Linguistics (Volume 1: Long Papers), volume 1, pp. 1073–1083, 2017. + +Kai Sheng Tai, Richard Socher, and Christopher D Manning. Improved semantic representations from tree-structured long short-term memory networks. 2015. + +Ashish Vaswani, Noam Shazeer, Niki Parmar, Jakob Uszkoreit, Llion Jones, Aidan N Gomez, Łukasz Kaiser, and Illia Polosukhin. Attention is all you need. In Advances in Neural Information Processing Systems, pp. 5998–6008, 2017. + +Oriol Vinyals, Meire Fortunato, and Navdeep Jaitly. Pointer networks. In Advances in Neural Information Processing Systems, pp. 2692–2700, 2015a. + +Oriol Vinyals, Lukasz Kaiser, Terry Koo, Slav Petrov, Ilya Sutskever, and Geoffrey Hinton. Grammar as a foreign language. In Advances in Neural Information Processing Systems, 2015b. + +Yao Wan, Zhou Zhao, Min Yang, Guandong Xu, Haochao Ying, Jian Wu, and Philip S Yu. Improving automatic source code summarization via deep reinforcement learning. In Proceedings of the 33rd ACM/IEEE International Conference on Automated Software Engineering, pp. 397–407. ACM, 2018. + +# A CODE SUMMARIZATION SAMPLES + +A.1 METHODDOC + +# C# Sample 1 + +public static bool TryConvertTo(object valueToConvert, Type resultType, IFormatProvider formatProvider, out object result){ result $=$ null; try{ result $=$ ConvertTo(valueToConvert, resultType, formatProvider); catch (InvalidCastException){ return false; catch (ArgumentException){ return false; } return true; +} + +# Ground truth + +sets result to valuetoconvert converted to resulttype considering formatprovider for custom conversions calling the parse method and calling convert . changetype . +converts the specified type to a primitive type . +sets result to resulttype +sets result to valuetoconvert converted to resulttype. + +BILSTM LSTM BILSTM+GNN $\mathbf { J } \to \mathbf { L S T N }$ M BILSTM+G $\mathbf { \ V N } \to \mathbf { L S T M } + \mathbf { \beta }$ POINTER + +# C# Sample 2 + +public virtual Task Init(string name, IProviderRuntime providerRuntime, IProviderConfiguration config){ Log $=$ providerRuntime.GetLogger(this.GetType().FullName); this.serializerSettings $=$ OrleansJsonSerializer.GetDefaultSerializerSettings(); return TaskDone.Done; +} + +Ground truth BILSTM $\bf \Pi \to L S T N$ I BILSTM+G $\mathbf { \partial } _ { \mathbf { i } } \mathbf { N N } \to \mathbf { L S T M }$ BILSTM+GN $\mathbf { N } \to \mathbf { L S T }$ M+POINTER + +initializes the storage provider +creates a grain object +initializes the provider provider +initialization function to initialize the specified provider. + +# C# Sample 3 + +public void NullParameter(){ TaskParameter t $=$ new TaskParameter(null); Assert.Null(t.WrappedParameter); Assert.Equal( TaskParameterType.Null , t.ParameterType ); ((INodePacketTranslatable) t).Translate( TranslationHelpers.GetWriteTranslator()); TaskParameter $\begin{array} { r l } { \pm 2 } & { { } = } \end{array}$ TaskParameter.FactoryForDeserialization( TranslationHelpers.GetReadTranslator()); Assert.Null(t2.WrappedParameter); Assert.Equal(TaskParameterType.Null, t2.ParameterType); +} + +Ground truth BILSTM LSTM BILSTM+G $\mathbf { N N } \to \mathbf { L S T M }$ BILSTM+GNN LSTM+POINTER verifies that construction and serialization with a null parameter is ok tests that the value is a value that is a value to the specified type verifies that construction with an parameter parameter verifies that construction and serialization with a parameter that is null + +# C# Sample 4 + +public override DbGeometryWellKnownValue CreateWellKnownValue(DbGeometry geometryValue){ geometryValue.CheckNull("geometryValue"); var spatialValue $=$ geometryValue.AsSpatialValue(); DbGeometryWellKnownValue result $=$ CreateWellKnownValue(spatialValue, + +() $= >$ SpatialExceptions.CouldNotCreateWellKnownGeometryValueNoSrid("geometryValue"), () $= >$ SpatialExceptions.CouldNotCreateWellKnownGeometryValueNoWkbOrWkt("geometryValue"), (srid, wkb, wkt) $= >$ new DbGeometryWellKnownValue() { CoordinateSystemId $=$ srid, WellKnownBinary $=$ wkb, WellKnownText $=$ wkt }); return result; } + +Ground truth BILSTM LSTM B $\mathbf { I L S T M + G N N } \to \mathbf { L S T M }$ BILSTM+GN $\Gamma \to \mathbf { L } \mathbf { S }$ TM+POINTER + +creates an instance of t:system.data.spatial.dbgeometry value using one or both of the standard well known spatial formats. +creates a t:system.data.spatial.dbgeography value based on the specified well known binary value . +creates a new t:system.data.spatial.dbgeography instance using the specified well known spatial formats . +creates a new instance of the t:system.data.spatial.dbgeometry value based on the provided geometry value and returns the resulting well as known spatial formats . + +# A.2 METHODNAMING + +# C# Sample 1 + +public bool _(D d) { return d ! $=$ null && d.Val == Val ; +} + +Ground truth equals BILSTM LSTM foo BILSTM+GNN LSTM equals BILSTM+ $\mathbf { G N N } \to \mathbf { L S T M } \mathbf { + P O I }$ NTER equals + +# C# Sample 2 + +internal void _(string switchName, Hashtable bag, string parameterName) { object obj $=$ bag[parameterName]; if(obj ! $=$ null){ int value $=$ (int) obj; AppendSwitchIfNotNull(switchName, value.ToString(CultureInfo.InvariantCulture)); +} + +Ground truth append switch with integer BILSTM LSTM set string BILSTM+GN $\ J \to \mathbf { L S T N }$ M append switch BILSTM $\mathbf { + G N N \to L S T M + P O }$ INTER append switch if not null + +# C# Sample 3 + +internal static string _(){ var currentPlatformString $=$ string.Empty; if (RuntimeInformation.IsOSPlatform(OSPlatform.Windows)){ currentPlatformString $=$ "WINDOWS"; } else if (RuntimeInformation.IsOSPlatform(OSPlatform.Linux)){ currentPlatformString $=$ "LINUX"; } else if ( RuntimeInformation.IsOSPlatform(OSPlatform.OSX)) { currentPlatformString $=$ "OSX"; } else { Assert.True(false, "unrecognized current platform"); } return currentPlatformString ; + +Ground truth BILSTM LSTM BILSTM+GNN LSTM BILSTM+ $\mathbf { G N N } \to \mathbf { L S T M + P O I N T }$ ER + +get os platform as string get name +get platform +get current platform string + +# C# Sample 4 + +public override DbGeometryWellKnownValue CreateWellKnownValue(DbGeometry geometryValue){ geometryValue.CheckNull("geometryValue"); var spatialValue $=$ geometryValue.AsSpatialValue(); DbGeometryWellKnownValue result $=$ CreateWellKnownValue(spatialValue, () $= >$ SpatialExceptions.CouldNotCreateWellKnownGeometryValueNoSrid("geometryValue"), () $= >$ SpatialExceptions.CouldNotCreateWellKnownGeometryValueNoWkbOrWkt("geometryValue"), (srid, wkb, wkt) $= >$ new DbGeometryWellKnownValue () { CoordinateSystemId $=$ srid , WellKnownBinary $=$ wkb , WellKnownText $=$ wkt }); return result; +} + +Ground truth BILSTM LSTM BILSTM+GN $\ J \to \mathbf { L S T N }$ BILSTM+G $\mathbf { N N } \to \mathbf { L S T M } +$ POINTER create well known value spatial geometry from xml geometry point get well known value + +# Java Sample 1 + +public static void _(String name, int expected, MetricsRecordBuilder rb) { Assert.assertEquals("Bad value for metric " $^ +$ name, expected, getIntCounter(name, rb)); +} + +Ground truth assert counter BILSTM LSTM assert email value BILSTM+GNN $\ J \to \mathbf { L S T N }$ M assert header BILSTM+GNN LSTM+POINTER assert int counter + +B NATURAL LANGUAGE SUMMARIZATION SAMPLES + +
Input: -LRB- CNN -RRB- Gunshots were fired at rapper Lil Wayne 's tour bus early Sunday inAtlanta .No one was injured in the shooting ,and no arrests have been made,Atlanta Police spokeswoman Elizabeth Espy said .Police are stillooking for suspects . Ofcers were called to aparking lot in Atlanta 's Buckhead neighborhood,Espy said .They arrived at 3:25 a.m. and locatedtwo tour buses that had been shot multiple times .The drivers of the buses said the incident occurredon Interstate 285 near Interstate 75,Espy said .Witnesses provided a limited description of the twovehicles suspected to be involved : a“ Corvette style vehicle ”and an SUV .Lil Wayne was in Atlantafor a performance at Compound nightclub Saturday night . CNN 's Carma Hassan contributed to thisreport .
Reference: rapper lil wayne not injured after shots fired at his tour bus on an atlanta interstate ,police say . no one has been arrested in the shooting
See et al. (2017) (+ Pointer): police are stillooking for suspects . the incident occurred on interstate285 near interstate 75 ,police say . witnesses provided a limited description of the two vehicles suspected to be involved : a“ corvette style vehicle ” and an suv .
See et al. (2017) (+ Pointer + Coverage): lil wayne 's tour bus was shot multiple times , police say: police are stillooking for suspects .they arrived at 3:25 a.m. and located two tour buses that hadbeen shot .
BILSTM+GNN → LSTM: the incident occurred on interstate %UNK% near interstate 75 . no onewas injured in the shooting ,and no arrests have been made ,atlanta police spokeswoman says .
BILSTM+GNN →LSTM+PoINTER gunshots fired at rapper lil wayne 's tour bus early sundayin atlanta ,police say . no one was injured in the shooting ,and no arrests have been made , policesay.
+ +
Input: Tottenham have held further discussions with Marseille over a potential deal for midfielderFlorian Thauvin .The 22-year-old has been left out of the squad for this weekend 's game withMetz as Marseille push for a f 15m sale .The winger,who can also play behind the striker, wasthe subject of enquiries from Spurs earlier in the year and has also been watched by Chelsea andValencia .Tottenham have held further talks with Ligue 1 side Marseille over a possible deal forFlorian Thauvin .Marseille are already resigned to losing Andre Ayew and Andre-Pierre Gignacwith English sides keen on both .Everton,Newcastle and Swansea,have allshown an interest inAyew,who is a free agent in the summer .
Reference: florian thauvin has been left out of marseille 's squad with metz . marseille are pushingfor a f 15m sale and tottenham are interested .the winger has also been watched by chelsea and laliga side valencia .
See et al. (2O17)(+ Pointer): tottenham have held further discussions with marseille over a potentialdeal for midfelder florian thauvin .the 22-year-old has been left out of the squad for this weekend 's game with metz as marseille push for a 15m sale .
See et al. (2017) (+ Pointer + Coverage): florian thauvin has been left out of the squad for thisweekend 's game with metz as marseille push for a 15m sale .the 22-year-old was the subject ofenquiries from spurs earlier in the year .
BILSTM+GNN →LSTM: the 22-year-old has been left out of the squad for this weekend 's gamewith metz .the 22-year-old has been left out of the squad for this weekend 's game with metz .thewinger has been left out of the squad for this weekend 's game with metz .
BILSTM+GNN → LSTM+PoINTER tottenham have held further discussions with marseille overa potential deal .the winger has been left out of the squad for this weekend 's game .tottenham haveheld further talks with marseille over a potential deal .
+ +# C CODE DATASETS INFORMATION + +# C.1 C# DATASET + +We extract the C# dataset from open-source projects on GitHub. Overall, our dataset contains 460,905 methods, 55,635 of which have a documentation comment. The dataset is split $8 5 - 5 - 1 0 \%$ . The projects and exact state of the repositories used is listed in Table 3 + +Table 3: Projects in our C# dataset. Ordered alphabetically. + +
NameGit SHADescription
Akka.NET6f32f6a7Actor-based Concurrent & Distributed Framework
AutoMapper19d6f7fcObject-to-Object Mapping Library
BenchmarkDotNet57005f05Benchmarking Library
CommonMark.NETf3d54530Markdown Parser
CoreCLRcc5dcbe6.NET Core Runtime
CoreFxec1671fd.NETFoundationalLibraries
Dapper3c7cde28Object Mapper Library
EntityFrameworkc4d9a269Object-Relational Mapper
Humanizer2b1c94c4String Manipulation and Formatting
Lean90ee6aaeAlgorithmic Trading Engine
Mono9b9e4f4b.NET Implementation
MsBuild7f95dc15Build Engine
Nancyde458a9bHTTP Service Framework
NLog49fdd08eLogging Library
Opserver9e4d3a40Monitoring System
orleansf89c5866Distributed Virtual Actor Model
Pollyf3d2973dResilience & Transient Fault Handling Library
Powershell9ac701dbCommand-line Shell
ravendb6437de30Document Database
roslyn8ca0a542Compiler & Code Analysis & Compilation
ServiceStack17f081b9Real-time web library
SignalR9b05bcb0Push Notification Framework
Wox13e6c5eeApplication Launcher
+ +# C.2 JAVA METHOD NAMING DATASETS + +We use the datasets and splits of Alon et al. (2018a) provided by their website. Upon scanning all methods in the dataset, the size of the corpora can be seen in Table 4. More information can be found at Alon et al. (2018a). + +# C.3 PYTHON METHOD DOCUMENTATION DATASET + +We use the dataset as split of Barone & Sennrich (2017) provided by their GitHub repository. Upon parsing the dataset, we get 106,065 training samples, 1,943 validation samples and 1,937 test samples. We note that $1 6 . 9 \%$ of the documentation samples in the validation set and $1 5 . 3 \%$ of the samples in test set have a sample with the identical natural language documentation on the training set. This eludes to a potential issue, described by Lopes et al. (2017). See Allamanis (2018) for a lengthier discussion of this issue. + +Table 4: The statistics of the extracted graphs from the Java method naming dataset of Alon et al. (2018a). + +
DatasetTrain SizeValid SizeTest Size
Java - Small691,50523,83756,952
+ +# C.4 GRAPH DATA STATISTICS + +Below we present the data characteristics of the graphs we use across the datasets. + +Table 5: Graph Statistics For Datasets. + +
DatasetAvg Num NodesAvg Num Edges
CNN/DM903.22532.9
C# Method Names125.2239.3
C# Documentation133.5265.9
Java-SmallMethod Names144.4251.6
\ No newline at end of file diff --git a/parse/train/H1ersoRqtm/H1ersoRqtm_content_list.json b/parse/train/H1ersoRqtm/H1ersoRqtm_content_list.json new file mode 100644 index 0000000000000000000000000000000000000000..e1109b629c10dea9a1419bc83ca074a18d7f0976 --- /dev/null +++ b/parse/train/H1ersoRqtm/H1ersoRqtm_content_list.json @@ -0,0 +1,1902 @@ +[ + { + "type": "text", + "text": "STRUCTURED NEURAL SUMMARIZATION ", + "text_level": 1, + "bbox": [ + 176, + 98, + 666, + 121 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Patrick Fernandes, Miltiadis Allamanis & Marc Brockschmidt ", + "bbox": [ + 183, + 143, + 620, + 159 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Microsoft Research \nCambridge, United Kingdom \n{t-pafern,miallama,mabrocks}@microsoft.com ", + "bbox": [ + 184, + 161, + 598, + 202 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "ABSTRACT ", + "text_level": 1, + "bbox": [ + 454, + 237, + 544, + 252 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Summarization of long sequences into a concise statement is a core problem in natural language processing, requiring non-trivial understanding of the input. Based on the promising results of graph neural networks on highly structured data, we develop a framework to extend existing sequence encoders with a graph component that can reason about long-distance relationships in weakly structured data such as text. In an extensive evaluation, we show that the resulting hybrid sequence-graph models outperform both pure sequence models as well as pure graph models on a range of summarization tasks. ", + "bbox": [ + 233, + 280, + 766, + 391 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "1 INTRODUCTION ", + "text_level": 1, + "bbox": [ + 176, + 444, + 336, + 460 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Summarization, the task of condensing a large and complex input into a smaller representation that retains the core semantics of the input, is a classical task for natural language processing systems. Automatic summarization requires a machine learning component to identify important entities and relationships between them, while ignoring redundancies and common concepts. ", + "bbox": [ + 174, + 484, + 825, + 540 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Current approaches to summarization are based on the sequence-to-sequence paradigm over the words of some text, with a sequence encoder — typically a recurrent neural network, but sometimes a 1D-CNN (Narayan et al., 2018) or using self-attention (McCann et al., 2018) — processing the input and a sequence decoder generating the output. Recent successful implementations of this paradigm have substantially improved performance by focusing on the decoder, extending it with an attention mechanism over the input sequence and copying facilities (See et al., 2017; McCann et al., 2018). However, while standard encoders (e.g. bidirectional LSTMs) theoretically have the ability to handle arbitrary long-distance relationships, in practice they often fail to correctly handle long texts and are easily distracted by simple noise (Jia & Liang, 2017). ", + "bbox": [ + 174, + 547, + 825, + 674 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "In this work, we focus on an improvement of sequence encoders that is compatible with a wide range of decoder choices. To mitigate the long-distance relationship problem, we draw inspiration from recent work on highly-structured objects (Li et al., 2015; Kipf & Welling, 2017; Gilmer et al., 2017; Allamanis et al., 2018; Cvitkovic et al., 2018). In this line of work, highly-structured data such as entity relationships, molecules and programs is modelled using graphs. Graph neural networks are then successfully applied to directly learn from these graph representations. Here, we propose to extend this idea to weakly-structured data such as natural language. Using existing tools, we can annotate (accepting some noise) such data with additional relationships (e.g. co-references) to obtain a graph. However, the sequential aspect of the input data is still rich in meaning, and thus we propose a hybrid model in which a standard sequence encoder generates rich input for a graph neural network. In our experiments, the resulting combination outperforms baselines that use pure sequence or pure graph-based representations. ", + "bbox": [ + 174, + 680, + 825, + 847 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Briefly, the contributions of our work are: 1. A framework that extends standard sequence encoder models with a graph component that leverages additional structure in sequence data. 2. Application of this extension to a range of existing sequence models and an extensive evaluation on three summarization tasks from the literature. 3. We release all used code and data at https://github.com/CoderPat/structured-neural-summarization. ", + "bbox": [ + 174, + 854, + 825, + 922 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "public void Add(string name, object value $=$ null, DbType? dbType $=$ null, ParameterDirection? direction $=$ null, int? size $=$ null, byte? precision $=$ null, byte? scale $=$ null) { parameters[Clean(name) $] =$ new ParamInfo{ Name $=$ name, Value $=$ value, ParameterDirection $=$ direction ?? ParameterDirection.Input, DbType $=$ dbType, Size $=$ size, Precision $=$ precision, Scale $=$ scale };} ", + "bbox": [ + 173, + 108, + 808, + 210 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "Ground truth: BILSTM LSTM: $\\mathbf { B I L S T M + G N N } \\to \\mathbf { L S T M } .$ : $\\mathbf { B I L S T M + G N N } \\to \\mathbf { L S T M + P O I N T E R }$ : ", + "bbox": [ + 183, + 224, + 426, + 275 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "add a parameter to this dynamic parameter list adds a new parameter to the specified parameter creates a new instance of the dynamic type specified add a parameter to a list of parameters ", + "bbox": [ + 442, + 224, + 753, + 275 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "Figure 1: An example from the dataset for the METHODDOC source code summarization task along with the outputs of a baseline and our models. In the METHODNAMING dataset, this method appears as a sample requiring to predict the name Add as a subtoken sequence of length 1. ", + "bbox": [ + 174, + 281, + 821, + 324 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "2 STRUCTURED SUMMARIZATION TASKS ", + "text_level": 1, + "bbox": [ + 176, + 359, + 531, + 376 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "In this work, we consider three summarization tasks with different properties. All tasks follow the common pattern of translating a long (structured) sequence into a shorter sequence while trying to preserve as much meaning as possible. The first two tasks are related to the summarization of source code (Figure 1), which is highly structured and thus can profit most from models that can take advantage of this structure; the final task is a classical natural language task illustrating that hybrid sequence-graph models are applicable for less structured inputs as well. ", + "bbox": [ + 174, + 397, + 825, + 481 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "METHODNAMING The aim of this task is to infer the name of a function (or method in objectoriented languages, such as Java, Python and C#) given its source code (Allamanis et al., 2016). Although method names are a single token, they are usually composed of one or more subtokens (split using snake case or camelCase) and thus, the method naming task can be cast as predicting a sequence of subtokens. Consequently, method names represent an “extreme” summary of the functionality of a given function (on average, the names in the Java dataset have only 2.9 subtokens). Notably, the vocabulary of tokens used in names is very large (due to abbreviations and domainspecific jargon), but this is mitigated by the fact that $33 \\%$ of subtokens in names can be copied directly from subtokens in the method’s source code. Finally, source code is highly structured input data with known semantics, which can be exploited to support name prediction. ", + "bbox": [ + 174, + 507, + 826, + 647 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "METHODDOC Similar to the first task, the aim of this task is to predict a succinct description of the functionality of a method given its source code (Barone & Sennrich, 2017). Such descriptions usually appear as documentation of methods (e.g. “docstrings” in Python or “JavaDocs” in Java). While the task shares many characteristics with the METHODNAMING task, the target sequence is substantially longer (on average 19.1 tokens in our C# dataset) and only $1 9 . 4 \\%$ of tokens in the documentation can be copied from the code. While method documentation is nearer to standard natural language than method names, it mixes project-specific jargon, code segments and often describes non-functional aspects of the code, such as performance characteristics and design considerations. ", + "bbox": [ + 174, + 674, + 825, + 785 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "NLSUMMARIZATION Finally, we consider the classic summarization of natural language as widely studied in NLP research. Specifically, we are interested in abstractive summarization, where given some text input (e.g. a news article) a machine learning model produces a novel natural language summary. Traditionally, NLP summarization methods treat text as a sequence of sentences and each one of them as a sequence of words (tokens). The input data has less explicitly defined structure than our first two tasks. However, we recast the task as a structured summarization problem by considering additional linguistic structure, including named entities and entity coreferences as inferred by existing NLP tools. ", + "bbox": [ + 174, + 811, + 825, + 924 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "3 MODEL ", + "text_level": 1, + "bbox": [ + 176, + 102, + 267, + 117 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "As discussed above, standard neural approaches to summarization follow the sequence-to-sequence framework. In this setting, most decoders only require a representation $^ { h }$ of the complete input sequence (e.g. the final state of an RNN) and per-token representations $h _ { t _ { i } }$ for each input token $t _ { i }$ These token representations are then used as the “memories” of an attention mechanism (Bahdanau et al., 2014; Luong et al., 2015) or a pointer network (Vinyals et al., 2015a). ", + "bbox": [ + 174, + 133, + 825, + 204 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "In this work, we propose an extension of sequence encoders that allows us to leverage known (or inferred) relationships among elements in the input data. To achieve that, we combine sequence encoders with graph neural networks (GNNs) (Li et al., 2015; Gilmer et al., 2017; Kipf & Welling, 2017). For this, we first use a standard sequential encoder (e.g. bidirectional RNNs) to obtain a pertoken representation $h _ { t _ { i } }$ , which we then feed into a GNN as the initial node representations. The resulting per-node (i.e. per-token) representations $\\boldsymbol { h } _ { t _ { i } } ^ { \\prime }$ can then be used by an unmodified decoder. Experimentally, we found this to surpass models that use either only the sequential structure or only the graph structure (see Sect. 4). We now discuss the different parts of our model in detail. ", + "bbox": [ + 173, + 210, + 826, + 321 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Gated Graph Neural Networks To process graphs, we follow Li et al. (2015) and briefly summarize the core concepts of GGNNs here. A graph $\\mathcal { G } = ( \\nu , \\pmb { \\varepsilon } , \\pmb { X } )$ is composed of a set of nodes $\\nu$ , node features $\\boldsymbol { X }$ , and a list of directed edge sets ${ \\pmb { \\mathcal { E } } } = ( { \\mathcal { E } } _ { 1 } , \\dots , { \\mathcal { E } } _ { K } )$ where $K$ is the number of edge types. Each $v \\in \\nu$ is associated with a real-valued vector $\\scriptstyle { \\mathbf { { \\mathit { x } } } } _ { \\mathit { v } }$ representing the features of the node (e.g., the embedding of a string label of that node), which is used for the initial state ${ h } _ { v } ^ { ( 0 ) }$ of a node. ", + "bbox": [ + 174, + 338, + 825, + 411 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Information is propagated through the graph using neural message passing (Gilmer et al., 2017). For this, every node $v$ sends messages to its neighbors by transforming its current representation $\\boldsymbol { h } _ { v } ^ { ( i ) }$ using an edge-type dependent function $f _ { k }$ . Here, $f _ { k }$ can be an arbitrary function; we use a simple linear layer. By computing all messages at the same time, all states can be updated simultaneously. In particular, a new state for a node $v$ is computed by aggregating all incoming messages as $m _ { v } ^ { ( i ) } = \\bar { g } ( \\{ f _ { k } ( h _ { u } ^ { ( i ) } ) \\} )$ there is an edge of type $k$ from $u$ to $v \\}$ ). $g$ is an aggregation function; we use elementwise summation for $g$ . Given the aggregated message $\\mathbf { \\it { m } } _ { v } ^ { ( i ) }$ and the current state vector $\\boldsymbol { h } _ { v } ^ { ( i ) }$ of node $v$ , we can compute the new state 1) = GRU(m(i)v , h(i)v ), where GRU is the recurrent cell function of a gated recurrent unit. These dynamics are rolled out for a fixed number of timesteps $T$ , and the state vectors resulting from the final step are used as output node representations, i.e., $\\mathrm { G N N } ( ( \\mathcal { V } , \\pmb { \\mathcal { E } } , X ) ) = \\{ \\pmb { h } _ { v } ^ { ( T ) } \\} _ { v \\in \\mathcal { V } }$ . ", + "bbox": [ + 173, + 417, + 826, + 588 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Sequence GNNs We now explain our novel combination of GGNNs and standard sequence encoders. As input, we take a sequence $\\boldsymbol { S } = \\left[ s _ { 1 } \\ldots s _ { N } \\right]$ and $K$ binary relationships $R _ { 1 } \\ldots R _ { K } \\in S \\times S$ between elements of the sequence. For example, $R _ { = }$ could be the equality relationship $\\left\\{ \\left( s _ { i } , s _ { j } \\right) \\right. \\mid$ $s _ { i } = s _ { j } \\}$ . The choice and construction of relationships is dataset-dependent, and will be discussed in detail in Sect. 4. Given any sequence encoder $\\mathcal { S E }$ that maps $S$ to per-element representations $\\left[ \\mathbf { e } _ { 1 } \\ldots \\mathbf { e } _ { N } \\right]$ and a sequence representation e (e.g. a bidirectional RNN), we can construct the sequence GNN $\\mathcal { S } \\mathcal { E } _ { G N N }$ by simply computing $\\mathbf { \\bar { e } } _ { 1 } ^ { \\prime } \\cdot \\cdot \\cdot \\mathbf { e } _ { N } ^ { \\prime } ] = \\mathbf { G N N } ( ( S , [ R _ { 1 } \\dots R _ { K } ] , [ \\mathbf { e } _ { 1 } \\dots \\mathbf { e } _ { N } ] ) )$ . To obtain a graph-level representation, we use the weighted averaging mechanism from Gilmer et al. (2017). Concretely, for each node $v$ in the graph, we compute a weight $\\sigma ( w ( \\pmb { h } _ { v } ^ { ( T ) } ) ) \\in [ 0 , 1 ]$ using a learnable function $w$ and the logistic sigmoid $\\sigma$ and compute a graph-level representation as $\\begin{array} { r } { \\hat { \\mathbf { e } } \\equiv \\sum _ { 1 \\leq i \\leq N } \\sigma ( w ( \\mathbf { e } _ { i } ^ { \\prime } ) ) \\cdot \\mathbb { N } ( \\mathbf { e } _ { i } ^ { \\prime } ) } \\end{array}$ , where $\\aleph$ is another learnable projection function. We found that best results were achieved by computing the final $\\mathbf { e ^ { \\prime } }$ as $W \\cdot ( \\mathbf { e } \\hat { \\mathbf { e } } )$ for some learnable matrix $W$ . ", + "bbox": [ + 173, + 603, + 826, + 775 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "This method can easily be extended to support additional nodes not present in the original sequence $S$ after running $\\mathcal { S E }$ (e.g., to accommodate meta-nodes representing sentences, or non-terminal nodes from a syntax tree). The initial node representation for these additional nodes can come from other sources, such as a simple embedding of their label. ", + "bbox": [ + 174, + 781, + 825, + 837 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Implementation Details. Processing large graphs of different shapes efficiently requires to overcome some engineering challenges. For example, the CNN/DM corpus has (on average) about 900 nodes per graph. To allow efficient computation, we use the trick of Allamanis et al. (2018) where all graphs in a minibatch are “flattened” into a single graph with multiple disconnected components. The varying graph sizes also represent a problem for the attention and copying mechanisms in the decoder, as they require to compute a softmax over a variable-sized list of memories. To handle this efficiently without padding, we associate each node in the (flattened) “batch” graph with the index of the sample in the minibatch from which the node originated. Then, using TensorFlow’s unsorted segment $\\mathbf { \\nabla } _ { \\cdot } \\star$ operations, we can perform an efficient and numerically stable softmax over the variable number of representations of the nodes of each graph. ", + "bbox": [ + 174, + 853, + 825, + 924 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "", + "bbox": [ + 174, + 103, + 825, + 174 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "4 EVALUATION ", + "text_level": 1, + "bbox": [ + 176, + 195, + 315, + 212 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "4.1 QUANTITATIVE EVALUATION ", + "text_level": 1, + "bbox": [ + 176, + 227, + 415, + 242 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "We evaluate Sequence GNNs on our three tasks by comparing them to models that use only sequence or graph information, as well as by comparing them to task-specific baselines. We discuss the three tasks, their respective baselines and how we present the data to the models (including the relationships considered in the graph component) next before analyzing the results. ", + "bbox": [ + 174, + 253, + 825, + 309 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "4.1.1 SETUP FOR METHODNAMING ", + "text_level": 1, + "bbox": [ + 176, + 327, + 433, + 340 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Datasets, Metrics, and Models. We consider two datasets for the METHODNAMING task. First, we consider the “Java (small)” dataset of Alon et al. (2018a), re-using the train-validation-test splits they have picked. We additionally generated a new dataset from 23 open-source C# projects mined from GitHub (see below for the reasons for this second dataset), removing any duplicates. More information about these datasets can be found in Appendix C. We follow earlier work on METHODNAMING (Allamanis et al., 2016; Alon et al., 2018a) and measure performance using the F1 score over the generated subtokens. However, since the task can be viewed as a form of (extreme) summarization, we also report ROUGE-2 and ROUGE-L scores (Lin, 2004), which we believe to be additional useful indicators for the quality of results. ROUGE-1 is omitted since it is equivalent to F1 score. We note that there is no widely accepted metric for this task and further work identifying the most appropriate metric is required. ", + "bbox": [ + 174, + 351, + 826, + 503 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "We compare to the current state of the art (Alon et al., 2018a), as well as a sequence-to-sequence implementation from the OpenNMT project (Klein et al.). Concretely, we combine two encoders (a bidirectional LSTM encoder with 1 layer and 256 hidden units, and its sequence GNN extension with 128 hidden units unrolled over 8 timesteps) with two decoders (an LSTM decoder with 1 layer and 256 hidden units with attention over the input sequence, and an extension using a pointer network-style copying mechanism (Vinyals et al., 2015a)). Additionally, we consider self-attention as an alternative to RNN-based sequence encoding architectures. For this, we use the Transformer (Vaswani et al., 2017) implementation in OpenNMT (i.e., using self-attention both for the decoder and the encoder) as a baseline and compare it to a version whose encoder is extended with a GNN component. ", + "bbox": [ + 174, + 511, + 825, + 636 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Data Representation Following the work of Allamanis et al. (2016); Alon et al. (2018a), we break up all identifier tokens (i.e. variables, methods, classes, etc.) in the source code into subtokens by splitting them according to camelCase and pascal case heuristics. This allows the models to extract information from the information-rich subtoken structure, and ensures that a copying mechanism in the decoder can directly copy relevant subtokens, something that we found to be very effective for this task. All models are provided with all (sub)tokens belonging to the source code of a method, including its declaration, with the actual method name replaced by a placeholder symbol. ", + "bbox": [ + 174, + 651, + 825, + 750 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "To construct a graph from the (sub)tokens, we implement a simplified form of the work of Allamanis et al. (2018). First, we introduce additional nodes for each (full) identifier token, and connect the constituent subtokens appearing in the input sequence using a INTOKEN edge; we additionally connect these nodes using a NEXTTOKEN edge. We also add nodes for the parse tree and use edges to indicate that one node is a CHILD of another. Finally, we add LASTLEXICALUSE edges to connect identifiers to their most (lexically) recent use in the source code. ", + "bbox": [ + 174, + 757, + 825, + 840 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "4.1.2 SETUP FOR METHODDOC ", + "text_level": 1, + "bbox": [ + 176, + 856, + 405, + 871 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Datasets, Metrics, and Models. We tried to evaluate on the Python dataset of Barone & Sennrich (2017) that contains pairs of method declarations and their documentation (“docstring”). However, following the work of Lopes et al. (2017), we found extensive duplication between different folds of the dataset and were only able to reach comparable results by substantially overfitting to the training data that overlapped with the test set. We have documented details in subsection C.3 and in Allamanis (2018), and decided to instead evaluate on our new dataset of 23 open-source C# projects from above, again removing duplicates and methods without documentation. Following Barone & Sennrich (2017), we measure the BLEU score for all models. However, we also report F1, ROUGE-2 and ROUGE-L scores, which should better reflect the summarization aspect of the task. We consider the same models as for the METHODNAMING task, using the same configuration, and use the same data representation. ", + "bbox": [ + 176, + 882, + 825, + 924 + ], + "page_idx": 3 + }, + { + "type": "image", + "img_path": "images/0f24ece1ecd933d082df2284a4d3190ac4a48d4c10138fc5c01fc0d240b7a43f.jpg", + "image_caption": [ + "Figure 2: (Partial) graph of an example input from the CNN/DM corpus. " + ], + "image_footnote": [], + "bbox": [ + 173, + 99, + 826, + 266 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "", + "bbox": [ + 174, + 333, + 825, + 445 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "4.1.3 SETUP FOR NLSUMMARIZATION ", + "text_level": 1, + "bbox": [ + 176, + 481, + 452, + 494 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Datasets, Metrics, and Models. We use the CNN/DM dataset (Hermann et al., 2015) using the exact data and split provided by See et al. (2017). The data is constructed from CNN and Daily Mail news articles along with a few sentences that summarize each article. To measure performance, we use the standard ROUGE metrics. We compare our model with the near-to-state-of-the-art work of See et al. (2017), who use a sequence-to-sequence model with attention and copying as basis, but have additionally substantially improved the decoder component. As our contribution is entirely on the encoder side and our model uses a standard sequence decoder, we are not expecting to outperform more recent models that introduce substantial novelty in the structure or training objective of the decoder (Chen & Bansal, 2018; Narayan et al., 2018). Again, we evaluate our contribution using an OpenNMT-based encoder/decoder combination. Concretely, we use a bidirectional LSTM encoder with 1 layer and 256 hidden units, and its sequence GNN extension with 128 hidden units unrolled over 8 timesteps. As decoder, we use an LSTM with 1 layer and 256 hidden units with attention over the input sequence, and an extension using a pointer network-style copying mechanism. ", + "bbox": [ + 173, + 512, + 825, + 694 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Data Representation We use Stanford CoreNLP (Manning et al., 2014) (version 3.9.1) to tokenize the text and provide the resulting tokens to the encoder. For the graph construction (Figure 2), we extract the named entities and run coreference resolution using CoreNLP. We connect tokens using a NEXT edge and introduce additional super-nodes for each sentence, connecting each token to the corresponding sentence-node using a IN edge. We also connect subsequent sentence-nodes using a NEXT edge. Then, for each multi-token named entity we create a new node, labeling it with the type of the entity and connecting it with all tokens referring to that entity using an IN edge. Finally, coreferences of entities are connected with a special REF edge. Figure 2 shows a partial graph for an article in the CNN/DM dataset. The goal of this graph construction process is to explicitly annotate important relationships that can be useful for summarization. We note that (a) in early efforts we experimented with adding dependency parse edges, but found that they do not provide significant benefits and (b) that since we retrieve the annotations from CoreNLP, they can contain errors and thus, the performance of the our method is influenced by the accuracy of the upstream annotators of named entities and coreferences. ", + "bbox": [ + 174, + 729, + 825, + 922 + ], + "page_idx": 4 + }, + { + "type": "table", + "img_path": "images/9736507d854d7ae3d25c3d066f5e08fa8e12e5b33e3fbe97e7e9b348dd2f9e0d.jpg", + "table_caption": [ + "Table 1: Evaluation results for all models and tasks. " + ], + "table_footnote": [], + "table_body": "
METHODNAMINGF1ROUGE-2ROUGE-L
Java
Alon et al. (2018a)43.0一1
SELFATT →SELFATT24.98.327.4
SELFATT+GNN →SELFATT44.520.943.4
BILSTM →LSTM35.817.939.7
BILSTM+GNN →LSTM44.721.143.1
BILSTM →LSTM+POINTER42.522.445.6
GNN →LSTM+POINTER50.524.848.9
BILSTM+GNN→LSTM+POINTER51.425.050.0
C#
SELFATT →SELFATT41.325.243.2
SELFATT+GNN→→SELFATT62.131.061.1
BILSTM →LSTM48.832.851.8
BILSTM+GNN →LSTM62.631.061.3
BILSTM →LSTM+POINTER57.229.760.4
GNN→LSTM+POINTER63.031.561.3
BILSTM+GNN→LSTM+POINTER63.431.962.4
METHODDOCF1ROUGE-2ROUGE-LBLEU
C#
SELFATT →SELFATT40.027.841.113.9
SELFATT+GNN→SELFATT37.625.637.921.4
BILSTM →LSTM35.215.330.810.0
BILSTM+GNN →LSTM41.128.941.022.5
BILSTM →LSTM+POINTER35.220.836.714.7
GNN →LSTM+POINTER38.925.637.117.7
BILSTM+GNN →→LSTM+POINTER (average pooling)43.229.041.021.3
BILSTM+GNN→LSTM+POINTER45.428.341.122.2
NLSUMMARIZATIONROUGE-1ROUGE-2ROUGE-L
CNN/DM
BILSTM →LSTM33.611.427.9
BILSTM+GNN→LSTM33.013.328.3
See et al. (2017) (+ Pointer)36.415.733.4
BILSTM →LSTM+POINTER35.913.930.3
BILSTM+GNN →LSTM+POINTER38.116.133.2
See et al. (2017) (+ Pointer + Coverage)39.517.336.4
", + "bbox": [ + 171, + 119, + 834, + 617 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "4.1.4 RESULTS & ANALYSIS ", + "text_level": 1, + "bbox": [ + 174, + 645, + 385, + 660 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "We show all results in Tab. 1. Results for models from the literature are taken from the respective papers and repeated here. Across all tasks, the results show the advantage of our hybrid sequence GNN encoders over pure sequence encoders. ", + "bbox": [ + 174, + 672, + 825, + 714 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "On METHODNAMING, we can see that all GNN-augmented models are able to outperform the current specialized state of the art, requiring only simple graph structure that can easily be obtained using existing parsers for a programming language. The results in performance between the different encoder and decoder configurations nicely show that their effects are largely orthogonal. ", + "bbox": [ + 174, + 722, + 825, + 777 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "On METHODDOC, the unmodified SELFATT $ s$ ELFATT model already performs quite well, and the augmentation with graph data only improves the BLEU score and worsens the results on ROUGE. Inspection of the results shows that this is due to the length of predictions. Whereas the ground truth data has on average 19 tokens in each result, SELFATT $ s$ ELFATT predicts on average 11 tokens, and SELFATT+GNN SELFATT 16 tokens. Additionally, we experimented with an ablation in which a model is only using graph information, e.g., a setting comparable to a simplification of the architecture of Allamanis et al. (2018). For this, we configured the GNN to use 128-dimensional representations and unrolled it for 10 timesteps, keeping the decoder configuration as for the other models. The results indicate that this configuration performs less well than a pure sequenced model. We speculate that this is mainly due to the fact that 10 timesteps are insufficient to propagate inforpublic static bool TryFormat(float value, Span destination, out int bytesWritten, StandardFormat format $=$ default) { return TryFormatFloatingPoint(value, destination, out bytesWritten, format); } ", + "bbox": [ + 174, + 785, + 825, + 924 + ], + "page_idx": 5 + }, + { + "type": "table", + "img_path": "images/8265df601c5ba5ef05dfa6c6e0c8c05477fe004786c78230eb7168e068ff09b8.jpg", + "table_caption": [ + "Table 2: Ablations on CNN/DM Corpus " + ], + "table_footnote": [], + "table_body": "
NLSUMMARIZATION (CNN/DM)ROUGE-1ROUGE-2ROUGE-L
See et al.(2017) (base)31.311.828.8
See et al. (2017) (+ Pointer)36.415.733.4
See et al.(2017) (+ Pointer + Coverage)39.517.336.4
BILSTM →LSTM33.611.427.9
BILSTM →LSTM+POINTER35.913.930.3
BILSTM →LSTM+PoINTER (+ coref/entity annotations)36.214.230.5
BILSTM+GNN →LSTM33.013.328.3
BILSTM+GNN→LSTM+POINTER (only sentence nodes)36.015.229.6
BILSTM+GNN →LSTM+POINTER (sentence nodes + eq edges)36.115.430.3
BILSTM+GNN→LSTM+POINTER38.116.133.2
", + "bbox": [ + 173, + 121, + 821, + 282 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "", + "bbox": [ + 176, + 309, + 767, + 361 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "Ground truth BILSTM LSTM BILST $\\mathbf { M } \\mathbf { + } \\mathbf { G N N } \\to \\mathbf { L S T M }$ B $\\mathbf { \\mathbf { \\mathbf { I L S T M + G N N } } } \\mathbf { \\mathbf { \\Phi } } \\to \\mathbf { \\mathbf { \\mathbf { L S T M + P O I N T E R } } }$ formats a single as a utf8 string formats a number of bytes in a utf8 string formats a timespan as a utf8 string formats a float as a utf8 string ", + "bbox": [ + 183, + 376, + 423, + 425 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "", + "bbox": [ + 441, + 375, + 686, + 426 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "Figure 3: An example from the dataset for the METHODDOC source code summarization task along with the outputs of a baseline and our models. ", + "bbox": [ + 173, + 433, + 821, + 462 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "mation across the whole graph, especially in combination with summation as aggregation function for messages in graph information propagation. ", + "bbox": [ + 174, + 501, + 823, + 529 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "Finally, on NLSUMMARIZATION, our experiments show that the same model suitable for tasks on highly structured code is competitive with specialized models for natural language tasks. While there is still a gap to the best configuration of See et al. (2017) (and an even larger one to more recent work in the area), we believe that this is entirely due to our simplistic decoder and training objective, and that our contribution can be combined with these advances. ", + "bbox": [ + 174, + 535, + 825, + 604 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "In Table 2 we show some ablations for NLSUMMARIZATION. As we use the same hyperparameters across all datasets and tasks, we additionally perform an experiment with the model of See et al. (2017) (as implemented in OpenNMT) but using our settings. The results achieved by these baselines trend to be a bit worse than the results reported in the original paper, which we believe is due to a lack of hyperparameter optimization for this task. We then evaluated how much the additional linguistic structure provided by CoreNLP helps. First, we add the coreference and entity annotations to the baseline $\\mathrm { B I L S T M } \\to \\mathrm { L S T M } + \\mathrm { P O I N T E R }$ model (by extending the embedding of tokens with an embedding of the entity information, and inserting fresh $\\mathrm { ~ \\omega ~ } _ { \\mathrm { i R E F 1 } } \\mathrm { ~ \\omega ~ }$ ”, . . . tokens at the sources/targets of co-references) and observe only minimal improvements. This suggests that our graph-based encoder is better-suited to exploit additional structured information compared to a biLSTM encoder. We then drop all linguistic structure information from our model, keeping only the sentence edges/nodes. This still improves on the baseline $\\mathrm { B I L S T M } \\to \\mathrm { L S T M } + \\mathrm { P O I I }$ NTER model (in the ROUGE-2 score), suggesting that the GNN still yields improvements in the absence of linguistic structure. Finally, we add long-range dependency edges by connecting tokens with equivalent string representations of their stems and observe further minor improvements, indicating that even using only purely syntactical information, without a semantic parse, can already provide gains. ", + "bbox": [ + 174, + 612, + 825, + 834 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "4.2 QUALITATIVE EVALUATION ", + "text_level": 1, + "bbox": [ + 176, + 864, + 405, + 878 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "We look at a few sample suggestions in our dataset across the tasks. Here we highlight some observations we make that point out interesting aspects and failure cases of our model. ", + "bbox": [ + 176, + 895, + 823, + 924 + ], + "page_idx": 6 + }, + { + "type": "table", + "img_path": "images/879ad0593963ca3e074145df80c378944913e18c84b695b9649622deeb04970e.jpg", + "table_caption": [ + "Figure 4: Sample natural language translations from the CNN-DM dataset. " + ], + "table_footnote": [], + "table_body": "
Input: Arsenal,Newcastle United and Southampton have checked on Caen midfelder N'golo Kante .Paris-born Kante is a defensive minded player who has impressed for Caen this season and they are wiling to sellfor around f 5milion.Marseille have been in constant contact with Caen over signing the 24-year-old who has similarities with Lassna Diarra and Claude Makelele in terms of stature and style .N'Golo Kante is attractinginterest froma host of PremierLeague clubs including Arsenal.Caen would be wiling to sellKante for aroundf5million.
Reference:n'golo kante is wanted by arsenal,newcastle and southampton.marseille are also keen on thef5mrated midfielder.kante has been compared to lassana diarra and claude makelele.click here for the latestpremier league news .
See et al.(2O17) (+ Pointer): arsenal,newcastle united and southampton have checked on caen midfieldern'golo kante .paris-born kante is attracting interest from a host of premier league clubs including arsenal .paris-born kante is attracting interest from a host of premier league clubs including arsenal
See et al. (2O17)(+ Pointer + Coverage): arsenal, newcastle united and southampton have checked on caenmidfielder n'golo kante.paris-born kante is a defensive minded player who has impressed for caen this season.marseille have been in constant contact with caen over signing the 24-year-old .
BILSTM+GNN →LSTM: marseille have been linked with caen midfielder %UNK% %UNK% . marseillehave been interested from a host of premier league clubs including arsenal .caen have been interested from ahost of premier league clubs including arsenal .
BILSTM+GNN →LSTM+PoINTER n'golo kante is attracting interest from a host of premier league clubs .marseilehave been in constant contact with caen over signing the24-year-old.the 24-year-old has similaritieswith lassana diarra and claude makelele in terms of stature .
", + "bbox": [ + 173, + 102, + 841, + 406 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "METHODDOC Figures 1 and 3 illustrate typical results of baselines and our model on the METHODDOC task (see Appendix A for more examples). The hardness of the task stems from the large number of distractors and the need to identify the most relevant parts of the input. In Figure 1, the token “parameter” and variations appears many times, and identifying the correct relationship is non-trivial, but is evidently eased by graph edges explicitly denoting these relationships. Similarly, in Figure 3, many variables are passed around, and the semantics of the method require understanding how information flows between them. ", + "bbox": [ + 174, + 464, + 825, + 561 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "NLSUMMARIZATION Figure 4 shows one sample summarization. More samples for this task can be found in Appendix B. First, we notice that the model produces natural-looking summaries with no noticeable negative impact on the fluency of the language over existing methods. Furthermore, the GNN-based model seems to capture the central named entity in the article and creates a summary centered around that entity. We hypothesize that the GNN component that links long-distance relationships helps capture and maintain a better “global” view of the article, allowing for better identification of central entities. Our model still suffers from repetition of information (see Appendix B), and so we believe that our model would also profit from advances such as taking coverage into account (See et al., 2017) or optimizing for ROUGE-L scores directly via reinforcement learning (Chen & Bansal, 2018; Narayan et al., 2018). ", + "bbox": [ + 174, + 583, + 826, + 723 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "5 RELATED WORK ", + "text_level": 1, + "bbox": [ + 176, + 750, + 344, + 765 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "Natural language processing research has studied summarization for a long time. Most related is work on abstractive summarization, in which the core content of a given text (usually a news article) is summarized in a novel and concise sentence. Chopra et al. (2016) and Nallapati et al. (2016) use deep learning models with attention on the input text to guide a decoder that generates a summary. See et al. (2017) and McCann et al. (2018) extend this idea with pointer networks (Vinyals et al., 2015a) to allow for copying tokens from the input text to the output summary. These approaches treat text as a simple token sequences, not explicitly exposing additional structure. In principle, deep sequence networks are known to be able to learn the inherent structure of natural language (e.g. in parsing (Vinyals et al., 2015b) and entity recognition (Lample et al., 2016)), but our experiments indicate that explicitly exposing this structure by separating concerns improves performance. ", + "bbox": [ + 174, + 785, + 825, + 924 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "Recent work in summarization has proposed improved training objectives for summarization, such as tracking coverage of the input document (See et al., 2017) or using reinforcement learning to directly identify actions in the decoder that improve target measures such as ROUGE-L (Chen & Bansal, 2018; Narayan et al., 2018). These objectives are orthogonal to the graph-augmented encoder discussed in this work, and we are interested in combining these efforts in future work. ", + "bbox": [ + 174, + 103, + 825, + 172 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "Exposing more language structure explicitly has been studied over the last years, with a focus on tree-based models (Tai et al., 2015). Very recently, first uses of graphs in natural language processing have been explored. Marcheggiani & Titov (2017) use graph convolutional networks to encode single sentences and assist machine translation. De Cao et al. (2018) create a graph over named entities over a set of documents to assist question answering. Closer to our work is the work of Liu et al. (2018), who use abstract meaning representation (AMR), in which the source document is first parsed into AMR graphs, before a summary graph is created, which is finally rendered in natural language. In contrast to that work we do not use AMRs but directly encode relatively simple relationships directly on the tokenized text, and do not treat summarization as a graph rewrite problem. Combining our encoder with AMRs to use richer graph structures may be a promising future direction. ", + "bbox": [ + 174, + 180, + 825, + 319 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "Finally, summarization in source code has also been studied in the forms of method naming, comment and documentation prediction. Method naming has been tackled with a series of models. For example, Allamanis et al. (2015) use a log-bilinear network to predict method names from features, and later extend this idea to use a convolutional attention network over the tokens of a method to predict the subtokens of names (Allamanis et al., 2016). Raychev et al. (2015) and Bichsel et al. (2016) use CRFs for a range of tasks on source code, including the inference of names for variables and methods. Recently, Alon et al. (2018b;a) extract and encode paths from the syntax tree of a program, setting the state of the art in accuracy on method naming. ", + "bbox": [ + 174, + 325, + 825, + 438 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "Linking text to code can have useful applications, such as code search (Gu et al., 2018), traceability (Guo et al., 2017), and detection of redundant method comments (Louis et al., 2018). Most approaches on source code either treat it as natural language (i.e., a token sequence), or use a language parser to explicitly expose its tree structure. For example, Barone & Sennrich (2017) use a simple sequenceto-sequence baseline, whereas Hu et al. (2017) summarize source code by linearizing the abstract syntax tree of the code and using a sequence-to-sequence model. Wan et al. (2018) instead directly operate on the tree structure using tree recurrent neural networks (Tai et al., 2015). The use of additional structure on related tasks on source code has been studied recently, for example in models that are conditioned on learned traversals of the syntax tree (Bielik et al., 2016) and in graph-based approaches (Allamanis et al., 2018; Cvitkovic et al., 2018). However, as noted by Liao et al. (2018), GNN-based approaches suffer from a tension between the ability to propagate information across large distances in a graph and the computational expense of the propagation function, which is linear in the number of graph edges per propagation step. ", + "bbox": [ + 174, + 444, + 825, + 625 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "6 DISCUSSION & CONCLUSIONS ", + "text_level": 1, + "bbox": [ + 176, + 645, + 459, + 661 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "We presented a framework for extending sequence encoders with a graph component that can leverage rich additional structure. In an evaluation on three different summarization tasks, we have shown that this augmentation improves the performance of a range of different sequence models across all tasks. We are excited about this initial progress and look forward to deeper integration of mixed sequence-graph modeling in a wide range of tasks across both formal and natural languages. The key insight, which we believe to be widely applicable, is that inductive biases induced by explicit relationship modeling are a simple way to boost the practical performance of existing deep learning systems. ", + "bbox": [ + 174, + 676, + 825, + 787 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "REFERENCES ", + "text_level": 1, + "bbox": [ + 174, + 102, + 287, + 118 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "Miltiadis Allamanis. The adverse effects of code duplication in machine learning models of code. arXiv preprint arXiv:1812.06469, 2018. 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Toward abstractive summarization using semantic representations. arXiv preprint arXiv:1805.10399, 2018. ", + "bbox": [ + 171, + 492, + 825, + 521 + ], + "page_idx": 10 + }, + { + "type": "text", + "text": "Cristina V Lopes, Petr Maj, Pedro Martins, Vaibhav Saini, Di Yang, Jakub Zitny, Hitesh Sajnani, and Jan Vitek. Dej´ avu: a map of code duplicates on github. \\` Proceedings of the ACM on Programming Languages, 1(OOPSLA):84, 2017. ", + "bbox": [ + 176, + 530, + 823, + 574 + ], + "page_idx": 10 + }, + { + "type": "text", + "text": "Annie Louis, Santanu Kumar Dash, Earl T Barr, and Charles Sutton. Deep learning to detect redundant method comments. arXiv preprint arXiv:1806.04616, 2018. ", + "bbox": [ + 173, + 583, + 823, + 613 + ], + "page_idx": 10 + }, + { + "type": "text", + "text": "Minh-Thang Luong, Hieu Pham, and Christopher D Manning. Effective approaches to attentionbased neural machine translation. arXiv preprint arXiv:1508.04025, 2015. ", + "bbox": [ + 173, + 621, + 825, + 651 + ], + "page_idx": 10 + }, + { + "type": "text", + "text": "Christopher Manning, Mihai Surdeanu, John Bauer, Jenny Finkel, Steven Bethard, and David McClosky. The stanford corenlp natural language processing toolkit. In Proceedings of 52nd annual meeting of the association for computational linguistics: system demonstrations, pp. 55–60, 2014. ", + "bbox": [ + 176, + 660, + 826, + 704 + ], + "page_idx": 10 + }, + { + "type": "text", + "text": "Diego Marcheggiani and Ivan Titov. Encoding sentences with graph convolutional networks for semantic role labeling. In Proceedings of the 2017 Conference on Empirical Methods in Natural Language Processing, pp. 1506–1515, 2017. ", + "bbox": [ + 176, + 713, + 825, + 756 + ], + "page_idx": 10 + }, + { + "type": "text", + "text": "Bryan McCann, Nitish Shirish Keskar, Caiming Xiong, and Richard Socher. The natural language decathlon: Multitask learning as question answering. arXiv preprint arXiv:1806.08730, 2018. ", + "bbox": [ + 171, + 765, + 823, + 795 + ], + "page_idx": 10 + }, + { + "type": "text", + "text": "Ramesh Nallapati, Bowen Zhou, Cicero dos Santos, Caglar Gulcehre, and Bing Xiang. Abstractive text summarization using sequence-to-sequence rnns and beyond. In Proceedings of The 20th SIGNLL Conference on Computational Natural Language Learning, pp. 280–290, 2016. ", + "bbox": [ + 176, + 804, + 823, + 848 + ], + "page_idx": 10 + }, + { + "type": "text", + "text": "Shashi Narayan, Shay B Cohen, and Mirella Lapata. Ranking sentences for extractive summarization with reinforcement learning. arXiv preprint arXiv:1802.08636, 2018. ", + "bbox": [ + 171, + 856, + 823, + 886 + ], + "page_idx": 10 + }, + { + "type": "text", + "text": "Veselin Raychev, Martin Vechev, and Andreas Krause. Predicting program properties from Big Code. In Principles of Programming Languages (POPL), 2015. ", + "bbox": [ + 174, + 895, + 823, + 924 + ], + "page_idx": 10 + }, + { + "type": "text", + "text": "Abigail See, Peter J Liu, and Christopher D Manning. Get to the point: Summarization with pointergenerator networks. In Proceedings of the 55th Annual Meeting of the Association for Computational Linguistics (Volume 1: Long Papers), volume 1, pp. 1073–1083, 2017. ", + "bbox": [ + 176, + 103, + 823, + 146 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": "Kai Sheng Tai, Richard Socher, and Christopher D Manning. Improved semantic representations from tree-structured long short-term memory networks. 2015. ", + "bbox": [ + 173, + 155, + 823, + 184 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": "Ashish Vaswani, Noam Shazeer, Niki Parmar, Jakob Uszkoreit, Llion Jones, Aidan N Gomez, Łukasz Kaiser, and Illia Polosukhin. Attention is all you need. In Advances in Neural Information Processing Systems, pp. 5998–6008, 2017. ", + "bbox": [ + 176, + 193, + 823, + 234 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": "Oriol Vinyals, Meire Fortunato, and Navdeep Jaitly. Pointer networks. In Advances in Neural Information Processing Systems, pp. 2692–2700, 2015a. ", + "bbox": [ + 173, + 244, + 823, + 273 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": "Oriol Vinyals, Lukasz Kaiser, Terry Koo, Slav Petrov, Ilya Sutskever, and Geoffrey Hinton. Grammar as a foreign language. In Advances in Neural Information Processing Systems, 2015b. ", + "bbox": [ + 174, + 281, + 823, + 310 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": "Yao Wan, Zhou Zhao, Min Yang, Guandong Xu, Haochao Ying, Jian Wu, and Philip S Yu. Improving automatic source code summarization via deep reinforcement learning. In Proceedings of the 33rd ACM/IEEE International Conference on Automated Software Engineering, pp. 397–407. ACM, 2018. ", + "bbox": [ + 174, + 319, + 826, + 376 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": "A CODE SUMMARIZATION SAMPLES", + "text_level": 1, + "bbox": [ + 176, + 102, + 493, + 118 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "A.1 METHODDOC ", + "bbox": [ + 176, + 133, + 315, + 147 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "C# Sample 1 ", + "text_level": 1, + "bbox": [ + 174, + 159, + 263, + 174 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "public static bool TryConvertTo(object valueToConvert, Type resultType, IFormatProvider formatProvider, out object result){ result $=$ null; try{ result $=$ ConvertTo(valueToConvert, resultType, formatProvider); catch (InvalidCastException){ return false; catch (ArgumentException){ return false; } return true; \n} ", + "bbox": [ + 173, + 184, + 836, + 333 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "Ground truth ", + "text_level": 1, + "bbox": [ + 184, + 349, + 271, + 362 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "sets result to valuetoconvert converted to resulttype considering formatprovider for custom conversions calling the parse method and calling convert . changetype . \nconverts the specified type to a primitive type . \nsets result to resulttype \nsets result to valuetoconvert converted to resulttype. ", + "bbox": [ + 441, + 349, + 818, + 425 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "BILSTM LSTM BILSTM+GNN $\\mathbf { J } \\to \\mathbf { L S T N }$ M BILSTM+G $\\mathbf { \\ V N } \\to \\mathbf { L S T M } + \\mathbf { \\beta }$ POINTER ", + "bbox": [ + 183, + 386, + 423, + 424 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "C# Sample 2 ", + "text_level": 1, + "bbox": [ + 174, + 438, + 264, + 452 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "public virtual Task Init(string name, IProviderRuntime providerRuntime, IProviderConfiguration config){ Log $=$ providerRuntime.GetLogger(this.GetType().FullName); this.serializerSettings $=$ OrleansJsonSerializer.GetDefaultSerializerSettings(); return TaskDone.Done; \n} ", + "bbox": [ + 174, + 460, + 895, + 535 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "Ground truth BILSTM $\\bf \\Pi \\to L S T N$ I BILSTM+G $\\mathbf { \\partial } _ { \\mathbf { i } } \\mathbf { N N } \\to \\mathbf { L S T M }$ BILSTM+GN $\\mathbf { N } \\to \\mathbf { L S T }$ M+POINTER ", + "bbox": [ + 181, + 551, + 423, + 602 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "initializes the storage provider \ncreates a grain object \ninitializes the provider provider \ninitialization function to initialize the specified provider. ", + "bbox": [ + 439, + 551, + 772, + 603 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "C# Sample 3 ", + "text_level": 1, + "bbox": [ + 174, + 609, + 264, + 625 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "public void NullParameter(){ TaskParameter t $=$ new TaskParameter(null); Assert.Null(t.WrappedParameter); Assert.Equal( TaskParameterType.Null , t.ParameterType ); ((INodePacketTranslatable) t).Translate( TranslationHelpers.GetWriteTranslator()); TaskParameter $\\begin{array} { r l } { \\pm 2 } & { { } = } \\end{array}$ TaskParameter.FactoryForDeserialization( TranslationHelpers.GetReadTranslator()); Assert.Null(t2.WrappedParameter); Assert.Equal(TaskParameterType.Null, t2.ParameterType); \n} ", + "bbox": [ + 174, + 633, + 746, + 770 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "Ground truth BILSTM LSTM BILSTM+G $\\mathbf { N N } \\to \\mathbf { L S T M }$ BILSTM+GNN LSTM+POINTER verifies that construction and serialization with a null parameter is ok tests that the value is a value that is a value to the specified type verifies that construction with an parameter parameter verifies that construction and serialization with a parameter that is null ", + "bbox": [ + 183, + 786, + 423, + 838 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "", + "bbox": [ + 439, + 786, + 852, + 838 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "C# Sample 4 ", + "text_level": 1, + "bbox": [ + 174, + 849, + 264, + 864 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "public override DbGeometryWellKnownValue CreateWellKnownValue(DbGeometry geometryValue){ geometryValue.CheckNull(\"geometryValue\"); var spatialValue $=$ geometryValue.AsSpatialValue(); DbGeometryWellKnownValue result $=$ CreateWellKnownValue(spatialValue, ", + "bbox": [ + 176, + 873, + 949, + 924 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "() $= >$ SpatialExceptions.CouldNotCreateWellKnownGeometryValueNoSrid(\"geometryValue\"), () $= >$ SpatialExceptions.CouldNotCreateWellKnownGeometryValueNoWkbOrWkt(\"geometryValue\"), (srid, wkb, wkt) $= >$ new DbGeometryWellKnownValue() { CoordinateSystemId $=$ srid, WellKnownBinary $=$ wkb, WellKnownText $=$ wkt }); return result; } ", + "bbox": [ + 174, + 104, + 957, + 194 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "Ground truth BILSTM LSTM B $\\mathbf { I L S T M + G N N } \\to \\mathbf { L S T M }$ BILSTM+GN $\\Gamma \\to \\mathbf { L } \\mathbf { S }$ TM+POINTER ", + "bbox": [ + 181, + 207, + 424, + 297 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "creates an instance of t:system.data.spatial.dbgeometry value using one or both of the standard well known spatial formats. \ncreates a t:system.data.spatial.dbgeography value based on the specified well known binary value . \ncreates a new t:system.data.spatial.dbgeography instance using the specified well known spatial formats . \ncreates a new instance of the t:system.data.spatial.dbgeometry value based on the provided geometry value and returns the resulting well as known spatial formats . ", + "bbox": [ + 441, + 208, + 841, + 321 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "A.2 METHODNAMING ", + "text_level": 1, + "bbox": [ + 176, + 343, + 343, + 358 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "C# Sample 1 ", + "text_level": 1, + "bbox": [ + 174, + 369, + 263, + 385 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "public bool _(D d) { return d ! $=$ null && d.Val == Val ; \n} ", + "bbox": [ + 174, + 392, + 500, + 430 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "Ground truth equals BILSTM LSTM foo BILSTM+GNN LSTM equals BILSTM+ $\\mathbf { G N N } \\to \\mathbf { L S T M } \\mathbf { + P O I }$ NTER equals ", + "bbox": [ + 181, + 445, + 496, + 496 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "C# Sample 2 ", + "text_level": 1, + "bbox": [ + 174, + 502, + 264, + 517 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "internal void _(string switchName, Hashtable bag, string parameterName) { object obj $=$ bag[parameterName]; if(obj ! $=$ null){ int value $=$ (int) obj; AppendSwitchIfNotNull(switchName, value.ToString(CultureInfo.InvariantCulture)); \n} ", + "bbox": [ + 173, + 526, + 818, + 626 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "Ground truth append switch with integer BILSTM LSTM set string BILSTM+GN $\\ J \\to \\mathbf { L S T N }$ M append switch BILSTM $\\mathbf { + G N N \\to L S T M + P O }$ INTER append switch if not null ", + "bbox": [ + 181, + 641, + 673, + 693 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "C# Sample 3 ", + "text_level": 1, + "bbox": [ + 174, + 699, + 264, + 713 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "internal static string _(){ var currentPlatformString $=$ string.Empty; if (RuntimeInformation.IsOSPlatform(OSPlatform.Windows)){ currentPlatformString $=$ \"WINDOWS\"; } else if (RuntimeInformation.IsOSPlatform(OSPlatform.Linux)){ currentPlatformString $=$ \"LINUX\"; } else if ( RuntimeInformation.IsOSPlatform(OSPlatform.OSX)) { currentPlatformString $=$ \"OSX\"; } else { Assert.True(false, \"unrecognized current platform\"); } return currentPlatformString ; ", + "bbox": [ + 200, + 723, + 758, + 914 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "Ground truth BILSTM LSTM BILSTM+GNN LSTM BILSTM+ $\\mathbf { G N N } \\to \\mathbf { L S T M + P O I N T }$ ER ", + "bbox": [ + 183, + 102, + 423, + 151 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "get os platform as string get name \nget platform \nget current platform string ", + "bbox": [ + 439, + 102, + 681, + 152 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "C# Sample 4 ", + "text_level": 1, + "bbox": [ + 174, + 159, + 264, + 174 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "public override DbGeometryWellKnownValue CreateWellKnownValue(DbGeometry geometryValue){ geometryValue.CheckNull(\"geometryValue\"); var spatialValue $=$ geometryValue.AsSpatialValue(); DbGeometryWellKnownValue result $=$ CreateWellKnownValue(spatialValue, () $= >$ SpatialExceptions.CouldNotCreateWellKnownGeometryValueNoSrid(\"geometryValue\"), () $= >$ SpatialExceptions.CouldNotCreateWellKnownGeometryValueNoWkbOrWkt(\"geometryValue\"), (srid, wkb, wkt) $= >$ new DbGeometryWellKnownValue () { CoordinateSystemId $=$ srid , WellKnownBinary $=$ wkb , WellKnownText $=$ wkt }); return result; \n} ", + "bbox": [ + 174, + 184, + 955, + 321 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "Ground truth BILSTM LSTM BILSTM+GN $\\ J \\to \\mathbf { L S T N }$ BILSTM+G $\\mathbf { N N } \\to \\mathbf { L S T M } +$ POINTER create well known value spatial geometry from xml geometry point get well known value ", + "bbox": [ + 181, + 338, + 423, + 388 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "", + "bbox": [ + 439, + 338, + 661, + 388 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "Java Sample 1 ", + "text_level": 1, + "bbox": [ + 174, + 396, + 276, + 411 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "public static void _(String name, int expected, MetricsRecordBuilder rb) { Assert.assertEquals(\"Bad value for metric \" $^ +$ name, expected, getIntCounter(name, rb)); \n} ", + "bbox": [ + 173, + 420, + 825, + 479 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "", + "bbox": [ + 174, + 472, + 184, + 483 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "Ground truth assert counter BILSTM LSTM assert email value BILSTM+GNN $\\ J \\to \\mathbf { L S T N }$ M assert header BILSTM+GNN LSTM+POINTER assert int counter ", + "bbox": [ + 183, + 498, + 602, + 550 + ], + "page_idx": 14 + }, + { + "type": "table", + "img_path": "images/17c8bb677ca1727c64543deca803078e74c9931cdd483d303dc153dacdc4aa06.jpg", + "table_caption": [ + "B NATURAL LANGUAGE SUMMARIZATION SAMPLES" + ], + "table_footnote": [], + "table_body": "
Input: -LRB- CNN -RRB- Gunshots were fired at rapper Lil Wayne 's tour bus early Sunday inAtlanta .No one was injured in the shooting ,and no arrests have been made,Atlanta Police spokeswoman Elizabeth Espy said .Police are stillooking for suspects . Ofcers were called to aparking lot in Atlanta 's Buckhead neighborhood,Espy said .They arrived at 3:25 a.m. and locatedtwo tour buses that had been shot multiple times .The drivers of the buses said the incident occurredon Interstate 285 near Interstate 75,Espy said .Witnesses provided a limited description of the twovehicles suspected to be involved : a“ Corvette style vehicle ”and an SUV .Lil Wayne was in Atlantafor a performance at Compound nightclub Saturday night . CNN 's Carma Hassan contributed to thisreport .
Reference: rapper lil wayne not injured after shots fired at his tour bus on an atlanta interstate ,police say . no one has been arrested in the shooting
See et al. (2017) (+ Pointer): police are stillooking for suspects . the incident occurred on interstate285 near interstate 75 ,police say . witnesses provided a limited description of the two vehicles suspected to be involved : a“ corvette style vehicle ” and an suv .
See et al. (2017) (+ Pointer + Coverage): lil wayne 's tour bus was shot multiple times , police say: police are stillooking for suspects .they arrived at 3:25 a.m. and located two tour buses that hadbeen shot .
BILSTM+GNN → LSTM: the incident occurred on interstate %UNK% near interstate 75 . no onewas injured in the shooting ,and no arrests have been made ,atlanta police spokeswoman says .
BILSTM+GNN →LSTM+PoINTER gunshots fired at rapper lil wayne 's tour bus early sundayin atlanta ,police say . no one was injured in the shooting ,and no arrests have been made , policesay.
", + "bbox": [ + 174, + 131, + 839, + 477 + ], + "page_idx": 15 + }, + { + "type": "table", + "img_path": "images/2582be1a64ce7532a8da18bfc0c3cf0f0d35e813ef43615062ab39e18a886462.jpg", + "table_caption": [], + "table_footnote": [], + "table_body": "
Input: Tottenham have held further discussions with Marseille over a potential deal for midfielderFlorian Thauvin .The 22-year-old has been left out of the squad for this weekend 's game withMetz as Marseille push for a f 15m sale .The winger,who can also play behind the striker, wasthe subject of enquiries from Spurs earlier in the year and has also been watched by Chelsea andValencia .Tottenham have held further talks with Ligue 1 side Marseille over a possible deal forFlorian Thauvin .Marseille are already resigned to losing Andre Ayew and Andre-Pierre Gignacwith English sides keen on both .Everton,Newcastle and Swansea,have allshown an interest inAyew,who is a free agent in the summer .
Reference: florian thauvin has been left out of marseille 's squad with metz . marseille are pushingfor a f 15m sale and tottenham are interested .the winger has also been watched by chelsea and laliga side valencia .
See et al. (2O17)(+ Pointer): tottenham have held further discussions with marseille over a potentialdeal for midfelder florian thauvin .the 22-year-old has been left out of the squad for this weekend 's game with metz as marseille push for a 15m sale .
See et al. (2017) (+ Pointer + Coverage): florian thauvin has been left out of the squad for thisweekend 's game with metz as marseille push for a 15m sale .the 22-year-old was the subject ofenquiries from spurs earlier in the year .
BILSTM+GNN →LSTM: the 22-year-old has been left out of the squad for this weekend 's gamewith metz .the 22-year-old has been left out of the squad for this weekend 's game with metz .thewinger has been left out of the squad for this weekend 's game with metz .
BILSTM+GNN → LSTM+PoINTER tottenham have held further discussions with marseille overa potential deal .the winger has been left out of the squad for this weekend 's game .tottenham haveheld further talks with marseille over a potential deal .
", + "bbox": [ + 174, + 510, + 839, + 868 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "C CODE DATASETS INFORMATION ", + "text_level": 1, + "bbox": [ + 176, + 102, + 473, + 118 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "C.1 C# DATASET ", + "text_level": 1, + "bbox": [ + 174, + 133, + 307, + 148 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "We extract the C# dataset from open-source projects on GitHub. Overall, our dataset contains 460,905 methods, 55,635 of which have a documentation comment. The dataset is split $8 5 - 5 - 1 0 \\%$ . The projects and exact state of the repositories used is listed in Table 3 ", + "bbox": [ + 174, + 160, + 825, + 203 + ], + "page_idx": 16 + }, + { + "type": "table", + "img_path": "images/8f3e3b14cf0c4e66fb1d5598a0c36f7ae9096fd31a267c2ba5b215c552d07249.jpg", + "table_caption": [ + "Table 3: Projects in our C# dataset. Ordered alphabetically. " + ], + "table_footnote": [], + "table_body": "
NameGit SHADescription
Akka.NET6f32f6a7Actor-based Concurrent & Distributed Framework
AutoMapper19d6f7fcObject-to-Object Mapping Library
BenchmarkDotNet57005f05Benchmarking Library
CommonMark.NETf3d54530Markdown Parser
CoreCLRcc5dcbe6.NET Core Runtime
CoreFxec1671fd.NETFoundationalLibraries
Dapper3c7cde28Object Mapper Library
EntityFrameworkc4d9a269Object-Relational Mapper
Humanizer2b1c94c4String Manipulation and Formatting
Lean90ee6aaeAlgorithmic Trading Engine
Mono9b9e4f4b.NET Implementation
MsBuild7f95dc15Build Engine
Nancyde458a9bHTTP Service Framework
NLog49fdd08eLogging Library
Opserver9e4d3a40Monitoring System
orleansf89c5866Distributed Virtual Actor Model
Pollyf3d2973dResilience & Transient Fault Handling Library
Powershell9ac701dbCommand-line Shell
ravendb6437de30Document Database
roslyn8ca0a542Compiler & Code Analysis & Compilation
ServiceStack17f081b9Real-time web library
SignalR9b05bcb0Push Notification Framework
Wox13e6c5eeApplication Launcher
", + "bbox": [ + 174, + 243, + 820, + 617 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "C.2 JAVA METHOD NAMING DATASETS ", + "text_level": 1, + "bbox": [ + 176, + 643, + 459, + 659 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "We use the datasets and splits of Alon et al. (2018a) provided by their website. Upon scanning all methods in the dataset, the size of the corpora can be seen in Table 4. More information can be found at Alon et al. (2018a). ", + "bbox": [ + 174, + 670, + 825, + 713 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "C.3 PYTHON METHOD DOCUMENTATION DATASET ", + "text_level": 1, + "bbox": [ + 174, + 731, + 542, + 746 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "We use the dataset as split of Barone & Sennrich (2017) provided by their GitHub repository. Upon parsing the dataset, we get 106,065 training samples, 1,943 validation samples and 1,937 test samples. We note that $1 6 . 9 \\%$ of the documentation samples in the validation set and $1 5 . 3 \\%$ of the samples in test set have a sample with the identical natural language documentation on the training set. This eludes to a potential issue, described by Lopes et al. (2017). See Allamanis (2018) for a lengthier discussion of this issue. ", + "bbox": [ + 173, + 757, + 826, + 813 + ], + "page_idx": 16 + }, + { + "type": "table", + "img_path": "images/f7c6b5866c07b5ce850ebf6cf50afa0b805a47d8f43b2da9868cff4166d9f0f8.jpg", + "table_caption": [ + "Table 4: The statistics of the extracted graphs from the Java method naming dataset of Alon et al. (2018a). " + ], + "table_footnote": [], + "table_body": "
DatasetTrain SizeValid SizeTest Size
Java - Small691,50523,83756,952
", + "bbox": [ + 320, + 877, + 678, + 921 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "", + "bbox": [ + 173, + 103, + 823, + 132 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "C.4 GRAPH DATA STATISTICS ", + "text_level": 1, + "bbox": [ + 174, + 148, + 395, + 164 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "Below we present the data characteristics of the graphs we use across the datasets. ", + "bbox": [ + 173, + 174, + 710, + 190 + ], + "page_idx": 17 + }, + { + "type": "table", + "img_path": "images/07b4d3250602bafe5f8460cf8adff0266382ea181ed4a4ad40d4fe86dc094a52.jpg", + "table_caption": [ + "Table 5: Graph Statistics For Datasets. " + ], + "table_footnote": [], + "table_body": "
DatasetAvg Num NodesAvg Num Edges
CNN/DM903.22532.9
C# Method Names125.2239.3
C# Documentation133.5265.9
Java-SmallMethod Names144.4251.6
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In an extensive evaluation, we show that the resulting hybrid sequence-graph", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 141, + 288, + 470, + 300 + ], + "spans": [ + { + "bbox": [ + 141, + 288, + 470, + 300 + ], + "score": 1.0, + "content": "models outperform both pure sequence models as well as pure graph models on a", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 141, + 300, + 265, + 311 + ], + "spans": [ + { + "bbox": [ + 141, + 300, + 265, + 311 + ], + "score": 1.0, + "content": "range of summarization tasks.", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 9.5, + "bbox_fs": [ + 141, + 223, + 471, + 311 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 352, + 206, + 365 + ], + "lines": [ + { + "bbox": [ + 105, + 351, + 208, + 368 + ], + "spans": [ + { + "bbox": [ + 105, + 351, + 208, + 368 + ], + "score": 1.0, + "content": "1 INTRODUCTION", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 14 + }, + { + "type": "text", + "bbox": [ + 107, + 384, + 505, + 428 + ], + "lines": [ + { + "bbox": [ + 105, + 384, + 506, + 398 + ], + "spans": [ + { + "bbox": [ + 105, + 384, + 506, + 398 + ], + "score": 1.0, + "content": "Summarization, the task of condensing a large and complex input into a smaller representation that", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 396, + 506, + 408 + ], + "spans": [ + { + "bbox": [ + 105, + 396, + 506, + 408 + ], + "score": 1.0, + "content": "retains the core semantics of the input, is a classical task for natural language processing systems.", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 407, + 505, + 420 + ], + "spans": [ + { + "bbox": [ + 105, + 407, + 505, + 420 + ], + "score": 1.0, + "content": "Automatic summarization requires a machine learning component to identify important entities and", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 418, + 430, + 430 + ], + "spans": [ + { + "bbox": [ + 105, + 418, + 430, + 430 + ], + "score": 1.0, + "content": "relationships between them, while ignoring redundancies and common concepts.", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 16.5, + "bbox_fs": [ + 105, + 384, + 506, + 430 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 434, + 505, + 534 + ], + "lines": [ + { + "bbox": [ + 106, + 435, + 505, + 447 + ], + "spans": [ + { + "bbox": [ + 106, + 435, + 505, + 447 + ], + "score": 1.0, + "content": "Current approaches to summarization are based on the sequence-to-sequence paradigm over the", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 446, + 505, + 458 + ], + "spans": [ + { + "bbox": [ + 105, + 446, + 505, + 458 + ], + "score": 1.0, + "content": "words of some text, with a sequence encoder — typically a recurrent neural network, but sometimes", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 456, + 505, + 470 + ], + "spans": [ + { + "bbox": [ + 105, + 456, + 505, + 470 + ], + "score": 1.0, + "content": "a 1D-CNN (Narayan et al., 2018) or using self-attention (McCann et al., 2018) — processing the input", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 466, + 505, + 481 + ], + "spans": [ + { + "bbox": [ + 105, + 466, + 505, + 481 + ], + "score": 1.0, + "content": "and a sequence decoder generating the output. Recent successful implementations of this paradigm", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 478, + 506, + 492 + ], + "spans": [ + { + "bbox": [ + 105, + 478, + 506, + 492 + ], + "score": 1.0, + "content": "have substantially improved performance by focusing on the decoder, extending it with an attention", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 490, + 506, + 501 + ], + "spans": [ + { + "bbox": [ + 106, + 490, + 506, + 501 + ], + "score": 1.0, + "content": "mechanism over the input sequence and copying facilities (See et al., 2017; McCann et al., 2018).", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 501, + 505, + 513 + ], + "spans": [ + { + "bbox": [ + 106, + 501, + 505, + 513 + ], + "score": 1.0, + "content": "However, while standard encoders (e.g. bidirectional LSTMs) theoretically have the ability to handle", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 511, + 505, + 524 + ], + "spans": [ + { + "bbox": [ + 105, + 511, + 505, + 524 + ], + "score": 1.0, + "content": "arbitrary long-distance relationships, in practice they often fail to correctly handle long texts and are", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 522, + 322, + 535 + ], + "spans": [ + { + "bbox": [ + 105, + 522, + 322, + 535 + ], + "score": 1.0, + "content": "easily distracted by simple noise (Jia & Liang, 2017).", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 23, + "bbox_fs": [ + 105, + 435, + 506, + 535 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 539, + 505, + 671 + ], + "lines": [ + { + "bbox": [ + 105, + 538, + 505, + 553 + ], + "spans": [ + { + "bbox": [ + 105, + 538, + 505, + 553 + ], + "score": 1.0, + "content": "In this work, we focus on an improvement of sequence encoders that is compatible with a wide range", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 550, + 506, + 563 + ], + "spans": [ + { + "bbox": [ + 105, + 550, + 506, + 563 + ], + "score": 1.0, + "content": "of decoder choices. To mitigate the long-distance relationship problem, we draw inspiration from", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 561, + 506, + 573 + ], + "spans": [ + { + "bbox": [ + 105, + 561, + 506, + 573 + ], + "score": 1.0, + "content": "recent work on highly-structured objects (Li et al., 2015; Kipf & Welling, 2017; Gilmer et al., 2017;", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 571, + 505, + 585 + ], + "spans": [ + { + "bbox": [ + 105, + 571, + 505, + 585 + ], + "score": 1.0, + "content": "Allamanis et al., 2018; Cvitkovic et al., 2018). In this line of work, highly-structured data such as", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 583, + 506, + 596 + ], + "spans": [ + { + "bbox": [ + 105, + 583, + 506, + 596 + ], + "score": 1.0, + "content": "entity relationships, molecules and programs is modelled using graphs. 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Using existing tools, we can", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 616, + 505, + 629 + ], + "spans": [ + { + "bbox": [ + 105, + 616, + 505, + 629 + ], + "score": 1.0, + "content": "annotate (accepting some noise) such data with additional relationships (e.g. co-references) to obtain", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 104, + 627, + 506, + 640 + ], + "spans": [ + { + "bbox": [ + 104, + 627, + 506, + 640 + ], + "score": 1.0, + "content": "a graph. 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In the METHODNAMING dataset, this method appears", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 245, + 437, + 258 + ], + "spans": [ + { + "bbox": [ + 105, + 245, + 437, + 258 + ], + "score": 1.0, + "content": "as a sample requiring to predict the name Add as a subtoken sequence of length 1.", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 17 + }, + { + "type": "title", + "bbox": [ + 108, + 285, + 325, + 298 + ], + "lines": [ + { + "bbox": [ + 105, + 284, + 326, + 299 + ], + "spans": [ + { + "bbox": [ + 105, + 284, + 326, + 299 + ], + "score": 1.0, + "content": "2 STRUCTURED SUMMARIZATION TASKS", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 19 + }, + { + "type": "text", + "bbox": [ + 107, + 315, + 505, + 381 + ], + "lines": [ + { + "bbox": [ + 106, + 316, + 505, + 327 + ], + "spans": [ + { + "bbox": [ + 106, + 316, + 505, + 327 + ], + "score": 1.0, + "content": "In this work, we consider three summarization tasks with different properties. All tasks follow the", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 104, + 325, + 506, + 340 + ], + "spans": [ + { + "bbox": [ + 104, + 325, + 506, + 340 + ], + "score": 1.0, + "content": "common pattern of translating a long (structured) sequence into a shorter sequence while trying", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 338, + 505, + 349 + ], + "spans": [ + { + "bbox": [ + 105, + 338, + 505, + 349 + ], + "score": 1.0, + "content": "to preserve as much meaning as possible. The first two tasks are related to the summarization of", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 348, + 505, + 361 + ], + "spans": [ + { + "bbox": [ + 105, + 348, + 505, + 361 + ], + "score": 1.0, + "content": "source code (Figure 1), which is highly structured and thus can profit most from models that can take", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 359, + 505, + 372 + ], + "spans": [ + { + "bbox": [ + 105, + 359, + 505, + 372 + ], + "score": 1.0, + "content": "advantage of this structure; the final task is a classical natural language task illustrating that hybrid", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 370, + 394, + 383 + ], + "spans": [ + { + "bbox": [ + 105, + 370, + 394, + 383 + ], + "score": 1.0, + "content": "sequence-graph models are applicable for less structured inputs as well.", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 22.5 + }, + { + "type": "text", + "bbox": [ + 107, + 402, + 506, + 513 + ], + "lines": [ + { + "bbox": [ + 105, + 402, + 507, + 415 + ], + "spans": [ + { + "bbox": [ + 105, + 402, + 507, + 415 + ], + "score": 1.0, + "content": "METHODNAMING The aim of this task is to infer the name of a function (or method in object-", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 414, + 507, + 426 + ], + "spans": [ + { + "bbox": [ + 105, + 414, + 507, + 426 + ], + "score": 1.0, + "content": "oriented languages, such as Java, Python and C#) given its source code (Allamanis et al., 2016).", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 425, + 505, + 438 + ], + "spans": [ + { + "bbox": [ + 106, + 425, + 505, + 438 + ], + "score": 1.0, + "content": "Although method names are a single token, they are usually composed of one or more subtokens (split", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 436, + 506, + 449 + ], + "spans": [ + { + "bbox": [ + 106, + 436, + 506, + 449 + ], + "score": 1.0, + "content": "using snake case or camelCase) and thus, the method naming task can be cast as predicting", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 104, + 447, + 506, + 459 + ], + "spans": [ + { + "bbox": [ + 104, + 447, + 506, + 459 + ], + "score": 1.0, + "content": "a sequence of subtokens. Consequently, method names represent an “extreme” summary of the", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 458, + 507, + 471 + ], + "spans": [ + { + "bbox": [ + 105, + 458, + 507, + 471 + ], + "score": 1.0, + "content": "functionality of a given function (on average, the names in the Java dataset have only 2.9 subtokens).", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 469, + 507, + 482 + ], + "spans": [ + { + "bbox": [ + 106, + 469, + 507, + 482 + ], + "score": 1.0, + "content": "Notably, the vocabulary of tokens used in names is very large (due to abbreviations and domain-", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 480, + 506, + 492 + ], + "spans": [ + { + "bbox": [ + 105, + 480, + 326, + 492 + ], + "score": 1.0, + "content": "specific jargon), but this is mitigated by the fact that", + "type": "text" + }, + { + "bbox": [ + 327, + 480, + 347, + 490 + ], + "score": 0.87, + "content": "33 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 347, + 480, + 506, + 492 + ], + "score": 1.0, + "content": "of subtokens in names can be copied", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 490, + 506, + 504 + ], + "spans": [ + { + "bbox": [ + 105, + 490, + 506, + 504 + ], + "score": 1.0, + "content": "directly from subtokens in the method’s source code. Finally, source code is highly structured input", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 501, + 426, + 514 + ], + "spans": [ + { + "bbox": [ + 105, + 501, + 426, + 514 + ], + "score": 1.0, + "content": "data with known semantics, which can be exploited to support name prediction.", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 30.5 + }, + { + "type": "text", + "bbox": [ + 107, + 534, + 505, + 622 + ], + "lines": [ + { + "bbox": [ + 106, + 533, + 505, + 546 + ], + "spans": [ + { + "bbox": [ + 106, + 533, + 505, + 546 + ], + "score": 1.0, + "content": "METHODDOC Similar to the first task, the aim of this task is to predict a succinct description of the", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 545, + 505, + 558 + ], + "spans": [ + { + "bbox": [ + 105, + 545, + 505, + 558 + ], + "score": 1.0, + "content": "functionality of a method given its source code (Barone & Sennrich, 2017). Such descriptions usually", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 556, + 506, + 569 + ], + "spans": [ + { + "bbox": [ + 105, + 556, + 506, + 569 + ], + "score": 1.0, + "content": "appear as documentation of methods (e.g. “docstrings” in Python or “JavaDocs” in Java). While the", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 567, + 504, + 579 + ], + "spans": [ + { + "bbox": [ + 105, + 567, + 504, + 579 + ], + "score": 1.0, + "content": "task shares many characteristics with the METHODNAMING task, the target sequence is substantially", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 578, + 505, + 591 + ], + "spans": [ + { + "bbox": [ + 105, + 578, + 338, + 591 + ], + "score": 1.0, + "content": "longer (on average 19.1 tokens in our C# dataset) and only", + "type": "text" + }, + { + "bbox": [ + 338, + 578, + 365, + 589 + ], + "score": 0.87, + "content": "1 9 . 4 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 365, + 578, + 505, + 591 + ], + "score": 1.0, + "content": "of tokens in the documentation can", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 589, + 505, + 601 + ], + "spans": [ + { + "bbox": [ + 105, + 589, + 505, + 601 + ], + "score": 1.0, + "content": "be copied from the code. While method documentation is nearer to standard natural language than", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 600, + 505, + 613 + ], + "spans": [ + { + "bbox": [ + 105, + 600, + 505, + 613 + ], + "score": 1.0, + "content": "method names, it mixes project-specific jargon, code segments and often describes non-functional", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 612, + 439, + 623 + ], + "spans": [ + { + "bbox": [ + 105, + 612, + 439, + 623 + ], + "score": 1.0, + "content": "aspects of the code, such as performance characteristics and design considerations.", + "type": "text" + } + ], + "index": 43 + } + ], + "index": 39.5 + }, + { + "type": "text", + "bbox": [ + 107, + 643, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 643, + 506, + 657 + ], + "spans": [ + { + "bbox": [ + 105, + 643, + 506, + 657 + ], + "score": 1.0, + "content": "NLSUMMARIZATION Finally, we consider the classic summarization of natural language as", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 655, + 505, + 667 + ], + "spans": [ + { + "bbox": [ + 105, + 655, + 505, + 667 + ], + "score": 1.0, + "content": "widely studied in NLP research. 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In the METHODNAMING dataset, this method appears", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 245, + 437, + 258 + ], + "spans": [ + { + "bbox": [ + 105, + 245, + 437, + 258 + ], + "score": 1.0, + "content": "as a sample requiring to predict the name Add as a subtoken sequence of length 1.", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 17, + "bbox_fs": [ + 105, + 222, + 505, + 258 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 285, + 325, + 298 + ], + "lines": [ + { + "bbox": [ + 105, + 284, + 326, + 299 + ], + "spans": [ + { + "bbox": [ + 105, + 284, + 326, + 299 + ], + "score": 1.0, + "content": "2 STRUCTURED SUMMARIZATION TASKS", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 19 + }, + { + "type": "text", + "bbox": [ + 107, + 315, + 505, + 381 + ], + "lines": [ + { + "bbox": [ + 106, + 316, + 505, + 327 + ], + "spans": [ + { + "bbox": [ + 106, + 316, + 505, + 327 + ], + "score": 1.0, + "content": "In this work, we consider three summarization tasks with different properties. All tasks follow the", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 104, + 325, + 506, + 340 + ], + "spans": [ + { + "bbox": [ + 104, + 325, + 506, + 340 + ], + "score": 1.0, + "content": "common pattern of translating a long (structured) sequence into a shorter sequence while trying", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 338, + 505, + 349 + ], + "spans": [ + { + "bbox": [ + 105, + 338, + 505, + 349 + ], + "score": 1.0, + "content": "to preserve as much meaning as possible. The first two tasks are related to the summarization of", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 348, + 505, + 361 + ], + "spans": [ + { + "bbox": [ + 105, + 348, + 505, + 361 + ], + "score": 1.0, + "content": "source code (Figure 1), which is highly structured and thus can profit most from models that can take", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 359, + 505, + 372 + ], + "spans": [ + { + "bbox": [ + 105, + 359, + 505, + 372 + ], + "score": 1.0, + "content": "advantage of this structure; the final task is a classical natural language task illustrating that hybrid", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 370, + 394, + 383 + ], + "spans": [ + { + "bbox": [ + 105, + 370, + 394, + 383 + ], + "score": 1.0, + "content": "sequence-graph models are applicable for less structured inputs as well.", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 22.5, + "bbox_fs": [ + 104, + 316, + 506, + 383 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 402, + 506, + 513 + ], + "lines": [ + { + "bbox": [ + 105, + 402, + 507, + 415 + ], + "spans": [ + { + "bbox": [ + 105, + 402, + 507, + 415 + ], + "score": 1.0, + "content": "METHODNAMING The aim of this task is to infer the name of a function (or method in object-", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 414, + 507, + 426 + ], + "spans": [ + { + "bbox": [ + 105, + 414, + 507, + 426 + ], + "score": 1.0, + "content": "oriented languages, such as Java, Python and C#) given its source code (Allamanis et al., 2016).", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 425, + 505, + 438 + ], + "spans": [ + { + "bbox": [ + 106, + 425, + 505, + 438 + ], + "score": 1.0, + "content": "Although method names are a single token, they are usually composed of one or more subtokens (split", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 436, + 506, + 449 + ], + "spans": [ + { + "bbox": [ + 106, + 436, + 506, + 449 + ], + "score": 1.0, + "content": "using snake case or camelCase) and thus, the method naming task can be cast as predicting", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 104, + 447, + 506, + 459 + ], + "spans": [ + { + "bbox": [ + 104, + 447, + 506, + 459 + ], + "score": 1.0, + "content": "a sequence of subtokens. Consequently, method names represent an “extreme” summary of the", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 458, + 507, + 471 + ], + "spans": [ + { + "bbox": [ + 105, + 458, + 507, + 471 + ], + "score": 1.0, + "content": "functionality of a given function (on average, the names in the Java dataset have only 2.9 subtokens).", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 469, + 507, + 482 + ], + "spans": [ + { + "bbox": [ + 106, + 469, + 507, + 482 + ], + "score": 1.0, + "content": "Notably, the vocabulary of tokens used in names is very large (due to abbreviations and domain-", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 480, + 506, + 492 + ], + "spans": [ + { + "bbox": [ + 105, + 480, + 326, + 492 + ], + "score": 1.0, + "content": "specific jargon), but this is mitigated by the fact that", + "type": "text" + }, + { + "bbox": [ + 327, + 480, + 347, + 490 + ], + "score": 0.87, + "content": "33 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 347, + 480, + 506, + 492 + ], + "score": 1.0, + "content": "of subtokens in names can be copied", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 490, + 506, + 504 + ], + "spans": [ + { + "bbox": [ + 105, + 490, + 506, + 504 + ], + "score": 1.0, + "content": "directly from subtokens in the method’s source code. Finally, source code is highly structured input", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 501, + 426, + 514 + ], + "spans": [ + { + "bbox": [ + 105, + 501, + 426, + 514 + ], + "score": 1.0, + "content": "data with known semantics, which can be exploited to support name prediction.", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 30.5, + "bbox_fs": [ + 104, + 402, + 507, + 514 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 534, + 505, + 622 + ], + "lines": [ + { + "bbox": [ + 106, + 533, + 505, + 546 + ], + "spans": [ + { + "bbox": [ + 106, + 533, + 505, + 546 + ], + "score": 1.0, + "content": "METHODDOC Similar to the first task, the aim of this task is to predict a succinct description of the", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 545, + 505, + 558 + ], + "spans": [ + { + "bbox": [ + 105, + 545, + 505, + 558 + ], + "score": 1.0, + "content": "functionality of a method given its source code (Barone & Sennrich, 2017). Such descriptions usually", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 556, + 506, + 569 + ], + "spans": [ + { + "bbox": [ + 105, + 556, + 506, + 569 + ], + "score": 1.0, + "content": "appear as documentation of methods (e.g. “docstrings” in Python or “JavaDocs” in Java). While the", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 567, + 504, + 579 + ], + "spans": [ + { + "bbox": [ + 105, + 567, + 504, + 579 + ], + "score": 1.0, + "content": "task shares many characteristics with the METHODNAMING task, the target sequence is substantially", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 578, + 505, + 591 + ], + "spans": [ + { + "bbox": [ + 105, + 578, + 338, + 591 + ], + "score": 1.0, + "content": "longer (on average 19.1 tokens in our C# dataset) and only", + "type": "text" + }, + { + "bbox": [ + 338, + 578, + 365, + 589 + ], + "score": 0.87, + "content": "1 9 . 4 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 365, + 578, + 505, + 591 + ], + "score": 1.0, + "content": "of tokens in the documentation can", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 589, + 505, + 601 + ], + "spans": [ + { + "bbox": [ + 105, + 589, + 505, + 601 + ], + "score": 1.0, + "content": "be copied from the code. While method documentation is nearer to standard natural language than", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 600, + 505, + 613 + ], + "spans": [ + { + "bbox": [ + 105, + 600, + 505, + 613 + ], + "score": 1.0, + "content": "method names, it mixes project-specific jargon, code segments and often describes non-functional", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 612, + 439, + 623 + ], + "spans": [ + { + "bbox": [ + 105, + 612, + 439, + 623 + ], + "score": 1.0, + "content": "aspects of the code, such as performance characteristics and design considerations.", + "type": "text" + } + ], + "index": 43 + } + ], + "index": 39.5, + "bbox_fs": [ + 105, + 533, + 506, + 623 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 643, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 643, + 506, + 657 + ], + "spans": [ + { + "bbox": [ + 105, + 643, + 506, + 657 + ], + "score": 1.0, + "content": "NLSUMMARIZATION Finally, we consider the classic summarization of natural language as", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 655, + 505, + 667 + ], + "spans": [ + { + "bbox": [ + 105, + 655, + 505, + 667 + ], + "score": 1.0, + "content": "widely studied in NLP research. Specifically, we are interested in abstractive summarization, where", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 106, + 666, + 506, + 678 + ], + "spans": [ + { + "bbox": [ + 106, + 666, + 506, + 678 + ], + "score": 1.0, + "content": "given some text input (e.g. a news article) a machine learning model produces a novel natural lan-", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 104, + 676, + 506, + 690 + ], + "spans": [ + { + "bbox": [ + 104, + 676, + 506, + 690 + ], + "score": 1.0, + "content": "guage summary. Traditionally, NLP summarization methods treat text as a sequence of sentences", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 105, + 688, + 505, + 700 + ], + "spans": [ + { + "bbox": [ + 105, + 688, + 505, + 700 + ], + "score": 1.0, + "content": "and each one of them as a sequence of words (tokens). The input data has less explicitly defined", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 105, + 699, + 504, + 711 + ], + "spans": [ + { + "bbox": [ + 105, + 699, + 504, + 711 + ], + "score": 1.0, + "content": "structure than our first two tasks. However, we recast the task as a structured summarization problem", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 105, + 709, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 105, + 709, + 505, + 722 + ], + "score": 1.0, + "content": "by considering additional linguistic structure, including named entities and entity coreferences as", + "type": "text" + } + ], + "index": 50 + }, + { + "bbox": [ + 105, + 721, + 232, + 733 + ], + "spans": [ + { + "bbox": [ + 105, + 721, + 232, + 733 + ], + "score": 1.0, + "content": "inferred by existing NLP tools.", + "type": "text" + } + ], + "index": 51 + } + ], + "index": 47.5, + "bbox_fs": [ + 104, + 643, + 506, + 733 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "title", + "bbox": [ + 108, + 81, + 164, + 93 + ], + "lines": [ + { + "bbox": [ + 104, + 79, + 167, + 96 + ], + "spans": [ + { + "bbox": [ + 104, + 79, + 167, + 96 + ], + "score": 1.0, + "content": "3 MODEL", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "text", + "bbox": [ + 107, + 106, + 505, + 162 + ], + "lines": [ + { + "bbox": [ + 106, + 106, + 505, + 119 + ], + "spans": [ + { + "bbox": [ + 106, + 106, + 505, + 119 + ], + "score": 1.0, + "content": "As discussed above, standard neural approaches to summarization follow the sequence-to-sequence", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 117, + 505, + 131 + ], + "spans": [ + { + "bbox": [ + 105, + 117, + 403, + 131 + ], + "score": 1.0, + "content": "framework. In this setting, most decoders only require a representation", + "type": "text" + }, + { + "bbox": [ + 403, + 118, + 412, + 128 + ], + "score": 0.73, + "content": "^ { h }", + "type": "inline_equation" + }, + { + "bbox": [ + 412, + 117, + 505, + 131 + ], + "score": 1.0, + "content": "of the complete input", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 128, + 503, + 141 + ], + "spans": [ + { + "bbox": [ + 105, + 128, + 395, + 141 + ], + "score": 1.0, + "content": "sequence (e.g. the final state of an RNN) and per-token representations", + "type": "text" + }, + { + "bbox": [ + 395, + 129, + 409, + 141 + ], + "score": 0.89, + "content": "h _ { t _ { i } }", + "type": "inline_equation" + }, + { + "bbox": [ + 410, + 128, + 495, + 141 + ], + "score": 1.0, + "content": "for each input token", + "type": "text" + }, + { + "bbox": [ + 495, + 129, + 503, + 140 + ], + "score": 0.84, + "content": "t _ { i }", + "type": "inline_equation" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 138, + 506, + 153 + ], + "spans": [ + { + "bbox": [ + 105, + 138, + 506, + 153 + ], + "score": 1.0, + "content": "These token representations are then used as the “memories” of an attention mechanism (Bahdanau", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 106, + 151, + 410, + 163 + ], + "spans": [ + { + "bbox": [ + 106, + 151, + 410, + 163 + ], + "score": 1.0, + "content": "et al., 2014; Luong et al., 2015) or a pointer network (Vinyals et al., 2015a).", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 3 + }, + { + "type": "text", + "bbox": [ + 106, + 167, + 506, + 255 + ], + "lines": [ + { + "bbox": [ + 105, + 167, + 505, + 180 + ], + "spans": [ + { + "bbox": [ + 105, + 167, + 505, + 180 + ], + "score": 1.0, + "content": "In this work, we propose an extension of sequence encoders that allows us to leverage known (or", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 179, + 506, + 191 + ], + "spans": [ + { + "bbox": [ + 106, + 179, + 506, + 191 + ], + "score": 1.0, + "content": "inferred) relationships among elements in the input data. To achieve that, we combine sequence", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 190, + 506, + 202 + ], + "spans": [ + { + "bbox": [ + 106, + 190, + 506, + 202 + ], + "score": 1.0, + "content": "encoders with graph neural networks (GNNs) (Li et al., 2015; Gilmer et al., 2017; Kipf & Welling,", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 200, + 507, + 214 + ], + "spans": [ + { + "bbox": [ + 105, + 200, + 507, + 214 + ], + "score": 1.0, + "content": "2017). For this, we first use a standard sequential encoder (e.g. bidirectional RNNs) to obtain a per-", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 106, + 212, + 506, + 224 + ], + "spans": [ + { + "bbox": [ + 106, + 212, + 191, + 224 + ], + "score": 1.0, + "content": "token representation", + "type": "text" + }, + { + "bbox": [ + 192, + 212, + 206, + 223 + ], + "score": 0.89, + "content": "h _ { t _ { i } }", + "type": "inline_equation" + }, + { + "bbox": [ + 206, + 212, + 506, + 224 + ], + "score": 1.0, + "content": ", which we then feed into a GNN as the initial node representations. The", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 222, + 507, + 235 + ], + "spans": [ + { + "bbox": [ + 105, + 222, + 309, + 235 + ], + "score": 1.0, + "content": "resulting per-node (i.e. per-token) representations", + "type": "text" + }, + { + "bbox": [ + 309, + 222, + 324, + 235 + ], + "score": 0.91, + "content": "\\boldsymbol { h } _ { t _ { i } } ^ { \\prime }", + "type": "inline_equation" + }, + { + "bbox": [ + 324, + 222, + 507, + 235 + ], + "score": 1.0, + "content": "can then be used by an unmodified decoder.", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 232, + 506, + 247 + ], + "spans": [ + { + "bbox": [ + 105, + 232, + 506, + 247 + ], + "score": 1.0, + "content": "Experimentally, we found this to surpass models that use either only the sequential structure or only", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 245, + 468, + 257 + ], + "spans": [ + { + "bbox": [ + 106, + 245, + 468, + 257 + ], + "score": 1.0, + "content": "the graph structure (see Sect. 4). We now discuss the different parts of our model in detail.", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 9.5 + }, + { + "type": "text", + "bbox": [ + 107, + 268, + 505, + 326 + ], + "lines": [ + { + "bbox": [ + 106, + 269, + 506, + 281 + ], + "spans": [ + { + "bbox": [ + 106, + 269, + 506, + 281 + ], + "score": 1.0, + "content": "Gated Graph Neural Networks To process graphs, we follow Li et al. (2015) and briefly summa-", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 280, + 506, + 292 + ], + "spans": [ + { + "bbox": [ + 106, + 280, + 305, + 292 + ], + "score": 1.0, + "content": "rize the core concepts of GGNNs here. A graph", + "type": "text" + }, + { + "bbox": [ + 305, + 280, + 368, + 291 + ], + "score": 0.91, + "content": "\\mathcal { G } = ( \\nu , \\pmb { \\varepsilon } , \\pmb { X } )", + "type": "inline_equation" + }, + { + "bbox": [ + 369, + 280, + 494, + 292 + ], + "score": 1.0, + "content": "is composed of a set of nodes", + "type": "text" + }, + { + "bbox": [ + 495, + 280, + 502, + 290 + ], + "score": 0.73, + "content": "\\nu", + "type": "inline_equation" + }, + { + "bbox": [ + 503, + 280, + 506, + 292 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 290, + 505, + 304 + ], + "spans": [ + { + "bbox": [ + 105, + 290, + 163, + 304 + ], + "score": 1.0, + "content": "node features", + "type": "text" + }, + { + "bbox": [ + 163, + 291, + 174, + 301 + ], + "score": 0.75, + "content": "\\boldsymbol { X }", + "type": "inline_equation" + }, + { + "bbox": [ + 174, + 290, + 302, + 304 + ], + "score": 1.0, + "content": ", and a list of directed edge sets", + "type": "text" + }, + { + "bbox": [ + 302, + 291, + 375, + 303 + ], + "score": 0.9, + "content": "{ \\pmb { \\mathcal { E } } } = ( { \\mathcal { E } } _ { 1 } , \\dots , { \\mathcal { E } } _ { K } )", + "type": "inline_equation" + }, + { + "bbox": [ + 375, + 290, + 403, + 304 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 404, + 291, + 415, + 301 + ], + "score": 0.83, + "content": "K", + "type": "inline_equation" + }, + { + "bbox": [ + 415, + 290, + 505, + 304 + ], + "score": 1.0, + "content": "is the number of edge", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 301, + 505, + 315 + ], + "spans": [ + { + "bbox": [ + 106, + 301, + 155, + 315 + ], + "score": 1.0, + "content": "types. Each", + "type": "text" + }, + { + "bbox": [ + 156, + 302, + 183, + 312 + ], + "score": 0.9, + "content": "v \\in \\nu", + "type": "inline_equation" + }, + { + "bbox": [ + 183, + 301, + 340, + 315 + ], + "score": 1.0, + "content": "is associated with a real-valued vector", + "type": "text" + }, + { + "bbox": [ + 341, + 304, + 353, + 313 + ], + "score": 0.85, + "content": "\\scriptstyle { \\mathbf { { \\mathit { x } } } } _ { \\mathit { v } }", + "type": "inline_equation" + }, + { + "bbox": [ + 353, + 301, + 505, + 315 + ], + "score": 1.0, + "content": "representing the features of the node", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 104, + 311, + 506, + 329 + ], + "spans": [ + { + "bbox": [ + 104, + 311, + 442, + 329 + ], + "score": 1.0, + "content": "(e.g., the embedding of a string label of that node), which is used for the initial state", + "type": "text" + }, + { + "bbox": [ + 443, + 312, + 461, + 327 + ], + "score": 0.92, + "content": "{ h } _ { v } ^ { ( 0 ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 461, + 311, + 506, + 329 + ], + "score": 1.0, + "content": "of a node.", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 16 + }, + { + "type": "text", + "bbox": [ + 106, + 331, + 506, + 466 + ], + "lines": [ + { + "bbox": [ + 105, + 330, + 507, + 345 + ], + "spans": [ + { + "bbox": [ + 105, + 330, + 507, + 345 + ], + "score": 1.0, + "content": "Information is propagated through the graph using neural message passing (Gilmer et al., 2017).", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 342, + 506, + 355 + ], + "spans": [ + { + "bbox": [ + 105, + 342, + 191, + 355 + ], + "score": 1.0, + "content": "For this, every node", + "type": "text" + }, + { + "bbox": [ + 191, + 344, + 198, + 352 + ], + "score": 0.74, + "content": "v", + "type": "inline_equation" + }, + { + "bbox": [ + 198, + 342, + 506, + 355 + ], + "score": 1.0, + "content": "sends messages to its neighbors by transforming its current representation", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 353, + 506, + 369 + ], + "spans": [ + { + "bbox": [ + 106, + 353, + 124, + 367 + ], + "score": 0.9, + "content": "\\boldsymbol { h } _ { v } ^ { ( i ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 124, + 353, + 289, + 369 + ], + "score": 1.0, + "content": "using an edge-type dependent function", + "type": "text" + }, + { + "bbox": [ + 290, + 356, + 300, + 367 + ], + "score": 0.88, + "content": "f _ { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 301, + 353, + 330, + 369 + ], + "score": 1.0, + "content": ". Here,", + "type": "text" + }, + { + "bbox": [ + 331, + 356, + 342, + 367 + ], + "score": 0.86, + "content": "f _ { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 342, + 353, + 506, + 369 + ], + "score": 1.0, + "content": "can be an arbitrary function; we use a", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 366, + 506, + 379 + ], + "spans": [ + { + "bbox": [ + 106, + 366, + 506, + 379 + ], + "score": 1.0, + "content": "simple linear layer. By computing all messages at the same time, all states can be updated simultane-", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 377, + 506, + 390 + ], + "spans": [ + { + "bbox": [ + 106, + 377, + 278, + 390 + ], + "score": 1.0, + "content": "ously. In particular, a new state for a node", + "type": "text" + }, + { + "bbox": [ + 278, + 379, + 285, + 387 + ], + "score": 0.76, + "content": "v", + "type": "inline_equation" + }, + { + "bbox": [ + 285, + 377, + 506, + 390 + ], + "score": 1.0, + "content": "is computed by aggregating all incoming messages as", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 107, + 384, + 509, + 407 + ], + "spans": [ + { + "bbox": [ + 107, + 388, + 194, + 403 + ], + "score": 0.93, + "content": "m _ { v } ^ { ( i ) } = \\bar { g } ( \\{ f _ { k } ( h _ { u } ^ { ( i ) } ) \\} )", + "type": "inline_equation" + }, + { + "bbox": [ + 194, + 384, + 290, + 407 + ], + "score": 1.0, + "content": "there is an edge of type", + "type": "text" + }, + { + "bbox": [ + 291, + 391, + 298, + 401 + ], + "score": 0.76, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 298, + 384, + 320, + 407 + ], + "score": 1.0, + "content": "from", + "type": "text" + }, + { + "bbox": [ + 321, + 393, + 328, + 401 + ], + "score": 0.62, + "content": "u", + "type": "inline_equation" + }, + { + "bbox": [ + 328, + 384, + 339, + 407 + ], + "score": 1.0, + "content": "to", + "type": "text" + }, + { + "bbox": [ + 339, + 392, + 350, + 403 + ], + "score": 0.72, + "content": "v \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 351, + 384, + 358, + 407 + ], + "score": 1.0, + "content": ").", + "type": "text" + }, + { + "bbox": [ + 358, + 393, + 365, + 402 + ], + "score": 0.74, + "content": "g", + "type": "inline_equation" + }, + { + "bbox": [ + 365, + 384, + 509, + 407 + ], + "score": 1.0, + "content": "is an aggregation function; we use", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 104, + 400, + 504, + 420 + ], + "spans": [ + { + "bbox": [ + 104, + 400, + 219, + 420 + ], + "score": 1.0, + "content": "elementwise summation for", + "type": "text" + }, + { + "bbox": [ + 220, + 407, + 226, + 416 + ], + "score": 0.73, + "content": "g", + "type": "inline_equation" + }, + { + "bbox": [ + 227, + 400, + 353, + 420 + ], + "score": 1.0, + "content": ". Given the aggregated message", + "type": "text" + }, + { + "bbox": [ + 354, + 402, + 374, + 415 + ], + "score": 0.92, + "content": "\\mathbf { \\it { m } } _ { v } ^ { ( i ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 375, + 400, + 486, + 420 + ], + "score": 1.0, + "content": "and the current state vector", + "type": "text" + }, + { + "bbox": [ + 487, + 403, + 504, + 417 + ], + "score": 0.9, + "content": "\\boldsymbol { h } _ { v } ^ { ( i ) }", + "type": "inline_equation" + } + ], + "index": 25 + }, + { + "bbox": [ + 101, + 412, + 510, + 439 + ], + "spans": [ + { + "bbox": [ + 101, + 415, + 140, + 435 + ], + "score": 1.0, + "content": "of node", + "type": "text" + }, + { + "bbox": [ + 140, + 421, + 146, + 429 + ], + "score": 0.67, + "content": "v", + "type": "inline_equation" + }, + { + "bbox": [ + 146, + 415, + 274, + 435 + ], + "score": 1.0, + "content": ", we can compute the new state", + "type": "text" + }, + { + "bbox": [ + 296, + 412, + 510, + 439 + ], + "score": 1.0, + "content": "1) = GRU(m(i)v , h(i)v ), where GRU is the recurrent", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 428, + 506, + 443 + ], + "spans": [ + { + "bbox": [ + 105, + 428, + 506, + 443 + ], + "score": 1.0, + "content": "cell function of a gated recurrent unit. These dynamics are rolled out for a fixed number of timesteps", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 440, + 507, + 454 + ], + "spans": [ + { + "bbox": [ + 106, + 441, + 115, + 451 + ], + "score": 0.76, + "content": "T", + "type": "inline_equation" + }, + { + "bbox": [ + 115, + 440, + 507, + 454 + ], + "score": 1.0, + "content": ", and the state vectors resulting from the final step are used as output node representations, i.e.,", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 449, + 243, + 468 + ], + "spans": [ + { + "bbox": [ + 106, + 451, + 237, + 466 + ], + "score": 0.91, + "content": "\\mathrm { G N N } ( ( \\mathcal { V } , \\pmb { \\mathcal { E } } , X ) ) = \\{ \\pmb { h } _ { v } ^ { ( T ) } \\} _ { v \\in \\mathcal { V } }", + "type": "inline_equation" + }, + { + "bbox": [ + 238, + 449, + 243, + 468 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 24 + }, + { + "type": "text", + "bbox": [ + 106, + 478, + 506, + 614 + ], + "lines": [ + { + "bbox": [ + 106, + 478, + 506, + 491 + ], + "spans": [ + { + "bbox": [ + 106, + 478, + 506, + 491 + ], + "score": 1.0, + "content": "Sequence GNNs We now explain our novel combination of GGNNs and standard sequence en-", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 489, + 504, + 501 + ], + "spans": [ + { + "bbox": [ + 105, + 489, + 252, + 501 + ], + "score": 1.0, + "content": "coders. As input, we take a sequence", + "type": "text" + }, + { + "bbox": [ + 253, + 489, + 315, + 501 + ], + "score": 0.95, + "content": "\\boldsymbol { S } = \\left[ s _ { 1 } \\ldots s _ { N } \\right]", + "type": "inline_equation" + }, + { + "bbox": [ + 316, + 489, + 333, + 501 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 333, + 489, + 343, + 499 + ], + "score": 0.83, + "content": "K", + "type": "inline_equation" + }, + { + "bbox": [ + 344, + 489, + 424, + 501 + ], + "score": 1.0, + "content": "binary relationships", + "type": "text" + }, + { + "bbox": [ + 425, + 489, + 504, + 500 + ], + "score": 0.9, + "content": "R _ { 1 } \\ldots R _ { K } \\in S \\times S", + "type": "inline_equation" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 500, + 504, + 514 + ], + "spans": [ + { + "bbox": [ + 105, + 500, + 306, + 514 + ], + "score": 1.0, + "content": "between elements of the sequence. For example,", + "type": "text" + }, + { + "bbox": [ + 307, + 501, + 322, + 511 + ], + "score": 0.88, + "content": "R _ { = }", + "type": "inline_equation" + }, + { + "bbox": [ + 322, + 500, + 462, + 514 + ], + "score": 1.0, + "content": "could be the equality relationship", + "type": "text" + }, + { + "bbox": [ + 463, + 500, + 504, + 512 + ], + "score": 0.89, + "content": "\\left\\{ \\left( s _ { i } , s _ { j } \\right) \\right. \\mid", + "type": "inline_equation" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 511, + 505, + 523 + ], + "spans": [ + { + "bbox": [ + 106, + 511, + 143, + 523 + ], + "score": 0.91, + "content": "s _ { i } = s _ { j } \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 144, + 511, + 505, + 523 + ], + "score": 1.0, + "content": ". The choice and construction of relationships is dataset-dependent, and will be discussed", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 522, + 505, + 534 + ], + "spans": [ + { + "bbox": [ + 105, + 522, + 309, + 534 + ], + "score": 1.0, + "content": "in detail in Sect. 4. Given any sequence encoder", + "type": "text" + }, + { + "bbox": [ + 310, + 522, + 324, + 532 + ], + "score": 0.83, + "content": "\\mathcal { S E }", + "type": "inline_equation" + }, + { + "bbox": [ + 325, + 522, + 369, + 534 + ], + "score": 1.0, + "content": "that maps", + "type": "text" + }, + { + "bbox": [ + 369, + 522, + 377, + 532 + ], + "score": 0.79, + "content": "S", + "type": "inline_equation" + }, + { + "bbox": [ + 378, + 522, + 505, + 534 + ], + "score": 1.0, + "content": "to per-element representations", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 107, + 533, + 507, + 546 + ], + "spans": [ + { + "bbox": [ + 107, + 533, + 151, + 545 + ], + "score": 0.92, + "content": "\\left[ \\mathbf { e } _ { 1 } \\ldots \\mathbf { e } _ { N } \\right]", + "type": "inline_equation" + }, + { + "bbox": [ + 151, + 533, + 507, + 546 + ], + "score": 1.0, + "content": "and a sequence representation e (e.g. a bidirectional RNN), we can construct the se-", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 104, + 543, + 507, + 557 + ], + "spans": [ + { + "bbox": [ + 104, + 543, + 162, + 557 + ], + "score": 1.0, + "content": "quence GNN", + "type": "text" + }, + { + "bbox": [ + 162, + 544, + 197, + 555 + ], + "score": 0.89, + "content": "\\mathcal { S } \\mathcal { E } _ { G N N }", + "type": "inline_equation" + }, + { + "bbox": [ + 197, + 543, + 289, + 557 + ], + "score": 1.0, + "content": "by simply computing", + "type": "text" + }, + { + "bbox": [ + 289, + 543, + 488, + 556 + ], + "score": 0.77, + "content": "\\mathbf { \\bar { e } } _ { 1 } ^ { \\prime } \\cdot \\cdot \\cdot \\mathbf { e } _ { N } ^ { \\prime } ] = \\mathbf { G N N } ( ( S , [ R _ { 1 } \\dots R _ { K } ] , [ \\mathbf { e } _ { 1 } \\dots \\mathbf { e } _ { N } ] ) )", + "type": "inline_equation" + }, + { + "bbox": [ + 488, + 543, + 507, + 557 + ], + "score": 1.0, + "content": ". To", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 555, + 507, + 567 + ], + "spans": [ + { + "bbox": [ + 105, + 555, + 507, + 567 + ], + "score": 1.0, + "content": "obtain a graph-level representation, we use the weighted averaging mechanism from Gilmer et al.", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 104, + 565, + 508, + 582 + ], + "spans": [ + { + "bbox": [ + 104, + 565, + 248, + 582 + ], + "score": 1.0, + "content": "(2017). 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In this setting, most decoders only require a representation", + "type": "text" + }, + { + "bbox": [ + 403, + 118, + 412, + 128 + ], + "score": 0.73, + "content": "^ { h }", + "type": "inline_equation" + }, + { + "bbox": [ + 412, + 117, + 505, + 131 + ], + "score": 1.0, + "content": "of the complete input", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 128, + 503, + 141 + ], + "spans": [ + { + "bbox": [ + 105, + 128, + 395, + 141 + ], + "score": 1.0, + "content": "sequence (e.g. the final state of an RNN) and per-token representations", + "type": "text" + }, + { + "bbox": [ + 395, + 129, + 409, + 141 + ], + "score": 0.89, + "content": "h _ { t _ { i } }", + "type": "inline_equation" + }, + { + "bbox": [ + 410, + 128, + 495, + 141 + ], + "score": 1.0, + "content": "for each input token", + "type": "text" + }, + { + "bbox": [ + 495, + 129, + 503, + 140 + ], + "score": 0.84, + "content": "t _ { i }", + "type": "inline_equation" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 138, + 506, + 153 + ], + "spans": [ + { + "bbox": [ + 105, + 138, + 506, + 153 + ], + "score": 1.0, + "content": "These token representations are then used as the “memories” of an attention mechanism (Bahdanau", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 106, + 151, + 410, + 163 + ], + "spans": [ + { + "bbox": [ + 106, + 151, + 410, + 163 + ], + "score": 1.0, + "content": "et al., 2014; Luong et al., 2015) or a pointer network (Vinyals et al., 2015a).", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 3, + "bbox_fs": [ + 105, + 106, + 506, + 163 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 167, + 506, + 255 + ], + "lines": [ + { + "bbox": [ + 105, + 167, + 505, + 180 + ], + "spans": [ + { + "bbox": [ + 105, + 167, + 505, + 180 + ], + "score": 1.0, + "content": "In this work, we propose an extension of sequence encoders that allows us to leverage known (or", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 179, + 506, + 191 + ], + "spans": [ + { + "bbox": [ + 106, + 179, + 506, + 191 + ], + "score": 1.0, + "content": "inferred) relationships among elements in the input data. To achieve that, we combine sequence", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 190, + 506, + 202 + ], + "spans": [ + { + "bbox": [ + 106, + 190, + 506, + 202 + ], + "score": 1.0, + "content": "encoders with graph neural networks (GNNs) (Li et al., 2015; Gilmer et al., 2017; Kipf & Welling,", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 200, + 507, + 214 + ], + "spans": [ + { + "bbox": [ + 105, + 200, + 507, + 214 + ], + "score": 1.0, + "content": "2017). For this, we first use a standard sequential encoder (e.g. bidirectional RNNs) to obtain a per-", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 106, + 212, + 506, + 224 + ], + "spans": [ + { + "bbox": [ + 106, + 212, + 191, + 224 + ], + "score": 1.0, + "content": "token representation", + "type": "text" + }, + { + "bbox": [ + 192, + 212, + 206, + 223 + ], + "score": 0.89, + "content": "h _ { t _ { i } }", + "type": "inline_equation" + }, + { + "bbox": [ + 206, + 212, + 506, + 224 + ], + "score": 1.0, + "content": ", which we then feed into a GNN as the initial node representations. The", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 222, + 507, + 235 + ], + "spans": [ + { + "bbox": [ + 105, + 222, + 309, + 235 + ], + "score": 1.0, + "content": "resulting per-node (i.e. per-token) representations", + "type": "text" + }, + { + "bbox": [ + 309, + 222, + 324, + 235 + ], + "score": 0.91, + "content": "\\boldsymbol { h } _ { t _ { i } } ^ { \\prime }", + "type": "inline_equation" + }, + { + "bbox": [ + 324, + 222, + 507, + 235 + ], + "score": 1.0, + "content": "can then be used by an unmodified decoder.", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 232, + 506, + 247 + ], + "spans": [ + { + "bbox": [ + 105, + 232, + 506, + 247 + ], + "score": 1.0, + "content": "Experimentally, we found this to surpass models that use either only the sequential structure or only", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 245, + 468, + 257 + ], + "spans": [ + { + "bbox": [ + 106, + 245, + 468, + 257 + ], + "score": 1.0, + "content": "the graph structure (see Sect. 4). We now discuss the different parts of our model in detail.", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 9.5, + "bbox_fs": [ + 105, + 167, + 507, + 257 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 268, + 505, + 326 + ], + "lines": [ + { + "bbox": [ + 106, + 269, + 506, + 281 + ], + "spans": [ + { + "bbox": [ + 106, + 269, + 506, + 281 + ], + "score": 1.0, + "content": "Gated Graph Neural Networks To process graphs, we follow Li et al. (2015) and briefly summa-", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 280, + 506, + 292 + ], + "spans": [ + { + "bbox": [ + 106, + 280, + 305, + 292 + ], + "score": 1.0, + "content": "rize the core concepts of GGNNs here. 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In particular, a new state for a node", + "type": "text" + }, + { + "bbox": [ + 278, + 379, + 285, + 387 + ], + "score": 0.76, + "content": "v", + "type": "inline_equation" + }, + { + "bbox": [ + 285, + 377, + 506, + 390 + ], + "score": 1.0, + "content": "is computed by aggregating all incoming messages as", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 107, + 384, + 509, + 407 + ], + "spans": [ + { + "bbox": [ + 107, + 388, + 194, + 403 + ], + "score": 0.93, + "content": "m _ { v } ^ { ( i ) } = \\bar { g } ( \\{ f _ { k } ( h _ { u } ^ { ( i ) } ) \\} )", + "type": "inline_equation" + }, + { + "bbox": [ + 194, + 384, + 290, + 407 + ], + "score": 1.0, + "content": "there is an edge of type", + "type": "text" + }, + { + "bbox": [ + 291, + 391, + 298, + 401 + ], + "score": 0.76, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 298, + 384, + 320, + 407 + ], + "score": 1.0, + "content": "from", + "type": "text" + }, + { + "bbox": [ + 321, + 393, + 328, + 401 + ], + "score": 0.62, + "content": "u", + "type": "inline_equation" + }, + { + "bbox": [ + 328, + 384, + 339, + 407 + ], + "score": 1.0, + "content": "to", + "type": "text" + }, + { + "bbox": [ + 339, + 392, + 350, + 403 + ], + "score": 0.72, + "content": "v \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 351, + 384, + 358, + 407 + ], + "score": 1.0, + "content": ").", + "type": "text" + }, + { + "bbox": [ + 358, + 393, + 365, + 402 + ], + "score": 0.74, + "content": "g", + "type": "inline_equation" + }, + { + "bbox": [ + 365, + 384, + 509, + 407 + ], + "score": 1.0, + "content": "is an aggregation function; we use", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 104, + 400, + 504, + 420 + ], + "spans": [ + { + "bbox": [ + 104, + 400, + 219, + 420 + ], + "score": 1.0, + "content": "elementwise summation for", + "type": "text" + }, + { + "bbox": [ + 220, + 407, + 226, + 416 + ], + "score": 0.73, + "content": "g", + "type": "inline_equation" + }, + { + "bbox": [ + 227, + 400, + 353, + 420 + ], + "score": 1.0, + "content": ". Given the aggregated message", + "type": "text" + }, + { + "bbox": [ + 354, + 402, + 374, + 415 + ], + "score": 0.92, + "content": "\\mathbf { \\it { m } } _ { v } ^ { ( i ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 375, + 400, + 486, + 420 + ], + "score": 1.0, + "content": "and the current state vector", + "type": "text" + }, + { + "bbox": [ + 487, + 403, + 504, + 417 + ], + "score": 0.9, + "content": "\\boldsymbol { h } _ { v } ^ { ( i ) }", + "type": "inline_equation" + } + ], + "index": 25 + }, + { + "bbox": [ + 101, + 412, + 510, + 439 + ], + "spans": [ + { + "bbox": [ + 101, + 415, + 140, + 435 + ], + "score": 1.0, + "content": "of node", + "type": "text" + }, + { + "bbox": [ + 140, + 421, + 146, + 429 + ], + "score": 0.67, + "content": "v", + "type": "inline_equation" + }, + { + "bbox": [ + 146, + 415, + 274, + 435 + ], + "score": 1.0, + "content": ", we can compute the new state", + "type": "text" + }, + { + "bbox": [ + 296, + 412, + 510, + 439 + ], + "score": 1.0, + "content": "1) = GRU(m(i)v , h(i)v ), where GRU is the recurrent", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 428, + 506, + 443 + ], + "spans": [ + { + "bbox": [ + 105, + 428, + 506, + 443 + ], + "score": 1.0, + "content": "cell function of a gated recurrent unit. These dynamics are rolled out for a fixed number of timesteps", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 440, + 507, + 454 + ], + "spans": [ + { + "bbox": [ + 106, + 441, + 115, + 451 + ], + "score": 0.76, + "content": "T", + "type": "inline_equation" + }, + { + "bbox": [ + 115, + 440, + 507, + 454 + ], + "score": 1.0, + "content": ", and the state vectors resulting from the final step are used as output node representations, i.e.,", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 449, + 243, + 468 + ], + "spans": [ + { + "bbox": [ + 106, + 451, + 237, + 466 + ], + "score": 0.91, + "content": "\\mathrm { G N N } ( ( \\mathcal { V } , \\pmb { \\mathcal { E } } , X ) ) = \\{ \\pmb { h } _ { v } ^ { ( T ) } \\} _ { v \\in \\mathcal { V } }", + "type": "inline_equation" + }, + { + "bbox": [ + 238, + 449, + 243, + 468 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 24, + "bbox_fs": [ + 101, + 330, + 510, + 468 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 478, + 506, + 614 + ], + "lines": [ + { + "bbox": [ + 106, + 478, + 506, + 491 + ], + "spans": [ + { + "bbox": [ + 106, + 478, + 506, + 491 + ], + "score": 1.0, + "content": "Sequence GNNs We now explain our novel combination of GGNNs and standard sequence en-", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 489, + 504, + 501 + ], + "spans": [ + { + "bbox": [ + 105, + 489, + 252, + 501 + ], + "score": 1.0, + "content": "coders. As input, we take a sequence", + "type": "text" + }, + { + "bbox": [ + 253, + 489, + 315, + 501 + ], + "score": 0.95, + "content": "\\boldsymbol { S } = \\left[ s _ { 1 } \\ldots s _ { N } \\right]", + "type": "inline_equation" + }, + { + "bbox": [ + 316, + 489, + 333, + 501 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 333, + 489, + 343, + 499 + ], + "score": 0.83, + "content": "K", + "type": "inline_equation" + }, + { + "bbox": [ + 344, + 489, + 424, + 501 + ], + "score": 1.0, + "content": "binary relationships", + "type": "text" + }, + { + "bbox": [ + 425, + 489, + 504, + 500 + ], + "score": 0.9, + "content": "R _ { 1 } \\ldots R _ { K } \\in S \\times S", + "type": "inline_equation" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 500, + 504, + 514 + ], + "spans": [ + { + "bbox": [ + 105, + 500, + 306, + 514 + ], + "score": 1.0, + "content": "between elements of the sequence. For example,", + "type": "text" + }, + { + "bbox": [ + 307, + 501, + 322, + 511 + ], + "score": 0.88, + "content": "R _ { = }", + "type": "inline_equation" + }, + { + "bbox": [ + 322, + 500, + 462, + 514 + ], + "score": 1.0, + "content": "could be the equality relationship", + "type": "text" + }, + { + "bbox": [ + 463, + 500, + 504, + 512 + ], + "score": 0.89, + "content": "\\left\\{ \\left( s _ { i } , s _ { j } \\right) \\right. \\mid", + "type": "inline_equation" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 511, + 505, + 523 + ], + "spans": [ + { + "bbox": [ + 106, + 511, + 143, + 523 + ], + "score": 0.91, + "content": "s _ { i } = s _ { j } \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 144, + 511, + 505, + 523 + ], + "score": 1.0, + "content": ". The choice and construction of relationships is dataset-dependent, and will be discussed", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 522, + 505, + 534 + ], + "spans": [ + { + "bbox": [ + 105, + 522, + 309, + 534 + ], + "score": 1.0, + "content": "in detail in Sect. 4. Given any sequence encoder", + "type": "text" + }, + { + "bbox": [ + 310, + 522, + 324, + 532 + ], + "score": 0.83, + "content": "\\mathcal { S E }", + "type": "inline_equation" + }, + { + "bbox": [ + 325, + 522, + 369, + 534 + ], + "score": 1.0, + "content": "that maps", + "type": "text" + }, + { + "bbox": [ + 369, + 522, + 377, + 532 + ], + "score": 0.79, + "content": "S", + "type": "inline_equation" + }, + { + "bbox": [ + 378, + 522, + 505, + 534 + ], + "score": 1.0, + "content": "to per-element representations", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 107, + 533, + 507, + 546 + ], + "spans": [ + { + "bbox": [ + 107, + 533, + 151, + 545 + ], + "score": 0.92, + "content": "\\left[ \\mathbf { e } _ { 1 } \\ldots \\mathbf { e } _ { N } \\right]", + "type": "inline_equation" + }, + { + "bbox": [ + 151, + 533, + 507, + 546 + ], + "score": 1.0, + "content": "and a sequence representation e (e.g. a bidirectional RNN), we can construct the se-", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 104, + 543, + 507, + 557 + ], + "spans": [ + { + "bbox": [ + 104, + 543, + 162, + 557 + ], + "score": 1.0, + "content": "quence GNN", + "type": "text" + }, + { + "bbox": [ + 162, + 544, + 197, + 555 + ], + "score": 0.89, + "content": "\\mathcal { S } \\mathcal { E } _ { G N N }", + "type": "inline_equation" + }, + { + "bbox": [ + 197, + 543, + 289, + 557 + ], + "score": 1.0, + "content": "by simply computing", + "type": "text" + }, + { + "bbox": [ + 289, + 543, + 488, + 556 + ], + "score": 0.77, + "content": "\\mathbf { \\bar { e } } _ { 1 } ^ { \\prime } \\cdot \\cdot \\cdot \\mathbf { e } _ { N } ^ { \\prime } ] = \\mathbf { G N N } ( ( S , [ R _ { 1 } \\dots R _ { K } ] , [ \\mathbf { e } _ { 1 } \\dots \\mathbf { e } _ { N } ] ) )", + "type": "inline_equation" + }, + { + "bbox": [ + 488, + 543, + 507, + 557 + ], + "score": 1.0, + "content": ". To", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 555, + 507, + 567 + ], + "spans": [ + { + "bbox": [ + 105, + 555, + 507, + 567 + ], + "score": 1.0, + "content": "obtain a graph-level representation, we use the weighted averaging mechanism from Gilmer et al.", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 104, + 565, + 508, + 582 + ], + "spans": [ + { + "bbox": [ + 104, + 565, + 248, + 582 + ], + "score": 1.0, + "content": "(2017). 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We found that best", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 601, + 470, + 615 + ], + "spans": [ + { + "bbox": [ + 105, + 601, + 286, + 615 + ], + "score": 1.0, + "content": "results were achieved by computing the final", + "type": "text" + }, + { + "bbox": [ + 287, + 602, + 296, + 612 + ], + "score": 0.85, + "content": "\\mathbf { e ^ { \\prime } }", + "type": "inline_equation" + }, + { + "bbox": [ + 297, + 601, + 308, + 615 + ], + "score": 1.0, + "content": "as", + "type": "text" + }, + { + "bbox": [ + 308, + 602, + 348, + 614 + ], + "score": 0.9, + "content": "W \\cdot ( \\mathbf { e } \\hat { \\mathbf { e } } )", + "type": "inline_equation" + }, + { + "bbox": [ + 348, + 601, + 454, + 615 + ], + "score": 1.0, + "content": "for some learnable matrix", + "type": "text" + }, + { + "bbox": [ + 455, + 603, + 466, + 612 + ], + "score": 0.73, + "content": "W", + "type": "inline_equation" + }, + { + "bbox": [ + 467, + 601, + 470, + 615 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 41 + } + ], + "index": 35.5, + "bbox_fs": [ + 104, + 478, + 508, + 615 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 619, + 505, + 663 + ], + "lines": [ + { + "bbox": [ + 105, + 619, + 505, + 632 + ], + "spans": [ + { + "bbox": [ + 105, + 619, + 505, + 632 + ], + "score": 1.0, + "content": "This method can easily be extended to support additional nodes not present in the original sequence", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 107, + 629, + 506, + 643 + ], + "spans": [ + { + "bbox": [ + 107, + 631, + 114, + 640 + ], + "score": 0.79, + "content": "S", + "type": "inline_equation" + }, + { + "bbox": [ + 115, + 629, + 169, + 643 + ], + "score": 1.0, + "content": "after running", + "type": "text" + }, + { + "bbox": [ + 169, + 630, + 183, + 641 + ], + "score": 0.79, + "content": "\\mathcal { S E }", + "type": "inline_equation" + }, + { + "bbox": [ + 184, + 629, + 506, + 643 + ], + "score": 1.0, + "content": "(e.g., to accommodate meta-nodes representing sentences, or non-terminal nodes", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 106, + 642, + 505, + 653 + ], + "spans": [ + { + "bbox": [ + 106, + 642, + 505, + 653 + ], + "score": 1.0, + "content": "from a syntax tree). The initial node representation for these additional nodes can come from other", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 652, + 312, + 664 + ], + "spans": [ + { + "bbox": [ + 105, + 652, + 312, + 664 + ], + "score": 1.0, + "content": "sources, such as a simple embedding of their label.", + "type": "text" + } + ], + "index": 45 + } + ], + "index": 43.5, + "bbox_fs": [ + 105, + 619, + 506, + 664 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 676, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 676, + 507, + 690 + ], + "spans": [ + { + "bbox": [ + 106, + 676, + 507, + 690 + ], + "score": 1.0, + "content": "Implementation Details. Processing large graphs of different shapes efficiently requires to over-", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 687, + 505, + 700 + ], + "spans": [ + { + "bbox": [ + 105, + 687, + 505, + 700 + ], + "score": 1.0, + "content": "come some engineering challenges. For example, the CNN/DM corpus has (on average) about 900", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 105, + 699, + 505, + 712 + ], + "spans": [ + { + "bbox": [ + 105, + 699, + 505, + 712 + ], + "score": 1.0, + "content": "nodes per graph. To allow efficient computation, we use the trick of Allamanis et al. (2018) where", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 105, + 708, + 506, + 723 + ], + "spans": [ + { + "bbox": [ + 105, + 708, + 506, + 723 + ], + "score": 1.0, + "content": "all graphs in a minibatch are “flattened” into a single graph with multiple disconnected components.", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 105, + 720, + 505, + 734 + ], + "spans": [ + { + "bbox": [ + 105, + 720, + 505, + 734 + ], + "score": 1.0, + "content": "The varying graph sizes also represent a problem for the attention and copying mechanisms in the", + "type": "text" + } + ], + "index": 50 + }, + { + "bbox": [ + 105, + 82, + 505, + 95 + ], + "spans": [ + { + "bbox": [ + 105, + 82, + 505, + 95 + ], + "score": 1.0, + "content": "decoder, as they require to compute a softmax over a variable-sized list of memories. To handle", + "type": "text", + "cross_page": true + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 93, + 505, + 105 + ], + "spans": [ + { + "bbox": [ + 106, + 93, + 505, + 105 + ], + "score": 1.0, + "content": "this efficiently without padding, we associate each node in the (flattened) “batch” graph with the", + "type": "text", + "cross_page": true + } + ], + "index": 1 + }, + { + "bbox": [ + 106, + 105, + 505, + 117 + ], + "spans": [ + { + "bbox": [ + 106, + 105, + 505, + 117 + ], + "score": 1.0, + "content": "index of the sample in the minibatch from which the node originated. Then, using TensorFlow’s", + "type": "text", + "cross_page": true + } + ], + "index": 2 + }, + { + "bbox": [ + 106, + 115, + 506, + 128 + ], + "spans": [ + { + "bbox": [ + 106, + 115, + 203, + 128 + ], + "score": 1.0, + "content": "unsorted segment", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 204, + 117, + 210, + 126 + ], + "score": 0.46, + "content": "\\mathbf { \\nabla } _ { \\cdot } \\star", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 211, + 115, + 506, + 128 + ], + "score": 1.0, + "content": "operations, we can perform an efficient and numerically stable softmax", + "type": "text", + "cross_page": true + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 126, + 392, + 140 + ], + "spans": [ + { + "bbox": [ + 105, + 126, + 392, + 140 + ], + "score": 1.0, + "content": "over the variable number of representations of the nodes of each graph.", + "type": "text", + "cross_page": true + } + ], + "index": 4 + } + ], + "index": 48, + "bbox_fs": [ + 105, + 676, + 507, + 734 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 82, + 505, + 138 + ], + "lines": [ + { + "bbox": [ + 105, + 82, + 505, + 95 + ], + "spans": [ + { + "bbox": [ + 105, + 82, + 505, + 95 + ], + "score": 1.0, + "content": "decoder, as they require to compute a softmax over a variable-sized list of memories. To handle", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 93, + 505, + 105 + ], + "spans": [ + { + "bbox": [ + 106, + 93, + 505, + 105 + ], + "score": 1.0, + "content": "this efficiently without padding, we associate each node in the (flattened) “batch” graph with the", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 106, + 105, + 505, + 117 + ], + "spans": [ + { + "bbox": [ + 106, + 105, + 505, + 117 + ], + "score": 1.0, + "content": "index of the sample in the minibatch from which the node originated. Then, using TensorFlow’s", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 106, + 115, + 506, + 128 + ], + "spans": [ + { + "bbox": [ + 106, + 115, + 203, + 128 + ], + "score": 1.0, + "content": "unsorted segment", + "type": "text" + }, + { + "bbox": [ + 204, + 117, + 210, + 126 + ], + "score": 0.46, + "content": "\\mathbf { \\nabla } _ { \\cdot } \\star", + "type": "inline_equation" + }, + { + "bbox": [ + 211, + 115, + 506, + 128 + ], + "score": 1.0, + "content": "operations, we can perform an efficient and numerically stable softmax", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 126, + 392, + 140 + ], + "spans": [ + { + "bbox": [ + 105, + 126, + 392, + 140 + ], + "score": 1.0, + "content": "over the variable number of representations of the nodes of each graph.", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 2 + }, + { + "type": "title", + "bbox": [ + 108, + 155, + 193, + 168 + ], + "lines": [ + { + "bbox": [ + 105, + 154, + 195, + 170 + ], + "spans": [ + { + "bbox": [ + 105, + 154, + 195, + 170 + ], + "score": 1.0, + "content": "4 EVALUATION", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 5 + }, + { + "type": "title", + "bbox": [ + 108, + 180, + 254, + 192 + ], + "lines": [ + { + "bbox": [ + 106, + 180, + 255, + 193 + ], + "spans": [ + { + "bbox": [ + 106, + 180, + 255, + 193 + ], + "score": 1.0, + "content": "4.1 QUANTITATIVE EVALUATION", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 6 + }, + { + "type": "text", + "bbox": [ + 107, + 201, + 505, + 245 + ], + "lines": [ + { + "bbox": [ + 105, + 201, + 505, + 214 + ], + "spans": [ + { + "bbox": [ + 105, + 201, + 505, + 214 + ], + "score": 1.0, + "content": "We evaluate Sequence GNNs on our three tasks by comparing them to models that use only sequence", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 212, + 505, + 224 + ], + "spans": [ + { + "bbox": [ + 105, + 212, + 505, + 224 + ], + "score": 1.0, + "content": "or graph information, as well as by comparing them to task-specific baselines. We discuss the three", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 224, + 505, + 236 + ], + "spans": [ + { + "bbox": [ + 106, + 224, + 505, + 236 + ], + "score": 1.0, + "content": "tasks, their respective baselines and how we present the data to the models (including the relationships", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 235, + 387, + 246 + ], + "spans": [ + { + "bbox": [ + 105, + 235, + 387, + 246 + ], + "score": 1.0, + "content": "considered in the graph component) next before analyzing the results.", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 8.5 + }, + { + "type": "title", + "bbox": [ + 108, + 259, + 265, + 270 + ], + "lines": [ + { + "bbox": [ + 106, + 258, + 267, + 271 + ], + "spans": [ + { + "bbox": [ + 106, + 258, + 267, + 271 + ], + "score": 1.0, + "content": "4.1.1 SETUP FOR METHODNAMING", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 11 + }, + { + "type": "text", + "bbox": [ + 107, + 278, + 506, + 399 + ], + "lines": [ + { + "bbox": [ + 105, + 278, + 506, + 291 + ], + "spans": [ + { + "bbox": [ + 105, + 278, + 506, + 291 + ], + "score": 1.0, + "content": "Datasets, Metrics, and Models. We consider two datasets for the METHODNAMING task. First,", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 289, + 505, + 302 + ], + "spans": [ + { + "bbox": [ + 105, + 289, + 505, + 302 + ], + "score": 1.0, + "content": "we consider the “Java (small)” dataset of Alon et al. (2018a), re-using the train-validation-test splits", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 301, + 505, + 312 + ], + "spans": [ + { + "bbox": [ + 106, + 301, + 505, + 312 + ], + "score": 1.0, + "content": "they have picked. We additionally generated a new dataset from 23 open-source C# projects mined", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 310, + 506, + 324 + ], + "spans": [ + { + "bbox": [ + 105, + 310, + 506, + 324 + ], + "score": 1.0, + "content": "from GitHub (see below for the reasons for this second dataset), removing any duplicates. More", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 321, + 507, + 335 + ], + "spans": [ + { + "bbox": [ + 105, + 321, + 507, + 335 + ], + "score": 1.0, + "content": "information about these datasets can be found in Appendix C. We follow earlier work on METHOD-", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 333, + 506, + 345 + ], + "spans": [ + { + "bbox": [ + 105, + 333, + 506, + 345 + ], + "score": 1.0, + "content": "NAMING (Allamanis et al., 2016; Alon et al., 2018a) and measure performance using the F1 score", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 344, + 506, + 356 + ], + "spans": [ + { + "bbox": [ + 106, + 344, + 506, + 356 + ], + "score": 1.0, + "content": "over the generated subtokens. However, since the task can be viewed as a form of (extreme) sum-", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 355, + 506, + 367 + ], + "spans": [ + { + "bbox": [ + 105, + 355, + 506, + 367 + ], + "score": 1.0, + "content": "marization, we also report ROUGE-2 and ROUGE-L scores (Lin, 2004), which we believe to be", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 365, + 506, + 378 + ], + "spans": [ + { + "bbox": [ + 105, + 365, + 506, + 378 + ], + "score": 1.0, + "content": "additional useful indicators for the quality of results. ROUGE-1 is omitted since it is equivalent to", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 104, + 375, + 506, + 391 + ], + "spans": [ + { + "bbox": [ + 104, + 375, + 506, + 391 + ], + "score": 1.0, + "content": "F1 score. We note that there is no widely accepted metric for this task and further work identifying", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 388, + 266, + 401 + ], + "spans": [ + { + "bbox": [ + 106, + 388, + 266, + 401 + ], + "score": 1.0, + "content": "the most appropriate metric is required.", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 17 + }, + { + "type": "text", + "bbox": [ + 107, + 405, + 505, + 504 + ], + "lines": [ + { + "bbox": [ + 106, + 405, + 505, + 417 + ], + "spans": [ + { + "bbox": [ + 106, + 405, + 505, + 417 + ], + "score": 1.0, + "content": "We compare to the current state of the art (Alon et al., 2018a), as well as a sequence-to-sequence", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 415, + 506, + 429 + ], + "spans": [ + { + "bbox": [ + 105, + 415, + 506, + 429 + ], + "score": 1.0, + "content": "implementation from the OpenNMT project (Klein et al.). Concretely, we combine two encoders (a", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 426, + 505, + 439 + ], + "spans": [ + { + "bbox": [ + 105, + 426, + 505, + 439 + ], + "score": 1.0, + "content": "bidirectional LSTM encoder with 1 layer and 256 hidden units, and its sequence GNN extension with", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 437, + 505, + 450 + ], + "spans": [ + { + "bbox": [ + 105, + 437, + 505, + 450 + ], + "score": 1.0, + "content": "128 hidden units unrolled over 8 timesteps) with two decoders (an LSTM decoder with 1 layer and 256", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 448, + 505, + 462 + ], + "spans": [ + { + "bbox": [ + 105, + 448, + 505, + 462 + ], + "score": 1.0, + "content": "hidden units with attention over the input sequence, and an extension using a pointer network-style", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 459, + 506, + 472 + ], + "spans": [ + { + "bbox": [ + 105, + 459, + 506, + 472 + ], + "score": 1.0, + "content": "copying mechanism (Vinyals et al., 2015a)). Additionally, we consider self-attention as an alternative", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 470, + 506, + 484 + ], + "spans": [ + { + "bbox": [ + 105, + 470, + 506, + 484 + ], + "score": 1.0, + "content": "to RNN-based sequence encoding architectures. For this, we use the Transformer (Vaswani et al.,", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 482, + 506, + 493 + ], + "spans": [ + { + "bbox": [ + 106, + 482, + 506, + 493 + ], + "score": 1.0, + "content": "2017) implementation in OpenNMT (i.e., using self-attention both for the decoder and the encoder)", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 104, + 492, + 479, + 506 + ], + "spans": [ + { + "bbox": [ + 104, + 492, + 479, + 506 + ], + "score": 1.0, + "content": "as a baseline and compare it to a version whose encoder is extended with a GNN component.", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 27 + }, + { + "type": "text", + "bbox": [ + 107, + 516, + 505, + 594 + ], + "lines": [ + { + "bbox": [ + 106, + 518, + 505, + 529 + ], + "spans": [ + { + "bbox": [ + 106, + 518, + 505, + 529 + ], + "score": 1.0, + "content": "Data Representation Following the work of Allamanis et al. (2016); Alon et al. (2018a), we break", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 527, + 505, + 541 + ], + "spans": [ + { + "bbox": [ + 105, + 527, + 505, + 541 + ], + "score": 1.0, + "content": "up all identifier tokens (i.e. variables, methods, classes, etc.) in the source code into subtokens by", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 540, + 505, + 551 + ], + "spans": [ + { + "bbox": [ + 105, + 540, + 505, + 551 + ], + "score": 1.0, + "content": "splitting them according to camelCase and pascal case heuristics. This allows the models", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 548, + 506, + 564 + ], + "spans": [ + { + "bbox": [ + 105, + 548, + 506, + 564 + ], + "score": 1.0, + "content": "to extract information from the information-rich subtoken structure, and ensures that a copying", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 559, + 506, + 574 + ], + "spans": [ + { + "bbox": [ + 105, + 559, + 506, + 574 + ], + "score": 1.0, + "content": "mechanism in the decoder can directly copy relevant subtokens, something that we found to be very", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 572, + 506, + 584 + ], + "spans": [ + { + "bbox": [ + 105, + 572, + 506, + 584 + ], + "score": 1.0, + "content": "effective for this task. All models are provided with all (sub)tokens belonging to the source code of a", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 583, + 498, + 595 + ], + "spans": [ + { + "bbox": [ + 105, + 583, + 498, + 595 + ], + "score": 1.0, + "content": "method, including its declaration, with the actual method name replaced by a placeholder symbol.", + "type": "text" + } + ], + "index": 38 + } + ], + "index": 35 + }, + { + "type": "text", + "bbox": [ + 107, + 600, + 505, + 666 + ], + "lines": [ + { + "bbox": [ + 105, + 599, + 505, + 612 + ], + "spans": [ + { + "bbox": [ + 105, + 599, + 505, + 612 + ], + "score": 1.0, + "content": "To construct a graph from the (sub)tokens, we implement a simplified form of the work of Allamanis", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 106, + 611, + 505, + 623 + ], + "spans": [ + { + "bbox": [ + 106, + 611, + 505, + 623 + ], + "score": 1.0, + "content": "et al. (2018). First, we introduce additional nodes for each (full) identifier token, and connect the", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 622, + 505, + 635 + ], + "spans": [ + { + "bbox": [ + 105, + 622, + 505, + 635 + ], + "score": 1.0, + "content": "constituent subtokens appearing in the input sequence using a INTOKEN edge; we additionally", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 632, + 505, + 645 + ], + "spans": [ + { + "bbox": [ + 105, + 632, + 505, + 645 + ], + "score": 1.0, + "content": "connect these nodes using a NEXTTOKEN edge. We also add nodes for the parse tree and use edges", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 644, + 505, + 656 + ], + "spans": [ + { + "bbox": [ + 105, + 644, + 505, + 656 + ], + "score": 1.0, + "content": "to indicate that one node is a CHILD of another. Finally, we add LASTLEXICALUSE edges to connect", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 106, + 655, + 365, + 666 + ], + "spans": [ + { + "bbox": [ + 106, + 655, + 365, + 666 + ], + "score": 1.0, + "content": "identifiers to their most (lexically) recent use in the source code.", + "type": "text" + } + ], + "index": 44 + } + ], + "index": 41.5 + }, + { + "type": "title", + "bbox": [ + 108, + 678, + 248, + 690 + ], + "lines": [ + { + "bbox": [ + 105, + 677, + 250, + 691 + ], + "spans": [ + { + "bbox": [ + 105, + 677, + 250, + 691 + ], + "score": 1.0, + "content": "4.1.2 SETUP FOR METHODDOC", + "type": "text" + } + ], + "index": 45 + } + ], + "index": 45 + }, + { + "type": "text", + "bbox": [ + 108, + 699, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 699, + 505, + 711 + ], + "spans": [ + { + "bbox": [ + 106, + 699, + 505, + 711 + ], + "score": 1.0, + "content": "Datasets, Metrics, and Models. We tried to evaluate on the Python dataset of Barone & Sennrich", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 709, + 507, + 723 + ], + "spans": [ + { + "bbox": [ + 105, + 709, + 507, + 723 + ], + "score": 1.0, + "content": "(2017) that contains pairs of method declarations and their documentation (“docstring”). However,", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 105, + 720, + 506, + 733 + ], + "spans": [ + { + "bbox": [ + 105, + 720, + 506, + 733 + ], + "score": 1.0, + "content": "following the work of Lopes et al. (2017), we found extensive duplication between different folds", + "type": "text" + } + ], + "index": 48 + } + ], + "index": 47 + } + ], + "page_idx": 3, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 108, + 27, + 293, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 26, + 294, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 26, + 294, + 38 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2019", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 302, + 752, + 308, + 759 + ], + "lines": [] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 82, + 505, + 138 + ], + "lines": [], + "index": 2, + "bbox_fs": [ + 105, + 82, + 506, + 140 + ], + "lines_deleted": true + }, + { + "type": "title", + "bbox": [ + 108, + 155, + 193, + 168 + ], + "lines": [ + { + "bbox": [ + 105, + 154, + 195, + 170 + ], + "spans": [ + { + "bbox": [ + 105, + 154, + 195, + 170 + ], + "score": 1.0, + "content": "4 EVALUATION", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 5 + }, + { + "type": "title", + "bbox": [ + 108, + 180, + 254, + 192 + ], + "lines": [ + { + "bbox": [ + 106, + 180, + 255, + 193 + ], + "spans": [ + { + "bbox": [ + 106, + 180, + 255, + 193 + ], + "score": 1.0, + "content": "4.1 QUANTITATIVE EVALUATION", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 6 + }, + { + "type": "text", + "bbox": [ + 107, + 201, + 505, + 245 + ], + "lines": [ + { + "bbox": [ + 105, + 201, + 505, + 214 + ], + "spans": [ + { + "bbox": [ + 105, + 201, + 505, + 214 + ], + "score": 1.0, + "content": "We evaluate Sequence GNNs on our three tasks by comparing them to models that use only sequence", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 212, + 505, + 224 + ], + "spans": [ + { + "bbox": [ + 105, + 212, + 505, + 224 + ], + "score": 1.0, + "content": "or graph information, as well as by comparing them to task-specific baselines. We discuss the three", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 224, + 505, + 236 + ], + "spans": [ + { + "bbox": [ + 106, + 224, + 505, + 236 + ], + "score": 1.0, + "content": "tasks, their respective baselines and how we present the data to the models (including the relationships", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 235, + 387, + 246 + ], + "spans": [ + { + "bbox": [ + 105, + 235, + 387, + 246 + ], + "score": 1.0, + "content": "considered in the graph component) next before analyzing the results.", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 8.5, + "bbox_fs": [ + 105, + 201, + 505, + 246 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 259, + 265, + 270 + ], + "lines": [ + { + "bbox": [ + 106, + 258, + 267, + 271 + ], + "spans": [ + { + "bbox": [ + 106, + 258, + 267, + 271 + ], + "score": 1.0, + "content": "4.1.1 SETUP FOR METHODNAMING", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 11 + }, + { + "type": "text", + "bbox": [ + 107, + 278, + 506, + 399 + ], + "lines": [ + { + "bbox": [ + 105, + 278, + 506, + 291 + ], + "spans": [ + { + "bbox": [ + 105, + 278, + 506, + 291 + ], + "score": 1.0, + "content": "Datasets, Metrics, and Models. We consider two datasets for the METHODNAMING task. First,", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 289, + 505, + 302 + ], + "spans": [ + { + "bbox": [ + 105, + 289, + 505, + 302 + ], + "score": 1.0, + "content": "we consider the “Java (small)” dataset of Alon et al. (2018a), re-using the train-validation-test splits", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 301, + 505, + 312 + ], + "spans": [ + { + "bbox": [ + 106, + 301, + 505, + 312 + ], + "score": 1.0, + "content": "they have picked. We additionally generated a new dataset from 23 open-source C# projects mined", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 310, + 506, + 324 + ], + "spans": [ + { + "bbox": [ + 105, + 310, + 506, + 324 + ], + "score": 1.0, + "content": "from GitHub (see below for the reasons for this second dataset), removing any duplicates. More", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 321, + 507, + 335 + ], + "spans": [ + { + "bbox": [ + 105, + 321, + 507, + 335 + ], + "score": 1.0, + "content": "information about these datasets can be found in Appendix C. We follow earlier work on METHOD-", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 333, + 506, + 345 + ], + "spans": [ + { + "bbox": [ + 105, + 333, + 506, + 345 + ], + "score": 1.0, + "content": "NAMING (Allamanis et al., 2016; Alon et al., 2018a) and measure performance using the F1 score", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 344, + 506, + 356 + ], + "spans": [ + { + "bbox": [ + 106, + 344, + 506, + 356 + ], + "score": 1.0, + "content": "over the generated subtokens. However, since the task can be viewed as a form of (extreme) sum-", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 355, + 506, + 367 + ], + "spans": [ + { + "bbox": [ + 105, + 355, + 506, + 367 + ], + "score": 1.0, + "content": "marization, we also report ROUGE-2 and ROUGE-L scores (Lin, 2004), which we believe to be", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 365, + 506, + 378 + ], + "spans": [ + { + "bbox": [ + 105, + 365, + 506, + 378 + ], + "score": 1.0, + "content": "additional useful indicators for the quality of results. ROUGE-1 is omitted since it is equivalent to", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 104, + 375, + 506, + 391 + ], + "spans": [ + { + "bbox": [ + 104, + 375, + 506, + 391 + ], + "score": 1.0, + "content": "F1 score. We note that there is no widely accepted metric for this task and further work identifying", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 388, + 266, + 401 + ], + "spans": [ + { + "bbox": [ + 106, + 388, + 266, + 401 + ], + "score": 1.0, + "content": "the most appropriate metric is required.", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 17, + "bbox_fs": [ + 104, + 278, + 507, + 401 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 405, + 505, + 504 + ], + "lines": [ + { + "bbox": [ + 106, + 405, + 505, + 417 + ], + "spans": [ + { + "bbox": [ + 106, + 405, + 505, + 417 + ], + "score": 1.0, + "content": "We compare to the current state of the art (Alon et al., 2018a), as well as a sequence-to-sequence", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 415, + 506, + 429 + ], + "spans": [ + { + "bbox": [ + 105, + 415, + 506, + 429 + ], + "score": 1.0, + "content": "implementation from the OpenNMT project (Klein et al.). Concretely, we combine two encoders (a", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 426, + 505, + 439 + ], + "spans": [ + { + "bbox": [ + 105, + 426, + 505, + 439 + ], + "score": 1.0, + "content": "bidirectional LSTM encoder with 1 layer and 256 hidden units, and its sequence GNN extension with", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 437, + 505, + 450 + ], + "spans": [ + { + "bbox": [ + 105, + 437, + 505, + 450 + ], + "score": 1.0, + "content": "128 hidden units unrolled over 8 timesteps) with two decoders (an LSTM decoder with 1 layer and 256", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 448, + 505, + 462 + ], + "spans": [ + { + "bbox": [ + 105, + 448, + 505, + 462 + ], + "score": 1.0, + "content": "hidden units with attention over the input sequence, and an extension using a pointer network-style", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 459, + 506, + 472 + ], + "spans": [ + { + "bbox": [ + 105, + 459, + 506, + 472 + ], + "score": 1.0, + "content": "copying mechanism (Vinyals et al., 2015a)). Additionally, we consider self-attention as an alternative", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 470, + 506, + 484 + ], + "spans": [ + { + "bbox": [ + 105, + 470, + 506, + 484 + ], + "score": 1.0, + "content": "to RNN-based sequence encoding architectures. For this, we use the Transformer (Vaswani et al.,", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 482, + 506, + 493 + ], + "spans": [ + { + "bbox": [ + 106, + 482, + 506, + 493 + ], + "score": 1.0, + "content": "2017) implementation in OpenNMT (i.e., using self-attention both for the decoder and the encoder)", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 104, + 492, + 479, + 506 + ], + "spans": [ + { + "bbox": [ + 104, + 492, + 479, + 506 + ], + "score": 1.0, + "content": "as a baseline and compare it to a version whose encoder is extended with a GNN component.", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 27, + "bbox_fs": [ + 104, + 405, + 506, + 506 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 516, + 505, + 594 + ], + "lines": [ + { + "bbox": [ + 106, + 518, + 505, + 529 + ], + "spans": [ + { + "bbox": [ + 106, + 518, + 505, + 529 + ], + "score": 1.0, + "content": "Data Representation Following the work of Allamanis et al. (2016); Alon et al. (2018a), we break", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 527, + 505, + 541 + ], + "spans": [ + { + "bbox": [ + 105, + 527, + 505, + 541 + ], + "score": 1.0, + "content": "up all identifier tokens (i.e. variables, methods, classes, etc.) in the source code into subtokens by", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 540, + 505, + 551 + ], + "spans": [ + { + "bbox": [ + 105, + 540, + 505, + 551 + ], + "score": 1.0, + "content": "splitting them according to camelCase and pascal case heuristics. This allows the models", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 548, + 506, + 564 + ], + "spans": [ + { + "bbox": [ + 105, + 548, + 506, + 564 + ], + "score": 1.0, + "content": "to extract information from the information-rich subtoken structure, and ensures that a copying", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 559, + 506, + 574 + ], + "spans": [ + { + "bbox": [ + 105, + 559, + 506, + 574 + ], + "score": 1.0, + "content": "mechanism in the decoder can directly copy relevant subtokens, something that we found to be very", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 572, + 506, + 584 + ], + "spans": [ + { + "bbox": [ + 105, + 572, + 506, + 584 + ], + "score": 1.0, + "content": "effective for this task. All models are provided with all (sub)tokens belonging to the source code of a", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 583, + 498, + 595 + ], + "spans": [ + { + "bbox": [ + 105, + 583, + 498, + 595 + ], + "score": 1.0, + "content": "method, including its declaration, with the actual method name replaced by a placeholder symbol.", + "type": "text" + } + ], + "index": 38 + } + ], + "index": 35, + "bbox_fs": [ + 105, + 518, + 506, + 595 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 600, + 505, + 666 + ], + "lines": [ + { + "bbox": [ + 105, + 599, + 505, + 612 + ], + "spans": [ + { + "bbox": [ + 105, + 599, + 505, + 612 + ], + "score": 1.0, + "content": "To construct a graph from the (sub)tokens, we implement a simplified form of the work of Allamanis", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 106, + 611, + 505, + 623 + ], + "spans": [ + { + "bbox": [ + 106, + 611, + 505, + 623 + ], + "score": 1.0, + "content": "et al. (2018). First, we introduce additional nodes for each (full) identifier token, and connect the", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 622, + 505, + 635 + ], + "spans": [ + { + "bbox": [ + 105, + 622, + 505, + 635 + ], + "score": 1.0, + "content": "constituent subtokens appearing in the input sequence using a INTOKEN edge; we additionally", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 632, + 505, + 645 + ], + "spans": [ + { + "bbox": [ + 105, + 632, + 505, + 645 + ], + "score": 1.0, + "content": "connect these nodes using a NEXTTOKEN edge. We also add nodes for the parse tree and use edges", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 644, + 505, + 656 + ], + "spans": [ + { + "bbox": [ + 105, + 644, + 505, + 656 + ], + "score": 1.0, + "content": "to indicate that one node is a CHILD of another. Finally, we add LASTLEXICALUSE edges to connect", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 106, + 655, + 365, + 666 + ], + "spans": [ + { + "bbox": [ + 106, + 655, + 365, + 666 + ], + "score": 1.0, + "content": "identifiers to their most (lexically) recent use in the source code.", + "type": "text" + } + ], + "index": 44 + } + ], + "index": 41.5, + "bbox_fs": [ + 105, + 599, + 505, + 666 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 678, + 248, + 690 + ], + "lines": [ + { + "bbox": [ + 105, + 677, + 250, + 691 + ], + "spans": [ + { + "bbox": [ + 105, + 677, + 250, + 691 + ], + "score": 1.0, + "content": "4.1.2 SETUP FOR METHODDOC", + "type": "text" + } + ], + "index": 45 + } + ], + "index": 45 + }, + { + "type": "text", + "bbox": [ + 108, + 699, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 699, + 505, + 711 + ], + "spans": [ + { + "bbox": [ + 106, + 699, + 505, + 711 + ], + "score": 1.0, + "content": "Datasets, Metrics, and Models. We tried to evaluate on the Python dataset of Barone & Sennrich", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 709, + 507, + 723 + ], + "spans": [ + { + "bbox": [ + 105, + 709, + 507, + 723 + ], + "score": 1.0, + "content": "(2017) that contains pairs of method declarations and their documentation (“docstring”). However,", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 105, + 720, + 506, + 733 + ], + "spans": [ + { + "bbox": [ + 105, + 720, + 506, + 733 + ], + "score": 1.0, + "content": "following the work of Lopes et al. (2017), we found extensive duplication between different folds", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 105, + 264, + 505, + 277 + ], + "spans": [ + { + "bbox": [ + 105, + 264, + 505, + 277 + ], + "score": 1.0, + "content": "of the dataset and were only able to reach comparable results by substantially overfitting to the", + "type": "text", + "cross_page": true + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 275, + 505, + 288 + ], + "spans": [ + { + "bbox": [ + 105, + 275, + 505, + 288 + ], + "score": 1.0, + "content": "training data that overlapped with the test set. We have documented details in subsection C.3 and in", + "type": "text", + "cross_page": true + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 286, + 505, + 299 + ], + "spans": [ + { + "bbox": [ + 105, + 286, + 505, + 299 + ], + "score": 1.0, + "content": "Allamanis (2018), and decided to instead evaluate on our new dataset of 23 open-source C# projects", + "type": "text", + "cross_page": true + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 298, + 506, + 310 + ], + "spans": [ + { + "bbox": [ + 105, + 298, + 506, + 310 + ], + "score": 1.0, + "content": "from above, again removing duplicates and methods without documentation. Following Barone &", + "type": "text", + "cross_page": true + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 308, + 506, + 321 + ], + "spans": [ + { + "bbox": [ + 105, + 308, + 506, + 321 + ], + "score": 1.0, + "content": "Sennrich (2017), we measure the BLEU score for all models. However, we also report F1, ROUGE-2", + "type": "text", + "cross_page": true + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 319, + 506, + 331 + ], + "spans": [ + { + "bbox": [ + 105, + 319, + 506, + 331 + ], + "score": 1.0, + "content": "and ROUGE-L scores, which should better reflect the summarization aspect of the task. We consider", + "type": "text", + "cross_page": true + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 330, + 505, + 343 + ], + "spans": [ + { + "bbox": [ + 105, + 330, + 505, + 343 + ], + "score": 1.0, + "content": "the same models as for the METHODNAMING task, using the same configuration, and use the same", + "type": "text", + "cross_page": true + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 341, + 187, + 354 + ], + "spans": [ + { + "bbox": [ + 105, + 341, + 187, + 354 + ], + "score": 1.0, + "content": "data representation.", + "type": "text", + "cross_page": true + } + ], + "index": 11 + } + ], + "index": 47, + "bbox_fs": [ + 105, + 699, + 507, + 733 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "image", + "bbox": [ + 106, + 79, + 506, + 211 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 106, + 79, + 506, + 211 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 106, + 79, + 506, + 211 + ], + "spans": [ + { + "bbox": [ + 106, + 79, + 506, + 211 + ], + "score": 0.968, + "type": "image", + "image_path": "0f24ece1ecd933d082df2284a4d3190ac4a48d4c10138fc5c01fc0d240b7a43f.jpg" + } + ] + } + ], + "index": 1, + "virtual_lines": [ + { + "bbox": [ + 106, + 79, + 506, + 123.0 + ], + "spans": [], + "index": 0 + }, + { + "bbox": [ + 106, + 123.0, + 506, + 167.0 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 106, + 167.0, + 506, + 211.0 + ], + "spans": [], + "index": 2 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 160, + 217, + 451, + 229 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 160, + 216, + 451, + 231 + ], + "spans": [ + { + "bbox": [ + 160, + 216, + 451, + 231 + ], + "score": 1.0, + "content": "Figure 2: (Partial) graph of an example input from the CNN/DM corpus.", + "type": "text" + } + ], + "index": 3 + } + ], + "index": 3 + } + ], + "index": 2.0 + }, + { + "type": "text", + "bbox": [ + 107, + 264, + 505, + 353 + ], + "lines": [ + { + "bbox": [ + 105, + 264, + 505, + 277 + ], + "spans": [ + { + "bbox": [ + 105, + 264, + 505, + 277 + ], + "score": 1.0, + "content": "of the dataset and were only able to reach comparable results by substantially overfitting to the", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 275, + 505, + 288 + ], + "spans": [ + { + "bbox": [ + 105, + 275, + 505, + 288 + ], + "score": 1.0, + "content": "training data that overlapped with the test set. We have documented details in subsection C.3 and in", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 286, + 505, + 299 + ], + "spans": [ + { + "bbox": [ + 105, + 286, + 505, + 299 + ], + "score": 1.0, + "content": "Allamanis (2018), and decided to instead evaluate on our new dataset of 23 open-source C# projects", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 298, + 506, + 310 + ], + "spans": [ + { + "bbox": [ + 105, + 298, + 506, + 310 + ], + "score": 1.0, + "content": "from above, again removing duplicates and methods without documentation. Following Barone &", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 308, + 506, + 321 + ], + "spans": [ + { + "bbox": [ + 105, + 308, + 506, + 321 + ], + "score": 1.0, + "content": "Sennrich (2017), we measure the BLEU score for all models. However, we also report F1, ROUGE-2", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 319, + 506, + 331 + ], + "spans": [ + { + "bbox": [ + 105, + 319, + 506, + 331 + ], + "score": 1.0, + "content": "and ROUGE-L scores, which should better reflect the summarization aspect of the task. 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We use the CNN/DM dataset (Hermann et al., 2015) using the", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 418, + 505, + 430 + ], + "spans": [ + { + "bbox": [ + 105, + 418, + 505, + 430 + ], + "score": 1.0, + "content": "exact data and split provided by See et al. (2017). The data is constructed from CNN and Daily Mail", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 429, + 506, + 442 + ], + "spans": [ + { + "bbox": [ + 105, + 429, + 506, + 442 + ], + "score": 1.0, + "content": "news articles along with a few sentences that summarize each article. To measure performance, we", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 439, + 506, + 452 + ], + "spans": [ + { + "bbox": [ + 106, + 439, + 506, + 452 + ], + "score": 1.0, + "content": "use the standard ROUGE metrics. 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METHODNAMINGF1ROUGE-2ROUGE-L
Java
Alon et al. (2018a)43.0一1
SELFATT →SELFATT24.98.327.4
SELFATT+GNN →SELFATT44.520.943.4
BILSTM →LSTM35.817.939.7
BILSTM+GNN →LSTM44.721.143.1
BILSTM →LSTM+POINTER42.522.445.6
GNN →LSTM+POINTER50.524.848.9
BILSTM+GNN→LSTM+POINTER51.425.050.0
C#
SELFATT →SELFATT41.325.243.2
SELFATT+GNN→→SELFATT62.131.061.1
BILSTM →LSTM48.832.851.8
BILSTM+GNN →LSTM62.631.061.3
BILSTM →LSTM+POINTER57.229.760.4
GNN→LSTM+POINTER63.031.561.3
BILSTM+GNN→LSTM+POINTER63.431.962.4
METHODDOCF1ROUGE-2ROUGE-LBLEU
C#
SELFATT →SELFATT40.027.841.113.9
SELFATT+GNN→SELFATT37.625.637.921.4
BILSTM →LSTM35.215.330.810.0
BILSTM+GNN →LSTM41.128.941.022.5
BILSTM →LSTM+POINTER35.220.836.714.7
GNN →LSTM+POINTER38.925.637.117.7
BILSTM+GNN →→LSTM+POINTER (average pooling)43.229.041.021.3
BILSTM+GNN→LSTM+POINTER45.428.341.122.2
NLSUMMARIZATIONROUGE-1ROUGE-2ROUGE-L
CNN/DM
BILSTM →LSTM33.611.427.9
BILSTM+GNN→LSTM33.013.328.3
See et al. (2017) (+ Pointer)36.415.733.4
BILSTM →LSTM+POINTER35.913.930.3
BILSTM+GNN →LSTM+POINTER38.116.133.2
See et al. (2017) (+ Pointer + Coverage)39.517.336.4
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The results indicate that this configuration performs less well than a pure sequenced model.", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 720, + 506, + 733 + ], + "spans": [ + { + "bbox": [ + 105, + 720, + 506, + 733 + ], + "score": 1.0, + "content": "We speculate that this is mainly due to the fact that 10 timesteps are insufficient to propagate infor-", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 16.5 + } + ], + "page_idx": 5, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 108, + 27, + 293, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 26, + 294, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 26, + 294, + 38 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2019", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 302, + 752, + 309, + 760 + ], + "lines": [ + { + "bbox": [ + 302, + 751, + 310, + 762 + ], + "spans": [ + { + "bbox": [ + 302, + 751, + 310, + 762 + ], + "score": 1.0, + "content": "6", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "table", + "bbox": [ + 105, + 95, + 511, + 489 + ], + "blocks": [ + { + "type": "table_caption", + "bbox": [ + 201, + 80, + 409, + 92 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 201, + 79, + 410, + 93 + ], + "spans": [ + { + "bbox": [ + 201, + 79, + 410, + 93 + ], + "score": 1.0, + "content": "Table 1: Evaluation results for all models and tasks.", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "table_body", + "bbox": [ + 105, + 95, + 511, + 489 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 105, + 95, + 511, + 489 + ], + "spans": [ + { + "bbox": [ + 105, + 95, + 511, + 489 + ], + "score": 0.98, + "html": "
METHODNAMINGF1ROUGE-2ROUGE-L
Java
Alon et al. (2018a)43.0一1
SELFATT →SELFATT24.98.327.4
SELFATT+GNN →SELFATT44.520.943.4
BILSTM →LSTM35.817.939.7
BILSTM+GNN →LSTM44.721.143.1
BILSTM →LSTM+POINTER42.522.445.6
GNN →LSTM+POINTER50.524.848.9
BILSTM+GNN→LSTM+POINTER51.425.050.0
C#
SELFATT →SELFATT41.325.243.2
SELFATT+GNN→→SELFATT62.131.061.1
BILSTM →LSTM48.832.851.8
BILSTM+GNN →LSTM62.631.061.3
BILSTM →LSTM+POINTER57.229.760.4
GNN→LSTM+POINTER63.031.561.3
BILSTM+GNN→LSTM+POINTER63.431.962.4
METHODDOCF1ROUGE-2ROUGE-LBLEU
C#
SELFATT →SELFATT40.027.841.113.9
SELFATT+GNN→SELFATT37.625.637.921.4
BILSTM →LSTM35.215.330.810.0
BILSTM+GNN →LSTM41.128.941.022.5
BILSTM →LSTM+POINTER35.220.836.714.7
GNN →LSTM+POINTER38.925.637.117.7
BILSTM+GNN →→LSTM+POINTER (average pooling)43.229.041.021.3
BILSTM+GNN→LSTM+POINTER45.428.341.122.2
NLSUMMARIZATIONROUGE-1ROUGE-2ROUGE-L
CNN/DM
BILSTM →LSTM33.611.427.9
BILSTM+GNN→LSTM33.013.328.3
See et al. (2017) (+ Pointer)36.415.733.4
BILSTM →LSTM+POINTER35.913.930.3
BILSTM+GNN →LSTM+POINTER38.116.133.2
See et al. (2017) (+ Pointer + Coverage)39.517.336.4
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NLSUMMARIZATION (CNN/DM)ROUGE-1ROUGE-2ROUGE-L
See et al.(2017) (base)31.311.828.8
See et al. (2017) (+ Pointer)36.415.733.4
See et al.(2017) (+ Pointer + Coverage)39.517.336.4
BILSTM →LSTM33.611.427.9
BILSTM →LSTM+POINTER35.913.930.3
BILSTM →LSTM+PoINTER (+ coref/entity annotations)36.214.230.5
BILSTM+GNN →LSTM33.013.328.3
BILSTM+GNN→LSTM+POINTER (only sentence nodes)36.015.229.6
BILSTM+GNN →LSTM+POINTER (sentence nodes + eq edges)36.115.430.3
BILSTM+GNN→LSTM+POINTER38.116.133.2
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NLSUMMARIZATION (CNN/DM)ROUGE-1ROUGE-2ROUGE-L
See et al.(2017) (base)31.311.828.8
See et al. (2017) (+ Pointer)36.415.733.4
See et al.(2017) (+ Pointer + Coverage)39.517.336.4
BILSTM →LSTM33.611.427.9
BILSTM →LSTM+POINTER35.913.930.3
BILSTM →LSTM+PoINTER (+ coref/entity annotations)36.214.230.5
BILSTM+GNN →LSTM33.013.328.3
BILSTM+GNN→LSTM+POINTER (only sentence nodes)36.015.229.6
BILSTM+GNN →LSTM+POINTER (sentence nodes + eq edges)36.115.430.3
BILSTM+GNN→LSTM+POINTER38.116.133.2
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While there", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 447, + 505, + 459 + ], + "spans": [ + { + "bbox": [ + 105, + 447, + 505, + 459 + ], + "score": 1.0, + "content": "is still a gap to the best configuration of See et al. (2017) (and an even larger one to more recent work", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 456, + 505, + 471 + ], + "spans": [ + { + "bbox": [ + 105, + 456, + 505, + 471 + ], + "score": 1.0, + "content": "in the area), we believe that this is entirely due to our simplistic decoder and training objective, and", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 468, + 345, + 480 + ], + "spans": [ + { + "bbox": [ + 105, + 468, + 345, + 480 + ], + "score": 1.0, + "content": "that our contribution can be combined with these advances.", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 22, + "bbox_fs": [ + 105, + 423, + 505, + 480 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 485, + 505, + 661 + ], + "lines": [ + { + "bbox": [ + 105, + 484, + 505, + 498 + ], + "spans": [ + { + "bbox": [ + 105, + 484, + 505, + 498 + ], + "score": 1.0, + "content": "In Table 2 we show some ablations for NLSUMMARIZATION. As we use the same hyperparameters", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 496, + 507, + 508 + ], + "spans": [ + { + "bbox": [ + 105, + 496, + 507, + 508 + ], + "score": 1.0, + "content": "across all datasets and tasks, we additionally perform an experiment with the model of See et al.", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 507, + 506, + 520 + ], + "spans": [ + { + "bbox": [ + 106, + 507, + 506, + 520 + ], + "score": 1.0, + "content": "(2017) (as implemented in OpenNMT) but using our settings. The results achieved by these baselines", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 518, + 506, + 531 + ], + "spans": [ + { + "bbox": [ + 106, + 518, + 506, + 531 + ], + "score": 1.0, + "content": "trend to be a bit worse than the results reported in the original paper, which we believe is due to a lack", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 529, + 505, + 541 + ], + "spans": [ + { + "bbox": [ + 105, + 529, + 505, + 541 + ], + "score": 1.0, + "content": "of hyperparameter optimization for this task. We then evaluated how much the additional linguistic", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 540, + 505, + 552 + ], + "spans": [ + { + "bbox": [ + 105, + 540, + 505, + 552 + ], + "score": 1.0, + "content": "structure provided by CoreNLP helps. First, we add the coreference and entity annotations to the", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 551, + 505, + 564 + ], + "spans": [ + { + "bbox": [ + 105, + 551, + 142, + 564 + ], + "score": 1.0, + "content": "baseline", + "type": "text" + }, + { + "bbox": [ + 142, + 551, + 279, + 562 + ], + "score": 0.33, + "content": "\\mathrm { B I L S T M } \\to \\mathrm { L S T M } + \\mathrm { P O I N T E R }", + "type": "inline_equation" + }, + { + "bbox": [ + 280, + 551, + 505, + 564 + ], + "score": 1.0, + "content": "model (by extending the embedding of tokens with an", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 562, + 506, + 576 + ], + "spans": [ + { + "bbox": [ + 105, + 562, + 327, + 576 + ], + "score": 1.0, + "content": "embedding of the entity information, and inserting fresh", + "type": "text" + }, + { + "bbox": [ + 327, + 562, + 365, + 574 + ], + "score": 0.75, + "content": "\\mathrm { ~ \\omega ~ } _ { \\mathrm { i R E F 1 } } \\mathrm { ~ \\omega ~ }", + "type": "inline_equation" + }, + { + "bbox": [ + 365, + 562, + 506, + 576 + ], + "score": 1.0, + "content": "”, . . . tokens at the sources/targets of", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 573, + 506, + 586 + ], + "spans": [ + { + "bbox": [ + 105, + 573, + 506, + 586 + ], + "score": 1.0, + "content": "co-references) and observe only minimal improvements. This suggests that our graph-based encoder", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 584, + 505, + 597 + ], + "spans": [ + { + "bbox": [ + 105, + 584, + 505, + 597 + ], + "score": 1.0, + "content": "is better-suited to exploit additional structured information compared to a biLSTM encoder. We then", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 106, + 595, + 506, + 608 + ], + "spans": [ + { + "bbox": [ + 106, + 595, + 506, + 608 + ], + "score": 1.0, + "content": "drop all linguistic structure information from our model, keeping only the sentence edges/nodes.", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 605, + 506, + 618 + ], + "spans": [ + { + "bbox": [ + 106, + 605, + 244, + 618 + ], + "score": 1.0, + "content": "This still improves on the baseline", + "type": "text" + }, + { + "bbox": [ + 244, + 606, + 356, + 617 + ], + "score": 0.55, + "content": "\\mathrm { B I L S T M } \\to \\mathrm { L S T M } + \\mathrm { P O I I }", + "type": "inline_equation" + }, + { + "bbox": [ + 357, + 605, + 506, + 618 + ], + "score": 1.0, + "content": "NTER model (in the ROUGE-2 score),", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 104, + 616, + 506, + 631 + ], + "spans": [ + { + "bbox": [ + 104, + 616, + 506, + 631 + ], + "score": 1.0, + "content": "suggesting that the GNN still yields improvements in the absence of linguistic structure. Finally, we", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 627, + 506, + 641 + ], + "spans": [ + { + "bbox": [ + 105, + 627, + 506, + 641 + ], + "score": 1.0, + "content": "add long-range dependency edges by connecting tokens with equivalent string representations of their", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 639, + 506, + 652 + ], + "spans": [ + { + "bbox": [ + 105, + 639, + 506, + 652 + ], + "score": 1.0, + "content": "stems and observe further minor improvements, indicating that even using only purely syntactical", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 650, + 369, + 663 + ], + "spans": [ + { + "bbox": [ + 105, + 650, + 369, + 663 + ], + "score": 1.0, + "content": "information, without a semantic parse, can already provide gains.", + "type": "text" + } + ], + "index": 40 + } + ], + "index": 32.5, + "bbox_fs": [ + 104, + 484, + 507, + 663 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 685, + 248, + 696 + ], + "lines": [ + { + "bbox": [ + 105, + 684, + 250, + 698 + ], + "spans": [ + { + "bbox": [ + 105, + 684, + 250, + 698 + ], + "score": 1.0, + "content": "4.2 QUALITATIVE EVALUATION", + "type": "text" + } + ], + "index": 41 + } + ], + "index": 41 + }, + { + "type": "text", + "bbox": [ + 108, + 709, + 504, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 709, + 506, + 722 + ], + "spans": [ + { + "bbox": [ + 106, + 709, + 506, + 722 + ], + "score": 1.0, + "content": "We look at a few sample suggestions in our dataset across the tasks. Here we highlight some obser-", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 720, + 432, + 732 + ], + "spans": [ + { + "bbox": [ + 105, + 720, + 432, + 732 + ], + "score": 1.0, + "content": "vations we make that point out interesting aspects and failure cases of our model.", + "type": "text" + } + ], + "index": 43 + } + ], + "index": 42.5, + "bbox_fs": [ + 105, + 709, + 506, + 732 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "table", + "bbox": [ + 106, + 81, + 515, + 322 + ], + "blocks": [ + { + "type": "table_body", + "bbox": [ + 106, + 81, + 515, + 322 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 106, + 81, + 515, + 322 + ], + "spans": [ + { + "bbox": [ + 106, + 81, + 515, + 322 + ], + "score": 0.775, + "html": "
Input: Arsenal,Newcastle United and Southampton have checked on Caen midfelder N'golo Kante .Paris-born Kante is a defensive minded player who has impressed for Caen this season and they are wiling to sellfor around f 5milion.Marseille have been in constant contact with Caen over signing the 24-year-old who has similarities with Lassna Diarra and Claude Makelele in terms of stature and style .N'Golo Kante is attractinginterest froma host of PremierLeague clubs including Arsenal.Caen would be wiling to sellKante for aroundf5million.
Reference:n'golo kante is wanted by arsenal,newcastle and southampton.marseille are also keen on thef5mrated midfielder.kante has been compared to lassana diarra and claude makelele.click here for the latestpremier league news .
See et al.(2O17) (+ Pointer): arsenal,newcastle united and southampton have checked on caen midfieldern'golo kante .paris-born kante is attracting interest from a host of premier league clubs including arsenal .paris-born kante is attracting interest from a host of premier league clubs including arsenal
See et al. (2O17)(+ Pointer + Coverage): arsenal, newcastle united and southampton have checked on caenmidfielder n'golo kante.paris-born kante is a defensive minded player who has impressed for caen this season.marseille have been in constant contact with caen over signing the 24-year-old .
BILSTM+GNN →LSTM: marseille have been linked with caen midfielder %UNK% %UNK% . marseillehave been interested from a host of premier league clubs including arsenal .caen have been interested from ahost of premier league clubs including arsenal .
BILSTM+GNN →LSTM+PoINTER n'golo kante is attracting interest from a host of premier league clubs .marseilehave been in constant contact with caen over signing the24-year-old.the 24-year-old has similaritieswith lassana diarra and claude makelele in terms of stature .
", + "type": "table", + "image_path": "879ad0593963ca3e074145df80c378944913e18c84b695b9649622deeb04970e.jpg" + } + ] + } + ], + "index": 1, + "virtual_lines": [ + { + "bbox": [ + 106, + 81, + 515, + 161.33333333333331 + ], + "spans": [], + "index": 0 + }, + { + "bbox": [ + 106, + 161.33333333333331, + 515, + 241.66666666666663 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 106, + 241.66666666666663, + 515, + 321.99999999999994 + ], + "spans": [], + "index": 2 + } + ] + }, + { + "type": "table_caption", + "bbox": [ + 155, + 330, + 454, + 343 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 154, + 330, + 456, + 344 + ], + "spans": [ + { + "bbox": [ + 154, + 330, + 456, + 344 + ], + "score": 1.0, + "content": "Figure 4: Sample natural language translations from the CNN-DM dataset.", + "type": "text" + } + ], + "index": 3 + } + ], + "index": 3 + } + ], + "index": 2.0 + }, + { + "type": "text", + "bbox": [ + 107, + 368, + 505, + 445 + ], + "lines": [ + { + "bbox": [ + 106, + 369, + 505, + 381 + ], + "spans": [ + { + "bbox": [ + 106, + 369, + 505, + 381 + ], + "score": 1.0, + "content": "METHODDOC Figures 1 and 3 illustrate typical results of baselines and our model on the", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 378, + 505, + 392 + ], + "spans": [ + { + "bbox": [ + 105, + 378, + 505, + 392 + ], + "score": 1.0, + "content": "METHODDOC task (see Appendix A for more examples). The hardness of the task stems from the", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 390, + 506, + 402 + ], + "spans": [ + { + "bbox": [ + 106, + 390, + 506, + 402 + ], + "score": 1.0, + "content": "large number of distractors and the need to identify the most relevant parts of the input. In Figure 1,", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 401, + 506, + 414 + ], + "spans": [ + { + "bbox": [ + 105, + 401, + 506, + 414 + ], + "score": 1.0, + "content": "the token “parameter” and variations appears many times, and identifying the correct relationship is", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 412, + 505, + 424 + ], + "spans": [ + { + "bbox": [ + 106, + 412, + 505, + 424 + ], + "score": 1.0, + "content": "non-trivial, but is evidently eased by graph edges explicitly denoting these relationships. Similarly, in", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 422, + 506, + 437 + ], + "spans": [ + { + "bbox": [ + 105, + 422, + 506, + 437 + ], + "score": 1.0, + "content": "Figure 3, many variables are passed around, and the semantics of the method require understanding", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 106, + 435, + 259, + 445 + ], + "spans": [ + { + "bbox": [ + 106, + 435, + 259, + 445 + ], + "score": 1.0, + "content": "how information flows between them.", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 7 + }, + { + "type": "text", + "bbox": [ + 107, + 462, + 506, + 573 + ], + "lines": [ + { + "bbox": [ + 105, + 462, + 505, + 475 + ], + "spans": [ + { + "bbox": [ + 105, + 462, + 505, + 475 + ], + "score": 1.0, + "content": "NLSUMMARIZATION Figure 4 shows one sample summarization. More samples for this task can", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 474, + 505, + 486 + ], + "spans": [ + { + "bbox": [ + 106, + 474, + 505, + 486 + ], + "score": 1.0, + "content": "be found in Appendix B. First, we notice that the model produces natural-looking summaries with", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 485, + 507, + 497 + ], + "spans": [ + { + "bbox": [ + 105, + 485, + 507, + 497 + ], + "score": 1.0, + "content": "no noticeable negative impact on the fluency of the language over existing methods. Furthermore,", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 495, + 507, + 509 + ], + "spans": [ + { + "bbox": [ + 105, + 495, + 507, + 509 + ], + "score": 1.0, + "content": "the GNN-based model seems to capture the central named entity in the article and creates a sum-", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 507, + 506, + 519 + ], + "spans": [ + { + "bbox": [ + 105, + 507, + 506, + 519 + ], + "score": 1.0, + "content": "mary centered around that entity. We hypothesize that the GNN component that links long-distance", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 518, + 506, + 529 + ], + "spans": [ + { + "bbox": [ + 106, + 518, + 506, + 529 + ], + "score": 1.0, + "content": "relationships helps capture and maintain a better “global” view of the article, allowing for better iden-", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 528, + 506, + 541 + ], + "spans": [ + { + "bbox": [ + 105, + 528, + 506, + 541 + ], + "score": 1.0, + "content": "tification of central entities. Our model still suffers from repetition of information (see Appendix B),", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 538, + 507, + 552 + ], + "spans": [ + { + "bbox": [ + 105, + 538, + 507, + 552 + ], + "score": 1.0, + "content": "and so we believe that our model would also profit from advances such as taking coverage into ac-", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 550, + 505, + 563 + ], + "spans": [ + { + "bbox": [ + 105, + 550, + 505, + 563 + ], + "score": 1.0, + "content": "count (See et al., 2017) or optimizing for ROUGE-L scores directly via reinforcement learning (Chen", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 562, + 262, + 573 + ], + "spans": [ + { + "bbox": [ + 106, + 562, + 262, + 573 + ], + "score": 1.0, + "content": "& Bansal, 2018; Narayan et al., 2018).", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 15.5 + }, + { + "type": "title", + "bbox": [ + 108, + 594, + 211, + 606 + ], + "lines": [ + { + "bbox": [ + 105, + 592, + 213, + 608 + ], + "spans": [ + { + "bbox": [ + 105, + 592, + 213, + 608 + ], + "score": 1.0, + "content": "5 RELATED WORK", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 21 + }, + { + "type": "text", + "bbox": [ + 107, + 622, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 621, + 506, + 634 + ], + "spans": [ + { + "bbox": [ + 106, + 621, + 506, + 634 + ], + "score": 1.0, + "content": "Natural language processing research has studied summarization for a long time. Most related is", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 633, + 506, + 646 + ], + "spans": [ + { + "bbox": [ + 106, + 633, + 506, + 646 + ], + "score": 1.0, + "content": "work on abstractive summarization, in which the core content of a given text (usually a news article)", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 643, + 506, + 657 + ], + "spans": [ + { + "bbox": [ + 105, + 643, + 506, + 657 + ], + "score": 1.0, + "content": "is summarized in a novel and concise sentence. Chopra et al. (2016) and Nallapati et al. (2016) use", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 654, + 507, + 668 + ], + "spans": [ + { + "bbox": [ + 105, + 654, + 507, + 668 + ], + "score": 1.0, + "content": "deep learning models with attention on the input text to guide a decoder that generates a summary.", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 665, + 506, + 678 + ], + "spans": [ + { + "bbox": [ + 105, + 665, + 506, + 678 + ], + "score": 1.0, + "content": "See et al. (2017) and McCann et al. (2018) extend this idea with pointer networks (Vinyals et al.,", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 676, + 506, + 690 + ], + "spans": [ + { + "bbox": [ + 105, + 676, + 506, + 690 + ], + "score": 1.0, + "content": "2015a) to allow for copying tokens from the input text to the output summary. These approaches", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 686, + 506, + 702 + ], + "spans": [ + { + "bbox": [ + 105, + 686, + 506, + 702 + ], + "score": 1.0, + "content": "treat text as a simple token sequences, not explicitly exposing additional structure. In principle, deep", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 698, + 506, + 712 + ], + "spans": [ + { + "bbox": [ + 105, + 698, + 506, + 712 + ], + "score": 1.0, + "content": "sequence networks are known to be able to learn the inherent structure of natural language (e.g. in", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 709, + 506, + 723 + ], + "spans": [ + { + "bbox": [ + 105, + 709, + 506, + 723 + ], + "score": 1.0, + "content": "parsing (Vinyals et al., 2015b) and entity recognition (Lample et al., 2016)), but our experiments", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 720, + 479, + 734 + ], + "spans": [ + { + "bbox": [ + 105, + 720, + 479, + 734 + ], + "score": 1.0, + "content": "indicate that explicitly exposing this structure by separating concerns improves performance.", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 26.5 + } + ], + "page_idx": 7, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 108, + 27, + 293, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 26, + 294, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 26, + 294, + 38 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2019", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 302, + 752, + 308, + 760 + ], + "lines": [ + { + "bbox": [ + 302, + 750, + 309, + 761 + ], + "spans": [ + { + "bbox": [ + 302, + 750, + 309, + 761 + ], + "score": 1.0, + "content": "8", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "table", + "bbox": [ + 106, + 81, + 515, + 322 + ], + "blocks": [ + { + "type": "table_body", + "bbox": [ + 106, + 81, + 515, + 322 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 106, + 81, + 515, + 322 + ], + "spans": [ + { + "bbox": [ + 106, + 81, + 515, + 322 + ], + "score": 0.775, + "html": "
Input: Arsenal,Newcastle United and Southampton have checked on Caen midfelder N'golo Kante .Paris-born Kante is a defensive minded player who has impressed for Caen this season and they are wiling to sellfor around f 5milion.Marseille have been in constant contact with Caen over signing the 24-year-old who has similarities with Lassna Diarra and Claude Makelele in terms of stature and style .N'Golo Kante is attractinginterest froma host of PremierLeague clubs including Arsenal.Caen would be wiling to sellKante for aroundf5million.
Reference:n'golo kante is wanted by arsenal,newcastle and southampton.marseille are also keen on thef5mrated midfielder.kante has been compared to lassana diarra and claude makelele.click here for the latestpremier league news .
See et al.(2O17) (+ Pointer): arsenal,newcastle united and southampton have checked on caen midfieldern'golo kante .paris-born kante is attracting interest from a host of premier league clubs including arsenal .paris-born kante is attracting interest from a host of premier league clubs including arsenal
See et al. (2O17)(+ Pointer + Coverage): arsenal, newcastle united and southampton have checked on caenmidfielder n'golo kante.paris-born kante is a defensive minded player who has impressed for caen this season.marseille have been in constant contact with caen over signing the 24-year-old .
BILSTM+GNN →LSTM: marseille have been linked with caen midfielder %UNK% %UNK% . marseillehave been interested from a host of premier league clubs including arsenal .caen have been interested from ahost of premier league clubs including arsenal .
BILSTM+GNN →LSTM+PoINTER n'golo kante is attracting interest from a host of premier league clubs .marseilehave been in constant contact with caen over signing the24-year-old.the 24-year-old has similaritieswith lassana diarra and claude makelele in terms of stature .
", + "type": "table", + "image_path": "879ad0593963ca3e074145df80c378944913e18c84b695b9649622deeb04970e.jpg" + } + ] + } + ], + "index": 1, + "virtual_lines": [ + { + "bbox": [ + 106, + 81, + 515, + 161.33333333333331 + ], + "spans": [], + "index": 0 + }, + { + "bbox": [ + 106, + 161.33333333333331, + 515, + 241.66666666666663 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 106, + 241.66666666666663, + 515, + 321.99999999999994 + ], + "spans": [], + "index": 2 + } + ] + }, + { + "type": "table_caption", + "bbox": [ + 155, + 330, + 454, + 343 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 154, + 330, + 456, + 344 + ], + "spans": [ + { + "bbox": [ + 154, + 330, + 456, + 344 + ], + "score": 1.0, + "content": "Figure 4: Sample natural language translations from the CNN-DM dataset.", + "type": "text" + } + ], + "index": 3 + } + ], + "index": 3 + } + ], + "index": 2.0 + }, + { + "type": "text", + "bbox": [ + 107, + 368, + 505, + 445 + ], + "lines": [ + { + "bbox": [ + 106, + 369, + 505, + 381 + ], + "spans": [ + { + "bbox": [ + 106, + 369, + 505, + 381 + ], + "score": 1.0, + "content": "METHODDOC Figures 1 and 3 illustrate typical results of baselines and our model on the", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 378, + 505, + 392 + ], + "spans": [ + { + "bbox": [ + 105, + 378, + 505, + 392 + ], + "score": 1.0, + "content": "METHODDOC task (see Appendix A for more examples). The hardness of the task stems from the", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 390, + 506, + 402 + ], + "spans": [ + { + "bbox": [ + 106, + 390, + 506, + 402 + ], + "score": 1.0, + "content": "large number of distractors and the need to identify the most relevant parts of the input. In Figure 1,", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 401, + 506, + 414 + ], + "spans": [ + { + "bbox": [ + 105, + 401, + 506, + 414 + ], + "score": 1.0, + "content": "the token “parameter” and variations appears many times, and identifying the correct relationship is", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 412, + 505, + 424 + ], + "spans": [ + { + "bbox": [ + 106, + 412, + 505, + 424 + ], + "score": 1.0, + "content": "non-trivial, but is evidently eased by graph edges explicitly denoting these relationships. Similarly, in", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 422, + 506, + 437 + ], + "spans": [ + { + "bbox": [ + 105, + 422, + 506, + 437 + ], + "score": 1.0, + "content": "Figure 3, many variables are passed around, and the semantics of the method require understanding", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 106, + 435, + 259, + 445 + ], + "spans": [ + { + "bbox": [ + 106, + 435, + 259, + 445 + ], + "score": 1.0, + "content": "how information flows between them.", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 7, + "bbox_fs": [ + 105, + 369, + 506, + 445 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 462, + 506, + 573 + ], + "lines": [ + { + "bbox": [ + 105, + 462, + 505, + 475 + ], + "spans": [ + { + "bbox": [ + 105, + 462, + 505, + 475 + ], + "score": 1.0, + "content": "NLSUMMARIZATION Figure 4 shows one sample summarization. More samples for this task can", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 474, + 505, + 486 + ], + "spans": [ + { + "bbox": [ + 106, + 474, + 505, + 486 + ], + "score": 1.0, + "content": "be found in Appendix B. First, we notice that the model produces natural-looking summaries with", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 485, + 507, + 497 + ], + "spans": [ + { + "bbox": [ + 105, + 485, + 507, + 497 + ], + "score": 1.0, + "content": "no noticeable negative impact on the fluency of the language over existing methods. Furthermore,", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 495, + 507, + 509 + ], + "spans": [ + { + "bbox": [ + 105, + 495, + 507, + 509 + ], + "score": 1.0, + "content": "the GNN-based model seems to capture the central named entity in the article and creates a sum-", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 507, + 506, + 519 + ], + "spans": [ + { + "bbox": [ + 105, + 507, + 506, + 519 + ], + "score": 1.0, + "content": "mary centered around that entity. We hypothesize that the GNN component that links long-distance", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 518, + 506, + 529 + ], + "spans": [ + { + "bbox": [ + 106, + 518, + 506, + 529 + ], + "score": 1.0, + "content": "relationships helps capture and maintain a better “global” view of the article, allowing for better iden-", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 528, + 506, + 541 + ], + "spans": [ + { + "bbox": [ + 105, + 528, + 506, + 541 + ], + "score": 1.0, + "content": "tification of central entities. Our model still suffers from repetition of information (see Appendix B),", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 538, + 507, + 552 + ], + "spans": [ + { + "bbox": [ + 105, + 538, + 507, + 552 + ], + "score": 1.0, + "content": "and so we believe that our model would also profit from advances such as taking coverage into ac-", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 550, + 505, + 563 + ], + "spans": [ + { + "bbox": [ + 105, + 550, + 505, + 563 + ], + "score": 1.0, + "content": "count (See et al., 2017) or optimizing for ROUGE-L scores directly via reinforcement learning (Chen", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 562, + 262, + 573 + ], + "spans": [ + { + "bbox": [ + 106, + 562, + 262, + 573 + ], + "score": 1.0, + "content": "& Bansal, 2018; Narayan et al., 2018).", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 15.5, + "bbox_fs": [ + 105, + 462, + 507, + 573 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 594, + 211, + 606 + ], + "lines": [ + { + "bbox": [ + 105, + 592, + 213, + 608 + ], + "spans": [ + { + "bbox": [ + 105, + 592, + 213, + 608 + ], + "score": 1.0, + "content": "5 RELATED WORK", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 21 + }, + { + "type": "text", + "bbox": [ + 107, + 622, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 621, + 506, + 634 + ], + "spans": [ + { + "bbox": [ + 106, + 621, + 506, + 634 + ], + "score": 1.0, + "content": "Natural language processing research has studied summarization for a long time. Most related is", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 633, + 506, + 646 + ], + "spans": [ + { + "bbox": [ + 106, + 633, + 506, + 646 + ], + "score": 1.0, + "content": "work on abstractive summarization, in which the core content of a given text (usually a news article)", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 643, + 506, + 657 + ], + "spans": [ + { + "bbox": [ + 105, + 643, + 506, + 657 + ], + "score": 1.0, + "content": "is summarized in a novel and concise sentence. Chopra et al. (2016) and Nallapati et al. (2016) use", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 654, + 507, + 668 + ], + "spans": [ + { + "bbox": [ + 105, + 654, + 507, + 668 + ], + "score": 1.0, + "content": "deep learning models with attention on the input text to guide a decoder that generates a summary.", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 665, + 506, + 678 + ], + "spans": [ + { + "bbox": [ + 105, + 665, + 506, + 678 + ], + "score": 1.0, + "content": "See et al. (2017) and McCann et al. (2018) extend this idea with pointer networks (Vinyals et al.,", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 676, + 506, + 690 + ], + "spans": [ + { + "bbox": [ + 105, + 676, + 506, + 690 + ], + "score": 1.0, + "content": "2015a) to allow for copying tokens from the input text to the output summary. These approaches", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 686, + 506, + 702 + ], + "spans": [ + { + "bbox": [ + 105, + 686, + 506, + 702 + ], + "score": 1.0, + "content": "treat text as a simple token sequences, not explicitly exposing additional structure. In principle, deep", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 698, + 506, + 712 + ], + "spans": [ + { + "bbox": [ + 105, + 698, + 506, + 712 + ], + "score": 1.0, + "content": "sequence networks are known to be able to learn the inherent structure of natural language (e.g. in", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 709, + 506, + 723 + ], + "spans": [ + { + "bbox": [ + 105, + 709, + 506, + 723 + ], + "score": 1.0, + "content": "parsing (Vinyals et al., 2015b) and entity recognition (Lample et al., 2016)), but our experiments", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 720, + 479, + 734 + ], + "spans": [ + { + "bbox": [ + 105, + 720, + 479, + 734 + ], + "score": 1.0, + "content": "indicate that explicitly exposing this structure by separating concerns improves performance.", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 26.5, + "bbox_fs": [ + 105, + 621, + 507, + 734 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 82, + 505, + 137 + ], + "lines": [ + { + "bbox": [ + 106, + 82, + 505, + 95 + ], + "spans": [ + { + "bbox": [ + 106, + 82, + 505, + 95 + ], + "score": 1.0, + "content": "Recent work in summarization has proposed improved training objectives for summarization, such", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 93, + 505, + 105 + ], + "spans": [ + { + "bbox": [ + 106, + 93, + 505, + 105 + ], + "score": 1.0, + "content": "as tracking coverage of the input document (See et al., 2017) or using reinforcement learning to", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 106, + 104, + 505, + 117 + ], + "spans": [ + { + "bbox": [ + 106, + 104, + 505, + 117 + ], + "score": 1.0, + "content": "directly identify actions in the decoder that improve target measures such as ROUGE-L (Chen &", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 106, + 115, + 506, + 128 + ], + "spans": [ + { + "bbox": [ + 106, + 115, + 506, + 128 + ], + "score": 1.0, + "content": "Bansal, 2018; Narayan et al., 2018). These objectives are orthogonal to the graph-augmented encoder", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 126, + 455, + 139 + ], + "spans": [ + { + "bbox": [ + 105, + 126, + 455, + 139 + ], + "score": 1.0, + "content": "discussed in this work, and we are interested in combining these efforts in future work.", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 2 + }, + { + "type": "text", + "bbox": [ + 107, + 143, + 505, + 253 + ], + "lines": [ + { + "bbox": [ + 105, + 142, + 506, + 157 + ], + "spans": [ + { + "bbox": [ + 105, + 142, + 506, + 157 + ], + "score": 1.0, + "content": "Exposing more language structure explicitly has been studied over the last years, with a focus on", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 153, + 505, + 168 + ], + "spans": [ + { + "bbox": [ + 105, + 153, + 505, + 168 + ], + "score": 1.0, + "content": "tree-based models (Tai et al., 2015). Very recently, first uses of graphs in natural language processing", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 165, + 506, + 178 + ], + "spans": [ + { + "bbox": [ + 105, + 165, + 506, + 178 + ], + "score": 1.0, + "content": "have been explored. Marcheggiani & Titov (2017) use graph convolutional networks to encode single", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 177, + 506, + 188 + ], + "spans": [ + { + "bbox": [ + 105, + 177, + 506, + 188 + ], + "score": 1.0, + "content": "sentences and assist machine translation. De Cao et al. (2018) create a graph over named entities over", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 187, + 506, + 200 + ], + "spans": [ + { + "bbox": [ + 105, + 187, + 506, + 200 + ], + "score": 1.0, + "content": "a set of documents to assist question answering. Closer to our work is the work of Liu et al. (2018),", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 197, + 506, + 211 + ], + "spans": [ + { + "bbox": [ + 105, + 197, + 506, + 211 + ], + "score": 1.0, + "content": "who use abstract meaning representation (AMR), in which the source document is first parsed into", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 209, + 506, + 222 + ], + "spans": [ + { + "bbox": [ + 105, + 209, + 506, + 222 + ], + "score": 1.0, + "content": "AMR graphs, before a summary graph is created, which is finally rendered in natural language. In", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 220, + 505, + 232 + ], + "spans": [ + { + "bbox": [ + 105, + 220, + 505, + 232 + ], + "score": 1.0, + "content": "contrast to that work we do not use AMRs but directly encode relatively simple relationships directly", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 231, + 506, + 244 + ], + "spans": [ + { + "bbox": [ + 105, + 231, + 506, + 244 + ], + "score": 1.0, + "content": "on the tokenized text, and do not treat summarization as a graph rewrite problem. Combining our", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 242, + 456, + 255 + ], + "spans": [ + { + "bbox": [ + 105, + 242, + 456, + 255 + ], + "score": 1.0, + "content": "encoder with AMRs to use richer graph structures may be a promising future direction.", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 9.5 + }, + { + "type": "text", + "bbox": [ + 107, + 258, + 505, + 347 + ], + "lines": [ + { + "bbox": [ + 106, + 259, + 506, + 271 + ], + "spans": [ + { + "bbox": [ + 106, + 259, + 506, + 271 + ], + "score": 1.0, + "content": "Finally, summarization in source code has also been studied in the forms of method naming, comment", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 269, + 506, + 281 + ], + "spans": [ + { + "bbox": [ + 106, + 269, + 506, + 281 + ], + "score": 1.0, + "content": "and documentation prediction. Method naming has been tackled with a series of models. For example,", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 281, + 505, + 293 + ], + "spans": [ + { + "bbox": [ + 106, + 281, + 505, + 293 + ], + "score": 1.0, + "content": "Allamanis et al. (2015) use a log-bilinear network to predict method names from features, and later", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 291, + 505, + 304 + ], + "spans": [ + { + "bbox": [ + 105, + 291, + 505, + 304 + ], + "score": 1.0, + "content": "extend this idea to use a convolutional attention network over the tokens of a method to predict the", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 303, + 505, + 315 + ], + "spans": [ + { + "bbox": [ + 106, + 303, + 505, + 315 + ], + "score": 1.0, + "content": "subtokens of names (Allamanis et al., 2016). Raychev et al. (2015) and Bichsel et al. (2016) use", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 314, + 506, + 326 + ], + "spans": [ + { + "bbox": [ + 106, + 314, + 506, + 326 + ], + "score": 1.0, + "content": "CRFs for a range of tasks on source code, including the inference of names for variables and methods.", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 324, + 505, + 338 + ], + "spans": [ + { + "bbox": [ + 105, + 324, + 505, + 338 + ], + "score": 1.0, + "content": "Recently, Alon et al. (2018b;a) extract and encode paths from the syntax tree of a program, setting", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 335, + 308, + 349 + ], + "spans": [ + { + "bbox": [ + 105, + 335, + 308, + 349 + ], + "score": 1.0, + "content": "the state of the art in accuracy on method naming.", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 18.5 + }, + { + "type": "text", + "bbox": [ + 107, + 352, + 505, + 495 + ], + "lines": [ + { + "bbox": [ + 105, + 352, + 505, + 366 + ], + "spans": [ + { + "bbox": [ + 105, + 352, + 505, + 366 + ], + "score": 1.0, + "content": "Linking text to code can have useful applications, such as code search (Gu et al., 2018), traceability", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 363, + 505, + 376 + ], + "spans": [ + { + "bbox": [ + 105, + 363, + 505, + 376 + ], + "score": 1.0, + "content": "(Guo et al., 2017), and detection of redundant method comments (Louis et al., 2018). Most approaches", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 374, + 505, + 388 + ], + "spans": [ + { + "bbox": [ + 105, + 374, + 505, + 388 + ], + "score": 1.0, + "content": "on source code either treat it as natural language (i.e., a token sequence), or use a language parser to", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 384, + 506, + 399 + ], + "spans": [ + { + "bbox": [ + 105, + 384, + 506, + 399 + ], + "score": 1.0, + "content": "explicitly expose its tree structure. For example, Barone & Sennrich (2017) use a simple sequence-", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 396, + 505, + 409 + ], + "spans": [ + { + "bbox": [ + 106, + 396, + 505, + 409 + ], + "score": 1.0, + "content": "to-sequence baseline, whereas Hu et al. (2017) summarize source code by linearizing the abstract", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 407, + 505, + 420 + ], + "spans": [ + { + "bbox": [ + 106, + 407, + 505, + 420 + ], + "score": 1.0, + "content": "syntax tree of the code and using a sequence-to-sequence model. Wan et al. (2018) instead directly", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 419, + 506, + 431 + ], + "spans": [ + { + "bbox": [ + 105, + 419, + 506, + 431 + ], + "score": 1.0, + "content": "operate on the tree structure using tree recurrent neural networks (Tai et al., 2015). The use of", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 429, + 505, + 441 + ], + "spans": [ + { + "bbox": [ + 106, + 429, + 505, + 441 + ], + "score": 1.0, + "content": "additional structure on related tasks on source code has been studied recently, for example in models", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 440, + 506, + 453 + ], + "spans": [ + { + "bbox": [ + 105, + 440, + 506, + 453 + ], + "score": 1.0, + "content": "that are conditioned on learned traversals of the syntax tree (Bielik et al., 2016) and in graph-based", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 451, + 506, + 465 + ], + "spans": [ + { + "bbox": [ + 105, + 451, + 506, + 465 + ], + "score": 1.0, + "content": "approaches (Allamanis et al., 2018; Cvitkovic et al., 2018). However, as noted by Liao et al. (2018),", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 461, + 505, + 475 + ], + "spans": [ + { + "bbox": [ + 105, + 461, + 505, + 475 + ], + "score": 1.0, + "content": "GNN-based approaches suffer from a tension between the ability to propagate information across", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 473, + 506, + 486 + ], + "spans": [ + { + "bbox": [ + 106, + 473, + 506, + 486 + ], + "score": 1.0, + "content": "large distances in a graph and the computational expense of the propagation function, which is linear", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 483, + 312, + 498 + ], + "spans": [ + { + "bbox": [ + 105, + 483, + 312, + 498 + ], + "score": 1.0, + "content": "in the number of graph edges per propagation step.", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 29 + }, + { + "type": "title", + "bbox": [ + 108, + 511, + 281, + 524 + ], + "lines": [ + { + "bbox": [ + 104, + 509, + 283, + 527 + ], + "spans": [ + { + "bbox": [ + 104, + 509, + 283, + 527 + ], + "score": 1.0, + "content": "6 DISCUSSION & CONCLUSIONS", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 36 + }, + { + "type": "text", + "bbox": [ + 107, + 536, + 505, + 624 + ], + "lines": [ + { + "bbox": [ + 105, + 535, + 505, + 550 + ], + "spans": [ + { + "bbox": [ + 105, + 535, + 505, + 550 + ], + "score": 1.0, + "content": "We presented a framework for extending sequence encoders with a graph component that can leverage", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 106, + 548, + 505, + 559 + ], + "spans": [ + { + "bbox": [ + 106, + 548, + 505, + 559 + ], + "score": 1.0, + "content": "rich additional structure. In an evaluation on three different summarization tasks, we have shown", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 558, + 506, + 571 + ], + "spans": [ + { + "bbox": [ + 105, + 558, + 506, + 571 + ], + "score": 1.0, + "content": "that this augmentation improves the performance of a range of different sequence models across", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 570, + 505, + 582 + ], + "spans": [ + { + "bbox": [ + 105, + 570, + 505, + 582 + ], + "score": 1.0, + "content": "all tasks. We are excited about this initial progress and look forward to deeper integration of mixed", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 580, + 505, + 593 + ], + "spans": [ + { + "bbox": [ + 105, + 580, + 505, + 593 + ], + "score": 1.0, + "content": "sequence-graph modeling in a wide range of tasks across both formal and natural languages. The", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 591, + 505, + 604 + ], + "spans": [ + { + "bbox": [ + 105, + 591, + 505, + 604 + ], + "score": 1.0, + "content": "key insight, which we believe to be widely applicable, is that inductive biases induced by explicit", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 602, + 505, + 615 + ], + "spans": [ + { + "bbox": [ + 105, + 602, + 505, + 615 + ], + "score": 1.0, + "content": "relationship modeling are a simple way to boost the practical performance of existing deep learning", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 106, + 614, + 143, + 626 + ], + "spans": [ + { + "bbox": [ + 106, + 614, + 143, + 626 + ], + "score": 1.0, + "content": "systems.", + "type": "text" + } + ], + "index": 44 + } + ], + "index": 40.5 + } + ], + "page_idx": 8, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 108, + 27, + 293, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 26, + 294, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 26, + 294, + 38 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2019", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 302, + 751, + 308, + 759 + ], + "lines": [ + { + "bbox": [ + 302, + 751, + 309, + 762 + ], + "spans": [ + { + "bbox": [ + 302, + 751, + 309, + 762 + ], + "score": 1.0, + "content": "9", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 82, + 505, + 137 + ], + "lines": [ + { + "bbox": [ + 106, + 82, + 505, + 95 + ], + "spans": [ + { + "bbox": [ + 106, + 82, + 505, + 95 + ], + "score": 1.0, + "content": "Recent work in summarization has proposed improved training objectives for summarization, such", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 93, + 505, + 105 + ], + "spans": [ + { + "bbox": [ + 106, + 93, + 505, + 105 + ], + "score": 1.0, + "content": "as tracking coverage of the input document (See et al., 2017) or using reinforcement learning to", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 106, + 104, + 505, + 117 + ], + "spans": [ + { + "bbox": [ + 106, + 104, + 505, + 117 + ], + "score": 1.0, + "content": "directly identify actions in the decoder that improve target measures such as ROUGE-L (Chen &", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 106, + 115, + 506, + 128 + ], + "spans": [ + { + "bbox": [ + 106, + 115, + 506, + 128 + ], + "score": 1.0, + "content": "Bansal, 2018; Narayan et al., 2018). 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Very recently, first uses of graphs in natural language processing", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 165, + 506, + 178 + ], + "spans": [ + { + "bbox": [ + 105, + 165, + 506, + 178 + ], + "score": 1.0, + "content": "have been explored. Marcheggiani & Titov (2017) use graph convolutional networks to encode single", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 177, + 506, + 188 + ], + "spans": [ + { + "bbox": [ + 105, + 177, + 506, + 188 + ], + "score": 1.0, + "content": "sentences and assist machine translation. De Cao et al. (2018) create a graph over named entities over", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 187, + 506, + 200 + ], + "spans": [ + { + "bbox": [ + 105, + 187, + 506, + 200 + ], + "score": 1.0, + "content": "a set of documents to assist question answering. Closer to our work is the work of Liu et al. (2018),", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 197, + 506, + 211 + ], + "spans": [ + { + "bbox": [ + 105, + 197, + 506, + 211 + ], + "score": 1.0, + "content": "who use abstract meaning representation (AMR), in which the source document is first parsed into", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 209, + 506, + 222 + ], + "spans": [ + { + "bbox": [ + 105, + 209, + 506, + 222 + ], + "score": 1.0, + "content": "AMR graphs, before a summary graph is created, which is finally rendered in natural language. In", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 220, + 505, + 232 + ], + "spans": [ + { + "bbox": [ + 105, + 220, + 505, + 232 + ], + "score": 1.0, + "content": "contrast to that work we do not use AMRs but directly encode relatively simple relationships directly", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 231, + 506, + 244 + ], + "spans": [ + { + "bbox": [ + 105, + 231, + 506, + 244 + ], + "score": 1.0, + "content": "on the tokenized text, and do not treat summarization as a graph rewrite problem. 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For example,", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 281, + 505, + 293 + ], + "spans": [ + { + "bbox": [ + 106, + 281, + 505, + 293 + ], + "score": 1.0, + "content": "Allamanis et al. (2015) use a log-bilinear network to predict method names from features, and later", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 291, + 505, + 304 + ], + "spans": [ + { + "bbox": [ + 105, + 291, + 505, + 304 + ], + "score": 1.0, + "content": "extend this idea to use a convolutional attention network over the tokens of a method to predict the", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 303, + 505, + 315 + ], + "spans": [ + { + "bbox": [ + 106, + 303, + 505, + 315 + ], + "score": 1.0, + "content": "subtokens of names (Allamanis et al., 2016). Raychev et al. (2015) and Bichsel et al. (2016) use", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 314, + 506, + 326 + ], + "spans": [ + { + "bbox": [ + 106, + 314, + 506, + 326 + ], + "score": 1.0, + "content": "CRFs for a range of tasks on source code, including the inference of names for variables and methods.", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 324, + 505, + 338 + ], + "spans": [ + { + "bbox": [ + 105, + 324, + 505, + 338 + ], + "score": 1.0, + "content": "Recently, Alon et al. (2018b;a) extract and encode paths from the syntax tree of a program, setting", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 335, + 308, + 349 + ], + "spans": [ + { + "bbox": [ + 105, + 335, + 308, + 349 + ], + "score": 1.0, + "content": "the state of the art in accuracy on method naming.", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 18.5, + "bbox_fs": [ + 105, + 259, + 506, + 349 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 352, + 505, + 495 + ], + "lines": [ + { + "bbox": [ + 105, + 352, + 505, + 366 + ], + "spans": [ + { + "bbox": [ + 105, + 352, + 505, + 366 + ], + "score": 1.0, + "content": "Linking text to code can have useful applications, such as code search (Gu et al., 2018), traceability", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 363, + 505, + 376 + ], + "spans": [ + { + "bbox": [ + 105, + 363, + 505, + 376 + ], + "score": 1.0, + "content": "(Guo et al., 2017), and detection of redundant method comments (Louis et al., 2018). Most approaches", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 374, + 505, + 388 + ], + "spans": [ + { + "bbox": [ + 105, + 374, + 505, + 388 + ], + "score": 1.0, + "content": "on source code either treat it as natural language (i.e., a token sequence), or use a language parser to", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 384, + 506, + 399 + ], + "spans": [ + { + "bbox": [ + 105, + 384, + 506, + 399 + ], + "score": 1.0, + "content": "explicitly expose its tree structure. For example, Barone & Sennrich (2017) use a simple sequence-", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 396, + 505, + 409 + ], + "spans": [ + { + "bbox": [ + 106, + 396, + 505, + 409 + ], + "score": 1.0, + "content": "to-sequence baseline, whereas Hu et al. (2017) summarize source code by linearizing the abstract", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 407, + 505, + 420 + ], + "spans": [ + { + "bbox": [ + 106, + 407, + 505, + 420 + ], + "score": 1.0, + "content": "syntax tree of the code and using a sequence-to-sequence model. Wan et al. (2018) instead directly", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 419, + 506, + 431 + ], + "spans": [ + { + "bbox": [ + 105, + 419, + 506, + 431 + ], + "score": 1.0, + "content": "operate on the tree structure using tree recurrent neural networks (Tai et al., 2015). The use of", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 429, + 505, + 441 + ], + "spans": [ + { + "bbox": [ + 106, + 429, + 505, + 441 + ], + "score": 1.0, + "content": "additional structure on related tasks on source code has been studied recently, for example in models", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 440, + 506, + 453 + ], + "spans": [ + { + "bbox": [ + 105, + 440, + 506, + 453 + ], + "score": 1.0, + "content": "that are conditioned on learned traversals of the syntax tree (Bielik et al., 2016) and in graph-based", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 451, + 506, + 465 + ], + "spans": [ + { + "bbox": [ + 105, + 451, + 506, + 465 + ], + "score": 1.0, + "content": "approaches (Allamanis et al., 2018; Cvitkovic et al., 2018). However, as noted by Liao et al. (2018),", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 461, + 505, + 475 + ], + "spans": [ + { + "bbox": [ + 105, + 461, + 505, + 475 + ], + "score": 1.0, + "content": "GNN-based approaches suffer from a tension between the ability to propagate information across", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 473, + 506, + 486 + ], + "spans": [ + { + "bbox": [ + 106, + 473, + 506, + 486 + ], + "score": 1.0, + "content": "large distances in a graph and the computational expense of the propagation function, which is linear", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 483, + 312, + 498 + ], + "spans": [ + { + "bbox": [ + 105, + 483, + 312, + 498 + ], + "score": 1.0, + "content": "in the number of graph edges per propagation step.", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 29, + "bbox_fs": [ + 105, + 352, + 506, + 498 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 511, + 281, + 524 + ], + "lines": [ + { + "bbox": [ + 104, + 509, + 283, + 527 + ], + "spans": [ + { + "bbox": [ + 104, + 509, + 283, + 527 + ], + "score": 1.0, + "content": "6 DISCUSSION & CONCLUSIONS", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 36 + }, + { + "type": "text", + "bbox": [ + 107, + 536, + 505, + 624 + ], + "lines": [ + { + "bbox": [ + 105, + 535, + 505, + 550 + ], + "spans": [ + { + "bbox": [ + 105, + 535, + 505, + 550 + ], + "score": 1.0, + "content": "We presented a framework for extending sequence encoders with a graph component that can leverage", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 106, + 548, + 505, + 559 + ], + "spans": [ + { + "bbox": [ + 106, + 548, + 505, + 559 + ], + "score": 1.0, + "content": "rich additional structure. In an evaluation on three different summarization tasks, we have shown", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 558, + 506, + 571 + ], + "spans": [ + { + "bbox": [ + 105, + 558, + 506, + 571 + ], + "score": 1.0, + "content": "that this augmentation improves the performance of a range of different sequence models across", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 570, + 505, + 582 + ], + "spans": [ + { + "bbox": [ + 105, + 570, + 505, + 582 + ], + "score": 1.0, + "content": "all tasks. We are excited about this initial progress and look forward to deeper integration of mixed", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 580, + 505, + 593 + ], + "spans": [ + { + "bbox": [ + 105, + 580, + 505, + 593 + ], + "score": 1.0, + "content": "sequence-graph modeling in a wide range of tasks across both formal and natural languages. The", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 591, + 505, + 604 + ], + "spans": [ + { + "bbox": [ + 105, + 591, + 505, + 604 + ], + "score": 1.0, + "content": "key insight, which we believe to be widely applicable, is that inductive biases induced by explicit", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 602, + 505, + 615 + ], + "spans": [ + { + "bbox": [ + 105, + 602, + 505, + 615 + ], + "score": 1.0, + "content": "relationship modeling are a simple way to boost the practical performance of existing deep learning", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 106, + 614, + 143, + 626 + ], + "spans": [ + { + "bbox": [ + 106, + 614, + 143, + 626 + ], + "score": 1.0, + "content": "systems.", + "type": "text" + } + ], + "index": 44 + } + ], + "index": 40.5, + "bbox_fs": [ + 105, + 535, + 506, + 626 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "title", + "bbox": [ + 107, + 81, + 176, + 94 + ], + "lines": [ + { + "bbox": [ + 106, + 82, + 176, + 94 + ], + "spans": [ + { + "bbox": [ + 106, + 82, + 176, + 94 + ], + "score": 1.0, + "content": "REFERENCES", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "text", + "bbox": [ + 105, + 100, + 505, + 123 + ], + "lines": [ + { + "bbox": [ + 106, + 100, + 506, + 113 + ], + "spans": [ + { + "bbox": [ + 106, + 100, + 506, + 113 + ], + "score": 1.0, + "content": "Miltiadis Allamanis. 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Input: -LRB- CNN -RRB- Gunshots were fired at rapper Lil Wayne 's tour bus early Sunday inAtlanta .No one was injured in the shooting ,and no arrests have been made,Atlanta Police spokeswoman Elizabeth Espy said .Police are stillooking for suspects . Ofcers were called to aparking lot in Atlanta 's Buckhead neighborhood,Espy said .They arrived at 3:25 a.m. and locatedtwo tour buses that had been shot multiple times .The drivers of the buses said the incident occurredon Interstate 285 near Interstate 75,Espy said .Witnesses provided a limited description of the twovehicles suspected to be involved : a“ Corvette style vehicle ”and an SUV .Lil Wayne was in Atlantafor a performance at Compound nightclub Saturday night . CNN 's Carma Hassan contributed to thisreport .
Reference: rapper lil wayne not injured after shots fired at his tour bus on an atlanta interstate ,police say . no one has been arrested in the shooting
See et al. (2017) (+ Pointer): police are stillooking for suspects . the incident occurred on interstate285 near interstate 75 ,police say . witnesses provided a limited description of the two vehicles suspected to be involved : a“ corvette style vehicle ” and an suv .
See et al. (2017) (+ Pointer + Coverage): lil wayne 's tour bus was shot multiple times , police say: police are stillooking for suspects .they arrived at 3:25 a.m. and located two tour buses that hadbeen shot .
BILSTM+GNN → LSTM: the incident occurred on interstate %UNK% near interstate 75 . no onewas injured in the shooting ,and no arrests have been made ,atlanta police spokeswoman says .
BILSTM+GNN →LSTM+PoINTER gunshots fired at rapper lil wayne 's tour bus early sundayin atlanta ,police say . no one was injured in the shooting ,and no arrests have been made , policesay.
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Input: Tottenham have held further discussions with Marseille over a potential deal for midfielderFlorian Thauvin .The 22-year-old has been left out of the squad for this weekend 's game withMetz as Marseille push for a f 15m sale .The winger,who can also play behind the striker, wasthe subject of enquiries from Spurs earlier in the year and has also been watched by Chelsea andValencia .Tottenham have held further talks with Ligue 1 side Marseille over a possible deal forFlorian Thauvin .Marseille are already resigned to losing Andre Ayew and Andre-Pierre Gignacwith English sides keen on both .Everton,Newcastle and Swansea,have allshown an interest inAyew,who is a free agent in the summer .
Reference: florian thauvin has been left out of marseille 's squad with metz . marseille are pushingfor a f 15m sale and tottenham are interested .the winger has also been watched by chelsea and laliga side valencia .
See et al. (2O17)(+ Pointer): tottenham have held further discussions with marseille over a potentialdeal for midfelder florian thauvin .the 22-year-old has been left out of the squad for this weekend 's game with metz as marseille push for a 15m sale .
See et al. (2017) (+ Pointer + Coverage): florian thauvin has been left out of the squad for thisweekend 's game with metz as marseille push for a 15m sale .the 22-year-old was the subject ofenquiries from spurs earlier in the year .
BILSTM+GNN →LSTM: the 22-year-old has been left out of the squad for this weekend 's gamewith metz .the 22-year-old has been left out of the squad for this weekend 's game with metz .thewinger has been left out of the squad for this weekend 's game with metz .
BILSTM+GNN → LSTM+PoINTER tottenham have held further discussions with marseille overa potential deal .the winger has been left out of the squad for this weekend 's game .tottenham haveheld further talks with marseille over a potential deal .
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Input: -LRB- CNN -RRB- Gunshots were fired at rapper Lil Wayne 's tour bus early Sunday inAtlanta .No one was injured in the shooting ,and no arrests have been made,Atlanta Police spokeswoman Elizabeth Espy said .Police are stillooking for suspects . Ofcers were called to aparking lot in Atlanta 's Buckhead neighborhood,Espy said .They arrived at 3:25 a.m. and locatedtwo tour buses that had been shot multiple times .The drivers of the buses said the incident occurredon Interstate 285 near Interstate 75,Espy said .Witnesses provided a limited description of the twovehicles suspected to be involved : a“ Corvette style vehicle ”and an SUV .Lil Wayne was in Atlantafor a performance at Compound nightclub Saturday night . CNN 's Carma Hassan contributed to thisreport .
Reference: rapper lil wayne not injured after shots fired at his tour bus on an atlanta interstate ,police say . no one has been arrested in the shooting
See et al. (2017) (+ Pointer): police are stillooking for suspects . the incident occurred on interstate285 near interstate 75 ,police say . witnesses provided a limited description of the two vehicles suspected to be involved : a“ corvette style vehicle ” and an suv .
See et al. (2017) (+ Pointer + Coverage): lil wayne 's tour bus was shot multiple times , police say: police are stillooking for suspects .they arrived at 3:25 a.m. and located two tour buses that hadbeen shot .
BILSTM+GNN → LSTM: the incident occurred on interstate %UNK% near interstate 75 . no onewas injured in the shooting ,and no arrests have been made ,atlanta police spokeswoman says .
BILSTM+GNN →LSTM+PoINTER gunshots fired at rapper lil wayne 's tour bus early sundayin atlanta ,police say . no one was injured in the shooting ,and no arrests have been made , policesay.
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Input: Tottenham have held further discussions with Marseille over a potential deal for midfielderFlorian Thauvin .The 22-year-old has been left out of the squad for this weekend 's game withMetz as Marseille push for a f 15m sale .The winger,who can also play behind the striker, wasthe subject of enquiries from Spurs earlier in the year and has also been watched by Chelsea andValencia .Tottenham have held further talks with Ligue 1 side Marseille over a possible deal forFlorian Thauvin .Marseille are already resigned to losing Andre Ayew and Andre-Pierre Gignacwith English sides keen on both .Everton,Newcastle and Swansea,have allshown an interest inAyew,who is a free agent in the summer .
Reference: florian thauvin has been left out of marseille 's squad with metz . marseille are pushingfor a f 15m sale and tottenham are interested .the winger has also been watched by chelsea and laliga side valencia .
See et al. (2O17)(+ Pointer): tottenham have held further discussions with marseille over a potentialdeal for midfelder florian thauvin .the 22-year-old has been left out of the squad for this weekend 's game with metz as marseille push for a 15m sale .
See et al. (2017) (+ Pointer + Coverage): florian thauvin has been left out of the squad for thisweekend 's game with metz as marseille push for a 15m sale .the 22-year-old was the subject ofenquiries from spurs earlier in the year .
BILSTM+GNN →LSTM: the 22-year-old has been left out of the squad for this weekend 's gamewith metz .the 22-year-old has been left out of the squad for this weekend 's game with metz .thewinger has been left out of the squad for this weekend 's game with metz .
BILSTM+GNN → LSTM+PoINTER tottenham have held further discussions with marseille overa potential deal .the winger has been left out of the squad for this weekend 's game .tottenham haveheld further talks with marseille over a potential deal .
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NameGit SHADescription
Akka.NET6f32f6a7Actor-based Concurrent & Distributed Framework
AutoMapper19d6f7fcObject-to-Object Mapping Library
BenchmarkDotNet57005f05Benchmarking Library
CommonMark.NETf3d54530Markdown Parser
CoreCLRcc5dcbe6.NET Core Runtime
CoreFxec1671fd.NETFoundationalLibraries
Dapper3c7cde28Object Mapper Library
EntityFrameworkc4d9a269Object-Relational Mapper
Humanizer2b1c94c4String Manipulation and Formatting
Lean90ee6aaeAlgorithmic Trading Engine
Mono9b9e4f4b.NET Implementation
MsBuild7f95dc15Build Engine
Nancyde458a9bHTTP Service Framework
NLog49fdd08eLogging Library
Opserver9e4d3a40Monitoring System
orleansf89c5866Distributed Virtual Actor Model
Pollyf3d2973dResilience & Transient Fault Handling Library
Powershell9ac701dbCommand-line Shell
ravendb6437de30Document Database
roslyn8ca0a542Compiler & Code Analysis & Compilation
ServiceStack17f081b9Real-time web library
SignalR9b05bcb0Push Notification Framework
Wox13e6c5eeApplication Launcher
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NameGit SHADescription
Akka.NET6f32f6a7Actor-based Concurrent & Distributed Framework
AutoMapper19d6f7fcObject-to-Object Mapping Library
BenchmarkDotNet57005f05Benchmarking Library
CommonMark.NETf3d54530Markdown Parser
CoreCLRcc5dcbe6.NET Core Runtime
CoreFxec1671fd.NETFoundationalLibraries
Dapper3c7cde28Object Mapper Library
EntityFrameworkc4d9a269Object-Relational Mapper
Humanizer2b1c94c4String Manipulation and Formatting
Lean90ee6aaeAlgorithmic Trading Engine
Mono9b9e4f4b.NET Implementation
MsBuild7f95dc15Build Engine
Nancyde458a9bHTTP Service Framework
NLog49fdd08eLogging Library
Opserver9e4d3a40Monitoring System
orleansf89c5866Distributed Virtual Actor Model
Pollyf3d2973dResilience & Transient Fault Handling Library
Powershell9ac701dbCommand-line Shell
ravendb6437de30Document Database
roslyn8ca0a542Compiler & Code Analysis & Compilation
ServiceStack17f081b9Real-time web library
SignalR9b05bcb0Push Notification Framework
Wox13e6c5eeApplication Launcher
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DatasetAvg Num NodesAvg Num Edges
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C# Method Names125.2239.3
C# Documentation133.5265.9
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C# Documentation133.5265.9
Java-SmallMethod Names144.4251.6
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METHODNAMINGF1ROUGE-2ROUGE-L
Java
Alon et al. (2018a)43.0一1
SELFATT →SELFATT24.98.327.4
SELFATT+GNN →SELFATT44.520.943.4
BILSTM →LSTM35.817.939.7
BILSTM+GNN →LSTM44.721.143.1
BILSTM →LSTM+POINTER42.522.445.6
GNN →LSTM+POINTER50.524.848.9
BILSTM+GNN→LSTM+POINTER51.425.050.0
C#
SELFATT →SELFATT41.325.243.2
SELFATT+GNN→→SELFATT62.131.061.1
BILSTM →LSTM48.832.851.8
BILSTM+GNN →LSTM62.631.061.3
BILSTM →LSTM+POINTER57.229.760.4
GNN→LSTM+POINTER63.031.561.3
BILSTM+GNN→LSTM+POINTER63.431.962.4
METHODDOCF1ROUGE-2ROUGE-LBLEU
C#
SELFATT →SELFATT40.027.841.113.9
SELFATT+GNN→SELFATT37.625.637.921.4
BILSTM →LSTM35.215.330.810.0
BILSTM+GNN →LSTM41.128.941.022.5
BILSTM →LSTM+POINTER35.220.836.714.7
GNN →LSTM+POINTER38.925.637.117.7
BILSTM+GNN →→LSTM+POINTER (average pooling)43.229.041.021.3
BILSTM+GNN→LSTM+POINTER45.428.341.122.2
NLSUMMARIZATIONROUGE-1ROUGE-2ROUGE-L
CNN/DM
BILSTM →LSTM33.611.427.9
BILSTM+GNN→LSTM33.013.328.3
See et al. (2017) (+ Pointer)36.415.733.4
BILSTM →LSTM+POINTER35.913.930.3
BILSTM+GNN →LSTM+POINTER38.116.133.2
See et al. (2017) (+ Pointer + Coverage)39.517.336.4
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NLSUMMARIZATION (CNN/DM)ROUGE-1ROUGE-2ROUGE-L
See et al.(2017) (base)31.311.828.8
See et al. (2017) (+ Pointer)36.415.733.4
See et al.(2017) (+ Pointer + Coverage)39.517.336.4
BILSTM →LSTM33.611.427.9
BILSTM →LSTM+POINTER35.913.930.3
BILSTM →LSTM+PoINTER (+ coref/entity annotations)36.214.230.5
BILSTM+GNN →LSTM33.013.328.3
BILSTM+GNN→LSTM+POINTER (only sentence nodes)36.015.229.6
BILSTM+GNN →LSTM+POINTER (sentence nodes + eq edges)36.115.430.3
BILSTM+GNN→LSTM+POINTER38.116.133.2
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Input: Arsenal,Newcastle United and Southampton have checked on Caen midfelder N'golo Kante .Paris-born Kante is a defensive minded player who has impressed for Caen this season and they are wiling to sellfor around f 5milion.Marseille have been in constant contact with Caen over signing the 24-year-old who has similarities with Lassna Diarra and Claude Makelele in terms of stature and style .N'Golo Kante is attractinginterest froma host of PremierLeague clubs including Arsenal.Caen would be wiling to sellKante for aroundf5million.
Reference:n'golo kante is wanted by arsenal,newcastle and southampton.marseille are also keen on thef5mrated midfielder.kante has been compared to lassana diarra and claude makelele.click here for the latestpremier league news .
See et al.(2O17) (+ Pointer): arsenal,newcastle united and southampton have checked on caen midfieldern'golo kante .paris-born kante is attracting interest from a host of premier league clubs including arsenal .paris-born kante is attracting interest from a host of premier league clubs including arsenal
See et al. (2O17)(+ Pointer + Coverage): arsenal, newcastle united and southampton have checked on caenmidfielder n'golo kante.paris-born kante is a defensive minded player who has impressed for caen this season.marseille have been in constant contact with caen over signing the 24-year-old .
BILSTM+GNN →LSTM: marseille have been linked with caen midfielder %UNK% %UNK% . marseillehave been interested from a host of premier league clubs including arsenal .caen have been interested from ahost of premier league clubs including arsenal .
BILSTM+GNN →LSTM+PoINTER n'golo kante is attracting interest from a host of premier league clubs .marseilehave been in constant contact with caen over signing the24-year-old.the 24-year-old has similaritieswith lassana diarra and claude makelele in terms of stature .
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Input: Tottenham have held further discussions with Marseille over a potential deal for midfielderFlorian Thauvin .The 22-year-old has been left out of the squad for this weekend 's game withMetz as Marseille push for a f 15m sale .The winger,who can also play behind the striker, wasthe subject of enquiries from Spurs earlier in the year and has also been watched by Chelsea andValencia .Tottenham have held further talks with Ligue 1 side Marseille over a possible deal forFlorian Thauvin .Marseille are already resigned to losing Andre Ayew and Andre-Pierre Gignacwith English sides keen on both .Everton,Newcastle and Swansea,have allshown an interest inAyew,who is a free agent in the summer .
Reference: florian thauvin has been left out of marseille 's squad with metz . marseille are pushingfor a f 15m sale and tottenham are interested .the winger has also been watched by chelsea and laliga side valencia .
See et al. (2O17)(+ Pointer): tottenham have held further discussions with marseille over a potentialdeal for midfelder florian thauvin .the 22-year-old has been left out of the squad for this weekend 's game with metz as marseille push for a 15m sale .
See et al. (2017) (+ Pointer + Coverage): florian thauvin has been left out of the squad for thisweekend 's game with metz as marseille push for a 15m sale .the 22-year-old was the subject ofenquiries from spurs earlier in the year .
BILSTM+GNN →LSTM: the 22-year-old has been left out of the squad for this weekend 's gamewith metz .the 22-year-old has been left out of the squad for this weekend 's game with metz .thewinger has been left out of the squad for this weekend 's game with metz .
BILSTM+GNN → LSTM+PoINTER tottenham have held further discussions with marseille overa potential deal .the winger has been left out of the squad for this weekend 's game .tottenham haveheld further talks with marseille over a potential deal .
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Input: -LRB- CNN -RRB- Gunshots were fired at rapper Lil Wayne 's tour bus early Sunday inAtlanta .No one was injured in the shooting ,and no arrests have been made,Atlanta Police spokeswoman Elizabeth Espy said .Police are stillooking for suspects . Ofcers were called to aparking lot in Atlanta 's Buckhead neighborhood,Espy said .They arrived at 3:25 a.m. and locatedtwo tour buses that had been shot multiple times .The drivers of the buses said the incident occurredon Interstate 285 near Interstate 75,Espy said .Witnesses provided a limited description of the twovehicles suspected to be involved : a“ Corvette style vehicle ”and an SUV .Lil Wayne was in Atlantafor a performance at Compound nightclub Saturday night . CNN 's Carma Hassan contributed to thisreport .
Reference: rapper lil wayne not injured after shots fired at his tour bus on an atlanta interstate ,police say . no one has been arrested in the shooting
See et al. (2017) (+ Pointer): police are stillooking for suspects . the incident occurred on interstate285 near interstate 75 ,police say . witnesses provided a limited description of the two vehicles suspected to be involved : a“ corvette style vehicle ” and an suv .
See et al. (2017) (+ Pointer + Coverage): lil wayne 's tour bus was shot multiple times , police say: police are stillooking for suspects .they arrived at 3:25 a.m. and located two tour buses that hadbeen shot .
BILSTM+GNN → LSTM: the incident occurred on interstate %UNK% near interstate 75 . no onewas injured in the shooting ,and no arrests have been made ,atlanta police spokeswoman says .
BILSTM+GNN →LSTM+PoINTER gunshots fired at rapper lil wayne 's tour bus early sundayin atlanta ,police say . no one was injured in the shooting ,and no arrests have been made , policesay.
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NameGit SHADescription
Akka.NET6f32f6a7Actor-based Concurrent & Distributed Framework
AutoMapper19d6f7fcObject-to-Object Mapping Library
BenchmarkDotNet57005f05Benchmarking Library
CommonMark.NETf3d54530Markdown Parser
CoreCLRcc5dcbe6.NET Core Runtime
CoreFxec1671fd.NETFoundationalLibraries
Dapper3c7cde28Object Mapper Library
EntityFrameworkc4d9a269Object-Relational Mapper
Humanizer2b1c94c4String Manipulation and Formatting
Lean90ee6aaeAlgorithmic Trading Engine
Mono9b9e4f4b.NET Implementation
MsBuild7f95dc15Build Engine
Nancyde458a9bHTTP Service Framework
NLog49fdd08eLogging Library
Opserver9e4d3a40Monitoring System
orleansf89c5866Distributed Virtual Actor Model
Pollyf3d2973dResilience & Transient Fault Handling Library
Powershell9ac701dbCommand-line Shell
ravendb6437de30Document Database
roslyn8ca0a542Compiler & Code Analysis & Compilation
ServiceStack17f081b9Real-time web library
SignalR9b05bcb0Push Notification Framework
Wox13e6c5eeApplication Launcher
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DatasetTrain SizeValid SizeTest Size
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Mosse1,3, Chengxu Zhuang2, Daniel Yamins1,2, Noah Goodman $^ { 1 , 2 }$ Department of Computer Science1, Psychology2, and Philosophy3 Stanford University {wumike, chengxuz, mmosse19, yamins, ngoodman}@stanford.edu + +# ABSTRACT + +Recent methods for learning unsupervised visual representations, dubbed contrastive learning, optimize the noise-contrastive estimation (NCE) bound on mutual information between two transformations of an image. NCE typically uses randomly sampled negative examples to normalize the objective, but this may often include many uninformative examples either because they are too easy or too hard to discriminate. Taking inspiration from metric learning, we show that choosing semi-hard negatives can yield stronger contrastive representations. To do this, we introduce a family of mutual information estimators that sample negatives conditionally – in a “ring” around each positive. We prove that these estimators remain lower-bounds of mutual information, with higher bias but lower variance than NCE. Experimentally, we find our approach, applied on top of existing models (IR, CMC, and MoCo) improves accuracy by $2 - 5 \%$ absolute points in each case, measured by linear evaluation on four standard image benchmarks. Moreover, we find continued benefits when transferring features to a variety of new image distributions from the Meta-Dataset collection and to a variety of downstream tasks such as object detection, instance segmentation, and key-point detection. + +# 1 INTRODUCTION + +Supervised learning has given rise to human-level performance in several visual tasks (Russakovsky et al., 2015; He et al., 2017), relying heavily on large image datasets paired with semantic annotations. These annotations vary in difficulty and cost, spanning from simple class labels to more granular descriptions like bounding boxes and key-points. As it is impractical to scale high quality annotations, this reliance on supervision poses a barrier to widespread adoption. While supervised pretraining is still the dominant approach in computer vision, recent studies using unsupervised “contrastive” objectives, have achieved remarkable results in the last two years, closing the gap to supervised baselines (Wu et al., 2018; Oord et al., 2018; Hjelm et al., 2018; Zhuang et al., 2019; Henaff et al., 2019; Misra & Maaten, 2020; He et al., 2019; Chen et al.,´ 2020a;b; Grill et al., 2020) + +Many contrastive algorithms are estimators of mutual information (Oord et al., 2018; Hjelm et al., 2018; Bachman et al., 2019), capturing the intuition that a good low-dimensional “representation” is one that linearizes the useful information embedded within a high-dimensional data point. In vision, these estimators maximize the similarity of encodings for two augmentations of the same image. This is trivial (e.g. assign all image pairs maximum similarity) unless the similarity function is normalized. This is typically done by comparing an image to “negative examples”, which a model must assign low similarity to. We hypothesize that how we choose these negatives greatly impacts the representation quality. With harder negatives, the encoder is encouraged to capture more granular information that may improve performance on downstream tasks. While research in contrastive learning has explored architectures, augmentations, and pretext tasks, there has been little attention given to the negative sampling procedure. Meanwhile, there is a rich body of work in deep metric learning showing semi-hard negative mining to improve the efficacy of triplet losses. Inspired by this, we hope to bring harder negative sampling to modern contrastive learning. + +Naively choosing difficult negatives may yield an objective that no longer bounds mutual information, removing a theoretical connection that is core to contrastive learning and has been shown to be important for downstream performance (Tian et al., $\boxed { 2 0 2 0 }$ . In this paper, we present a new estimator of mutual information based on the popular noise-contrastive estimator (NCE) that supports sampling negatives from conditional distributions. We summarize our contributions below: + +1. We prove our Conditional-NCE (CNCE) objective to lower bound mutual information. Further, we show that although CNCE is a looser bound than NCE, it has lower variance. This motivates its value for representation learning. +2. We use CNCE to generalize contrastive algorithms that utilize a memory structure like IR, CMC, and MoCo to sample semi-hard negatives in just a few lines of code and minimal compute overhead. +3. We find that the naive strategy of sampling hard negatives throughout training can be detrimental. We then show that slowly introducing harder negatives yields good performance. +4. On four image classification benchmarks, we find improvements of $2 - 5 \%$ absolute points. We also find consistent improvements (1) when transferring features to new image datasets and (2) in object detection, instance segmentation, and key-point detection. + +# 2 BACKGROUND + +We focus on exemplar-based contrastive objectives, where examples are compared to one another to learn a representation. Many of these objectives (Hjelm et al., 2018; Wu et al., 2018; Bachman et al., 2019; Tian et al., 2019; Chen et al., 2020a) are equivalent to NCE (Oord et al., 2018; Poole et al., $\boxed { 2 0 1 9 }$ , a popular lower bound on the mutual information, denoted by $\overline { { \mathcal { T } } }$ , between two random variables. This connection is well-known and stated in several works (Chen et al., 2020a; Tschannen et al., 2019; Tian et al., 2020; Wu et al., 2020). To review, recall: + +$$ +\mathscr { Z } ( X ; Y ) \geq \mathscr { Z } _ { \mathrm { N C E } } ( X ; Y ) = \mathbf { E } _ { x _ { i } , y _ { i } \sim p ( x , y ) } \mathbf { E } _ { y _ { 1 : k } \sim p ( y ) } \left[ \log \frac { e ^ { f _ { \theta } ( x _ { i } , y _ { i } ) } } { \frac { 1 } { k + 1 } \sum _ { j \in \{ i , 1 : k \} } e ^ { f _ { \theta } ( x _ { i } , y _ { j } ) } } \right] +$$ + +where $x , y$ are realizations of two random variables, $X$ and $Y$ , and $f _ { \theta } : X \times Y \to \mathbf { R }$ is a similarity function. We call $y _ { 1 : k } = \{ y _ { 1 } , . . . y _ { k } \}$ negative examples, being other realizations of $Y$ . + +Suppose the two random variables in Eq. $^ 1$ are both transformations of a common random variable $X$ . Let $\tau$ be a family of transformations where each member $t$ is a composition of cropping, color jittering, gaussian blurring, among others $( | \mathrm { W u \ e t \ a l . } | ) { | 2 0 1 8 } ; { | \mathrm { B a c h m a n \ e t \ a l . } | 2 0 1 9 } ; { | \mathrm { C h e n \ e t \ a l . } | } , | \overline { { 2 0 2 0 \mathrm { a } } } )$ We call a transformed input $t ( x )$ a “view” of $x$ . Let $\overline { { p ( t ) } }$ denote a distribution over $\tau$ , a common choice being uniform. Next, introduce an encoder $g _ { \boldsymbol \theta } : X \to \mathbf { S } ^ { n - 1 }$ that maps an example to a $\mathrm { L _ { 2 } }$ -normalized representation. Suppose we have a dataset $\mathcal { D } = \{ x _ { i } \} _ { i = 1 } ^ { n }$ of $n$ values for $X$ sampled from a distribution $p ( x )$ . Then, the contrastive objective for the $i$ -th example is: + +$$ +\mathcal { L } ( \boldsymbol { x } _ { i } ) = \mathbf { E } _ { t , t ^ { \prime } , t _ { 1 : k } \sim p ( t ) } \mathbf { E } _ { \boldsymbol { x } _ { 1 : k } \sim p ( \boldsymbol { x } ) } \left[ \log \frac { e ^ { g _ { \theta } ( t ( \boldsymbol { x } _ { i } ) ) ^ { T } g _ { \theta } ( t ^ { \prime } ( \boldsymbol { x } _ { i } ) ) / \tau } } { \frac { 1 } { k + 1 } \sum _ { j \in \{ i , 1 : k \} } e ^ { g _ { \theta } ( t ( \boldsymbol { x } _ { i } ) ) ^ { T } g _ { \theta } ( t _ { j } ( \boldsymbol { x } _ { j } ) ) / \tau } } \right] +$$ + +where $\tau$ is a temperature. The equivalence of Eq. $\displaystyle { 2 }$ to NCE is immediate given $f _ { \theta } ( x , y ) ~ =$ $g _ { \theta } ( x ) ^ { T } g _ { \theta } ( y ) / \tau$ . Maximizing Eq. $\bigstar$ chooses an embedding that pulls two views of the same example together while pushing two views of distinct examples apart. A drawback to this framework is that the number of negatives $k$ must be large to faithfully approximate the true partition. In practice, $k$ is limited by memory. Recent innovations have focused on tackling this challenge: + +Instance Discrimination $\mathbb { W } \mathrm { u } \mathrm { ~ e t ~ a l . } \backslash \mathbb { Z } 0 1 8 \}$ , or IR, introduces a memory bank of $n$ entries to cache embeddings of each example throughout training. Since every epoch we observe each example once, the memory bank will save the embedding of the view of the $i$ -th example observed last epoch in its $i$ - th entry. Representations stored in the memory bank are removed from the automatic differentiation tape, but in return, we can choose a large $k$ by querying $M$ . A follow up work, Contrastive Multiview Coding $( [ \mathrm { T i a n ~ e t ~ a l . } ] [ 2 0 1 9 )$ , or CMC, decomposes an image into two color modalities. Then, CMC sums two IR losses where the memory banks for each modality are swapped. + +Momentum Contrast $( \mathbb { H } { \mathrm { e ~ e t ~ a l . } } ) , \lfloor 2 0 1 9 \rangle$ , or MoCo, observed that the representations stored in the memory bank grow stale, since possibly thousands of optimization steps pass before updating an entry again. So, MoCo makes two important changes. First, it replaces the memory bank with a first-in first-out (FIFO) queue of size $k$ . During each minibatch, representations are cached into the queue while the most stale ones are removed. Second, MoCo introduces a second (momentum) encoder $g _ { \theta ^ { \prime } } ^ { \prime }$ as a copy of $g _ { \theta }$ . The primary encoder $g _ { \theta }$ is used to embed one view of $x _ { i }$ whereas the momentum encoder is used to embed the other. Again, gradients are not propagated to $g _ { \theta ^ { \prime } } ^ { \prime }$ . + +In this work, we focus on contrastive algorithms that utilize a memory structure that we repurpose in Sec. $\boxed { 4 }$ to efficiently sample hard negatives from. In Sec. $\textcircled { 7 }$ we briefly discuss generalizations to contrastive algorithms that do not use a memory structure. + +# 3 CONDITIONAL NOISE CONTRASTIVE ESTIMATION + +In NCE, the negative examples are sampled i.i.d. from the marginal distribution, $p ( y )$ . Indeed, the existing proof that NCE lower bounds mutual information $\pm { \overline { { \mathrm { ( P o o l e ~ e t ~ a l . } } } } , \pm { \overline { { 2 0 1 9 } } } )$ assumes this to be true. However, choosing negatives in this manner may not be the best choice for learning a good representation. For instance, prior work in metric learning has shown the effectiveness of semihard negative mining in optimizing triplet losses (Wu et al., 2017; Yuan et al., 2017; Schroff et al., $\boxed { 2 0 1 5 }$ . We similarly wish to exploit choosing semi-hard negatives in NCE conditional on the current example but to do so in a manner that preserves the lower bound on mutual information. + +In presenting the theory, we assume two random variables $X$ and $Y$ , deriving a general bound; we will return to the contrastive learning setting in Sec. $\boxed { 4 }$ To begin, in Eq. $\bar { \bigtriangledown }$ suppose we sample negatives from a distribution $q ( y | x )$ conditional on a value $x \sim p ( x )$ rather than the marginal $p ( \bar { y } )$ , which is independent of $X$ . Ideally, we would like to freely choose $q ( y | x )$ to be any distribution but not all choices preserve a bound on mutual information1. This does not, however, imply that we can only sample negatives from $\upsilon ( y ) ~ ( { \mathrm { P o o l e ~ e t ~ a l . } } ) ~ [ { \underline { { { 2 0 1 9 } } } } \vert ~ [ { \mathrm { O o r d ~ e t ~ a l . } } ] [ { \underline { { { 2 0 1 8 } } } } )$ . One of our contributions is to formally define a family of conditional distributions $\mathcal { Q }$ such that for any $q ( y | x ) \in \mathcal { Q }$ , drawing negative examples from $q$ defines an estimator that lower bounds $\mathcal { I } ( X ; Y )$ . We call this new bound the Conditional Noise Contrastive Estimator, or CNCE. We first prove CNCE to be a bound: + +Theorem 3.1. (The Conditional NCE bound) Define $d$ -dimensional random variables $X$ and $Y$ by a joint distribution $p ( x , y )$ and let $Y _ { 1 } , . . . , Y _ { k }$ be i.i.d. copies of $Y$ with the marginal distribution $p ( y )$ . Fix any function $f : ( X , Y ) \to \mathbf { R }$ , any realization $x$ of $X$ , and let $c = \mathbf { E } _ { y \sim p ( y ) } [ e ^ { f ( x , y ) } ] ,$ , the expected exponentiated similarity. Pick a set $B \subset \mathbf { R }$ strictly lower-bounded by c. Assume the pulled back set $S _ { B } \sp { \bullet } = \{ y | e \sp { f ( x , y ) } \in B \}$ has non-zero probability (i.e. $p ( S _ { B } ) > 0 ,$ ). For $A _ { 1 } , \ldots , A _ { k }$ in the Borel $\sigma$ -algebra over $\mathbf { R } ^ { d }$ , define $A = A _ { 1 } \times \ldots \times A _ { k }$ and let + +$$ +q ( ( Y _ { 1 } , \dots , Y _ { k } ) \in A | X = x ) = \prod _ { j = 1 } ^ { k } p ( A _ { j } | S _ { B } ) . +$$ + +Proof. To show $\mathcal { T } _ { \mathrm { C N C E } } \leq \mathcal { T } _ { \mathrm { N C E } }$ , we show $\begin{array} { r } { { \bf E } _ { p } [ \log \sum _ { j = 1 } ^ { k } e ^ { f ( x , y _ { j } ) } ] < { \bf E } _ { q } [ \log \sum _ { j = 1 } ^ { k } e ^ { f ( x , y _ { j } ) } ] } \end{array}$ . To see this, apply Jensen’s to the left-hand side of $\begin{array} { r } { \log \mathbf { E } _ { p } [ \sum _ { j = 1 } ^ { k } e ^ { f ( x , y _ { j } ) } ] < \log \sum _ { j = 1 } ^ { k } e ^ { f ( x , y _ { j } ) } } \end{array}$ , which holds if $y _ { j } \in S _ { B }$ for $j = 1 , \dotsc , k$ , and then take the expectation $\mathbf { E } _ { q }$ of both sides. The last inequality holds by monoticity of log, linearity of expectation, and the fact that ${ \bf E } _ { p } [ e ^ { f ( x , y _ { j } ) } ] \le e ^ { f ( x , y _ { j } ) }$ . + +Theorem Intuition. For intuition, although using arbitrary negative distributions in NCE does not bound mutual information, we have found a restricted class of distributions $\mathcal { Q }$ where every member $q ( y | x )$ “subsets the support” of the distribution $p ( y )$ . That is, given some fixed value $x$ , we have defined $q ( y | x )$ to constrain the support of $p ( y )$ to a set $S _ { B }$ whose members are “close” to $x$ as measured by the similarity function $f$ . For every element $y \in S _ { B }$ , the distribution $q ( y | x )$ wants to assign to it the same probability as $p ( y )$ . However, as $q ( y | x )$ is not defined outside of $S _ { B }$ , we must renormalize it to sum to one (hence $\begin{array} { r } { p ( A _ { j } | S _ { B } ) = \frac { p ( A _ { j } \cap S _ { B } ) } { p ( S _ { B } ) } ) } \end{array}$ . Intuitively, $q ( y | x )$ cannot change $p ( y )$ too much: it must redistribute mass proportionally. The primary distinction then, is the smaller support of $q ( y | x )$ , which forces samples from it to be harder for $f$ to distinguish from $x$ . Thm. 3.1 shows that substituting $q ( y | x )$ for $p ( y )$ in NCE still bounds mutual information. + +![](images/5cd68e2041e12a3bc85e25a2911d9fb4c19f8d06288b74bef3503bf2086665ed.jpg) +Figure 1: Visual illustration of Ring Discrimination. Black: view of example $x _ { i }$ ; gray: second view of $x _ { i }$ ; red: negative samples; gray area: distribution $q ( x | t ( x _ { i } ) )$ . In subfigure (c), the negative samples are annealed to be closer to $t ( x _ { i } )$ through training. In other words, the support of $q$ shrinks. + +Theorem Example 3.1. We give a concrete example for the choice $B$ that will be used in Sec. 4. For any realization $x$ , suppose we define two similarity thresholds $\omega _ { \ell } , \omega _ { u } \in \mathbf { R }$ where $c < \omega _ { \ell } < \omega _ { u }$ . Then, choose $B = [ w _ { \ell } , w _ { u } ]$ . In this case, the set $S _ { B }$ , which defines the support of the distribution $q ( y | x )$ , contains values of $y$ that are not “too-close” to $x$ but not “too-far”. In contrastive learning, we might pick these similarity thresholds to vary the difficulty of negative samples. + +Interestingly, Thm. $\boxed { 3 . 1 }$ states that CNCE is looser than NCE, which raises the question: when is a looser bound useful? In reply, we show that while CNCE is a more biased estimator than NCE, in return it has lower variance. Intuitively, because $q ( y | x )$ is the result of restricting $p ( y )$ to a smaller support, samples from $q ( y | x )$ have less opportunity to deviate, hence lower variance. Formally: + +Theorem 3.2. (Bias and Variance Tradeoff) Pick any $x , y \sim p ( x , y )$ . Fix the distribution $q ( y _ { 1 : k } | x )$ as stated in Theorem 3.1 Define a new random variable $\begin{array} { r } { Z ( y _ { 1 : k } ) = \log \left( \frac { e ^ { f ( x , y ) } } { \frac { 1 } { k } \sum _ { j = 1 } ^ { k } e ^ { f ( x , y _ { j } ) } } \right) } \end{array}$ representing the normalized similarity. By Theorem $\boxed { 3 . 1 }$ the expressions ${ \bf E } _ { p ( y _ { 1 : k } ) } [ Z ]$ and ${ \mathbf { E } } _ { q \left( y _ { 1 : k } \mid x \right) } [ Z ]$ are estimators for $\mathcal { I } ( X ; Y )$ . Suppose that the set $S _ { B }$ is chosen to ensure $W a r _ { q ( y _ { 1 : k } | x ) } [ Z ] \ \leq$ $V a r _ { \tilde { q } ( y _ { 1 : k } | x ) } [ Z ]$ , where $\tilde { q } ( A ) ~ = ~ p ( A |$ complement of $S _ { B }$ ). That is, we assume the variance of the normalized similarity when using $y _ { 1 : k } ~ \in ~ S _ { B }$ is smaller than when using $y _ { 1 : k } ~ \notin ~ S _ { B }$ . Then $\mathrm { B i a s } _ { p ( y _ { 1 : k } ) } ( Z ) \leq \mathrm { B i a s } _ { q ( y _ { 1 : k } | x ) } ( Z )$ and $\operatorname { V a r } _ { p ( y _ { 1 : k } ) } ( Z ) \geq \operatorname { V a r } _ { q ( y _ { 1 : k } | x ) } ( Z )$ . + +The proof can be found in $\mathrm { S e c . } \ \mathrm { A . } 2 . \ \mathrm { T h m . } \ 3 . 2$ provides one answer to our question of looseness. In stochastic optimization, a lower variance objective may lead to better local optima. For representation learning, using CNCE to sample more difficult negatives may (1) encourage the representation to distinguish fine-grained features useful in transfer tasks, and (2) provide less noisy gradients. + +# 4 RING DISCRIMINATION + +We have shown CNCE to be a new bound on the mutual information that uses hard negative samples. Now we wish to apply CNCE to contrastive learning where the two random variables are again transformations of a single variable $X$ . In this setting, for a fixed $x _ { i } \sim p ( x )$ , the CNCE distribution is written as $q ( x | t ( x _ { i } ) )$ for some transform $t \in \mathcal { T }$ . Samples from $x \stackrel { \cdot \cdot } { \sim } q ( x | t ( x _ { i } ) )$ will be such that the exponentiated distance, $\exp \{ g _ { \theta } ( t ( x _ { i } ) ) ^ { T } g _ { \theta } ( t ^ { \prime } ( x ) ) \}$ , is at least a minimum value $c$ . As in Example 3.1, we will choose $B = [ \omega _ { \ell } , \omega _ { u } ]$ , a closed interval in $\mathbf { R }$ defined by two thresholds. + +Picking thresholds. We pick the thresholds conditioned on the $i$ -th example in the dataset, hence each example has a different set $B$ . We first describe how to pick the upper threshold $\omega _ { u }$ . Given the $i$ -example $x _ { i }$ , we pick a number $u \in [ 0 , 1 0 0 ]$ representing an upper “percentile”. We consider each example $x$ in the dataset to be in the support $S _ { B }$ if and only if the (exponentiated) distance between the embedding of $x _ { i }$ and $x$ , or $\exp \{ \bar { g _ { \boldsymbol { \theta } } } ( t ( x _ { i } ) ) ^ { T } g _ { \boldsymbol { \theta } } ( t ^ { \prime } ( x ) ) \}$ , is below the $u$ -th percentile for all $x \in \mathcal { D }$ . Call this maximum distance $\omega _ { u }$ . In other words, we construct $q ( x | t ( x _ { i } ) )$ such that we ignore examples from the dataset whose embedding dot producted with the embedding of $x _ { i }$ is above $\omega _ { u }$ . (Note that $u = 1 0 0$ recovers NCE.) For a small enough choice of $u$ , the upper similarity threshold $\omega _ { u }$ will be greater than $c$ (defined in Thm. $3 . 1$ as the expected distance with respect to $p ( x ) )$ ), and the samples from $q ( x | t ( x _ { i } ) )$ will be harder negatives to discriminate from $x _ { i }$ . + +In picking the lower threshold $\omega _ { \ell }$ , one could choose it to be 0, so $B = [ 0 , \omega _ { u } )$ . However, picking the closest examples to $t ( x _ { i } )$ as its negative examples may be inappropriate, as these examples might be better suited as positive views rather than negatives (Zhuang et al., 2019; Xie et al., 2020). As an extreme case, if the same image is included in the dataset twice, we would not like to select it as a negative example for itself. Furthermore, choosing negatives “too close” to the current instance may result in representations that pick up on fine-grain details only, ignoring larger semantic concepts. This suggests removing examples from $q ( x | t ( x _ { i } ) )$ we consider “too close” to $x _ { i }$ . To do this, we pick a lower percentile $0 \leq \ell < u$ . For each example $x \in \mathcal { D }$ , we say it is in $S _ { B }$ if $\exp \{ g _ { \theta } ( t ( x _ { i } ) ) ^ { T } g _ { \theta } ( t ^ { \prime } ( x ) ) \}$ is below $\omega _ { u }$ and also if it is above the $\ell$ -th percentile of all distances with respect to $\mathcal { D }$ . Call this minimum distance $\omega _ { \ell }$ . Fig. 2 visualizes this whole procedure. + +![](images/b9475398b61f033925ccb174c0553ebb227c591ac73f17496792590fbf7e7ce0.jpg) +Figure 2: Defining the CNCE distribution $q ( x | t ( x _ { i } ) )$ . By choosing a lower and upper percentile $\ell$ and $u$ , we implicitly define similarity thresholds $\omega _ { \ell }$ and $\omega _ { u }$ to construct a support of valid negative examples, $S _ { B }$ , which in turn, defines the distribution $q \big ( \boldsymbol { x } | t ( \boldsymbol { x } _ { i } ) \big )$ . + +# Algorithm 1: MoCoRing + +# g q , g k : e n c o d e r n e t w o r k s +# m: momentum ; t : t e m p e r a t u r e +# u : r i n g u p p e r p e r c e n t i l e +# l : r i n g l o w e r p e r c e n t i l e +$\operatorname { t x } 1 = \operatorname { a u g } \left( \mathbf { x } \right)$ # random a u g m e n t a t i o n +t x 2 = aug ( x ) +emb1=norm ( g q ( t x 1 ) ) +emb2=norm ( g k ( t x 2 ) ) . d e t a c h ( ) +d p s =sum ( t x 1 t x 2 ) / t # d o t p r o d u c t +# s o r t f r o m c l o s e s t t o f a r t h e s t n e g +a l l $\mathrm { | } \mathbf { p } \mathbf { s } = \mathbf { s } \mathbf { o } \mathbf { r } \mathbf { t }$ ( emb1@queue . T / t ) +# f i n d i n d i c e s o f t h r e s h o l d s +i x ${ \bf \Pi } _ { - } 1 = 1 * 1 \mathbf { e } \mathbf { n }$ ( q u e u e ) +i x $\mathbf { u = u * l e n }$ ( q u e u e ) +r i n g d p $s =$ a l l d p s [ : , i x l : i x u ] +# n o n p a r a m e t r i c s o f t m a x +l o s s=dps + logsumexp ( r i n g d p s ) +l o s s . b a c k w a r d ( ) +s t e p ( g q . p a r a m s ) +# moco u p d a t e s +g k . params $\mathbf { \mu } = \mathbf { \mu } _ { \mathbf { m } * \mathbf { g } } \mathbf { \Lambda } _ { - } \mathbf { k }$ . p a r a m s + $( 1 - \mathrm { m } ) * .$ g q . p a r a m s +enqueue ( queue , emb2 ) ; dequeue ( queue ) +# t h r e s h o l d u p d a t e s +a n n e a l ( w l ) ; a n n e a l ( w u ) + +Ring Discrimination. Having defined $\omega _ { \ell }$ and $\omega _ { u }$ , we have a practical method of choosing $B$ , and thus $S _ { B }$ to define $q ( \bar { x } | t ( x _ { i } ) )$ for $i$ -th example. Intuitively, we construct a conditional distribution for negative examples that are (1) not too easy since their representations are fairly similar to that of $x _ { i }$ and (2) not too hard since we remove the “closest” instances to $x _ { i }$ from $S _ { B }$ . We call this algorithm Ring Discrimination, or Ring, inspired by the shape of negative set (see Fig. 1). + +Ring can be easily added to popular contrastive algorithms. For IR and CMC, this amounts to simply sampling entries in the memory bank that fall within the $\ell$ -th to $u$ -th percentile of all distances to the current example view (in representation space). Similarly, for MoCo, we sample from a subset of the queue (chosen to be in the $\ell$ -th to $u$ -th percentile), preserving the FIFO ordering. In our experiments, we refer to these as IRing, CMCRing, MoCoRing, respectively. Alg. $\perp$ shows PyTorch-like pseudocode for MoCoRing. One of the strengths of this approach is the simplicity: the algorithm requires only a few lines of code on top of existing implementations. + +Annealing Policy. Naively using hard negatives can collapse to a poor representation, especially if we choose the upper threshold, $\omega _ { u }$ , to be very small early in training. At the start of training, the encoder $g _ { \theta }$ is randomly initialized and cannot guarantee that elements in the $\ell$ -th to $u$ -th percentile are properly calibrated: if the representations are near random, choosing negatives that are close in embedding distance may detrimentally exclude those examples that are “actually” close. This could lock in poor local minima. To avoid this, we propose to use an annealing policy that reduces $\omega _ { u }$ (and thus the size of the support $S _ { B }$ ) throughout training. Early in training we choose $\omega _ { u }$ to be large. Over many epochs, we slowly decrease $\omega _ { u }$ closer to $\omega _ { l }$ , thereby selecting more difficult negatives. We explored several annealing policies and found a linear schedule to be well-performing and simple (see Sec. $\mathbf { G } )$ . In our experiments, annealing is shown to be crucial: being too aggressive with negatives early in training produced representations that performed poorly on downstream tasks. + +# 5 EXPERIMENTS + +We explore our method applied to IR, CMC, and MoCo in four commonly used visual datasets. As in prior work (Wu et al., 2018; Zhuang et al., 2019; He et al., 2019; Misra & Maaten, 2020; Henaff ´ et al., 2019; Kolesnikov et al., 2019; Donahue & Simonyan, 2019; Bachman et al., 2019; Tian et al., 2019; Chen et al., 2020a), we evaluate each method by linear classification on frozen embeddings. That is, we optimize a contrastive objective on a pretraining dataset to learn a representation; then, using a transfer dataset, we fit logistic regression on representations only. A better representation would contain more “object-centric” information, thereby achieving a higher classification score. + +Training Details. We pick the upper percentile $u = 1 0$ and the lower percentile $\ell = 1$ although we anneal $u$ starting from 100. We resize input images to be 256 by 256 pixels, and normalize them using dataset mean and standard deviation. The temperature $\tau$ is set to 0.07. We use a composition of a 224 by 224-pixel random crop, random color jittering, random horizontal flip, and random grayscale conversion as our augmentation family $\tau$ . We use a ResNet-18 encoder with a output dimension of 128. For CMC, we use two ResNet-18 encoders, doubling the number of parameters. For linear classification, we treat the pre-pool output (size $5 1 2 \times 7 \times 7 )$ after the last convolutional layer as the input to the logistic regression. Note that this setup is equivalent to using a linear projection head $( \mathbf { \overline { { C h e n \ e t \ a l . } } } ) , \mathbf { \overline { { 2 0 2 0 a } } } , \mathbf { \overline { { b } } } )$ . In pretraining, we use SGD with learning rate 0.03, momentum 0.9 and weight decay 1e-4 for 300 epochs and batch size 256 (128 for CMC). We drop the learning rate twice by a factor of 10 on epochs 200 and 250. In transfer, we use SGD with learning rate 0.01, momentum 0.9, and no weight decay for 100 epochs without dropping learning rate. These hyperparameters were taken from $\boxed { \mathrm { W u \ e t \ a l . } } \textcircled { 2 0 1 8 }$ and used in all of Table $\bar { 1 }$ for a consistent comparison. We found normalizing hyperparameters to be important for a fair comparison as many competing algorithms use different hyperparameters. For a state-of-the-art comparison, see Table 5. + +
ModelLinear EvaluationModelLinear EvaluationModelLinear EvaluationModelLinear Evaluation
IR81.2IR60.4IR61.4IR43.2
IRing83.9 (+2.7)IRing62.3 (+1.9)IRing64.3 (+2.9)IRing48.4 (+5.2)
CMC*85.6CMC*56.0CMC*63.8CMC*48.2
CMCRing*87.6 (+2.0)CMCRing*56.0 (+0.0)CMCRing*66.4 (+2.6)CMCRing*50.4 (+2.2)
MoCo83.1MoCo59.1MoCo63.8MoCo52.8
MoCoRing86.1 (+3.0)MoCoRing61.5 (+2.4)MoCoRing65.2 (+1.4)MoCoRing54.6 (+1.8)
LA83.9LA61.4LA63.0LA48.0
+ +(a) CIFAR10 (b) CIFAR100 (c) STL10 (d) ImageNet + +Table 1: Comparison of contrastive algorithms on four image domains. Superscript $( ^ { * } )$ indicates models that use twice as many parameters as others e.g. CMC has “L” and “ab” encoders. + +The results for CIFAR10, CIFAR100, STL10, and ImageNet are in Table $\bigstar$ Overall, IR, CMC, and MoCo all benefit from using more difficult negatives as shown by $2 - 5 \%$ absolute points of improvement across the four datasets. While we find different contrastive objectives to perform best in each dataset, the improvements from Ring are consistent: the Ring variant outperforms the base for every model and every dataset. We also include as a baseline Local Aggregation, or LA (Zhuang et al., 2019), a popular contrastive algorithm (see ${ \mathrm { S e c . } } \mathrm { H } )$ that implicitly uses hard negatives without annealing. We find our methods to outperform LA by up to $4 \%$ absolute. + +Ablations: Annealing and Upper Boundary. Having found good performance with Ring Discrimination, we want to assess the importance of the individual components that comprise Ring. We focus on the annealing policy and the exclusion of very close negatives from $S _ { B }$ . Concretely, we measure the transfer accuracy of (1) IRing without annealing and (2) IRing with an lower percentile $\ell = 0$ , thereby excluding no close negatives. That is, $S _ { B }$ contains all examples in the dataset with representation similarity less than the $\omega _ { u }$ (a “ball” instead of a “ring”). Table $2$ compares these ablations to IR and full IRing on CIFAR10 and ImageNet classification transfer. We observe that both ablations result in worse transfer accuracy, with proper annealing being especially important to prevent convergence to bad minima. We also find even with $\ell = 0$ , IRing outperforms IR, suggesting both removing negatives that are “too close” and “too far” contribute to the improved representation quality. + +Table 2: Lesioning the effects of annealing and choice of $\ell$ . + +
ModelLinear Eval.
IR81.2
IRing83.9
IRing (No Anneal)81.4
IRing(l=0)82.1
(a) CIFAR10
ModelLinear Eval.
IR43.2
IRing48.4
IRing (No Anneal)41.3
IRing(e=0)47.3
+ +(b) ImageNet + +Transferring Features. Thus far we have only evaluated the learned representations on unseen examples from the training distribution. As the goal of unsupervised learning is to capture general representations, we are also interested in their performance on new, unseen distributions. To gauge this, we use the same linear classification paradigm on a suite of image datasets from the “Meta Dataset” collection (Triantafillou et al., 2019) that have been used before in contrastive literature (Chen et al., 2020a). All representations were trained on CIFAR10. For each transfer dataset, we compute mean and variance from a training split to normalize input images, which we found important for generalization to new visual domains. + +Table 3: Transferring CIFAR10 embeddings to various image distributions. + +
ModelAircraftCUBirdsDTDFungiMNISTFashionMNISTTrafficSignVGGFlowerMSCOCO
IR40.917.939.22.796.991.797.168.152.4
IRing40.6 (-0.3)17.9 (+0.0)39.5 (+0.3)3.4(+0.7)97.8 (+0.9)91.6 (+0.1)98.8 (+1.7)68.5 (+0.4)52.5 (+0.1)
MoCo41.518.039.73.196.990.997.364.552.0
MoCoRing41.6(+0.1)18.6 (+0.6)39.5 (-0.2)3.6(+0.5)97.9 (+1.0)91.3 (+0.4)99.3 (+2.0)69.1 (+4.6)52.6 (+0.6)
CMC40.115.838.34.397.591.594.667.151.4
CMCRing40.8 (+0.7)16.8 (+1.0)40.6(+2.3)4.2 (-0.1)97.9 (+0.4)92.1 (+0.6)97.1( (+2.5)69.1 (+2.0)52.1 (+0.7)
LA41.317.839.02.397.292.398.266.952.3
+ +We find in Table $\textcircled { 3 }$ that the Ring models are competitive with the non-Ring analogues, with increases in transfer accuracies of 0.5 to $2 \%$ absolute. Most notable are the TrafficSign and VGGFlower datasets in which Ring models surpass others by a larger margin. We also observe that IRing largely outperforms LA. This suggests the features learned with more difficult negatives are not only useful for the training distribution but may also be transferrable to many visual datasets. + +More Downstream Tasks. Object classification is a popular transfer task, but we want our learned representations to capture holistic knowledge about the contents of an image. We must thus evaluate performance on transfer tasks such as detection and segmentation that require different kinds of visual information. We study four additional downstream tasks: object detection on COCO (Lin et al., $\boxed { 2 0 1 4 }$ and Pascal VOC’07 (Everingham et al., 2010), instance segmentation on COCO, and keypoint detection on COCO. In all cases, we employ embeddings trained on ImageNet with a ResNet-18 encoder. We base these experiments after those found in $\boxed { \mathrm { H e ~ e t ~ a l . } } \textcircled { 2 0 1 9 }$ with the same hyperparameters. However, we use a smaller backbone (ResNet-18 versus ResNet-50) and we freeze its parameters instead of finetuning them. We adapt code from Detectron2 (Wu et al., 2019). + +Table 4: Evaluation of ImageNet representations using four visual transfer tasks. + +
COCO: Object DetectionCOCO: Inst. SegmentationCOCO: Keypoint DetectionVOC: Object Detection
Arch.Mask R-CNN,R18-FPN,1x scheduleR-CNN,R18-FPNFaster R-CNN, R18-C4
ModelApbbAPAPApmkAPAPApkpAP5APApbbAPAP
IR8.619.06.68.517.47.434.663.032.95.514.53.3
IRing10.922.98.711.020.99.637.266.135.77.620.34.4
MoCo6.014.34.010.821.49.737.666.536.97.317.94.1
MoCoRing9.420.37.612.022.910.838.767.737.98.022.14.8
LA10.222.08.110.020.39.036.365.335.17.620.04.3
+ +We find IRing outperforms IR by around 2.3 points in COCO object detection, 2.5 points in COCO Instance Segmentation, 2.6 points in COCO keypoint detection, and 2.1 points in VOC object detection. Similarly, MoCoRing finds consistent improvements of 1-3 points over MoCo on the four tasks. Future work can investigate orthogonal directions of using larger encoders (e.g. ResNet-50) and finetuning ResNet parameters for these individual tasks. + +# 6 RELATED WORK + +Several of the ideas in Ring Discrimination relate to existing work. Below, we explore these connections, and at the same time, place our work in a fast-paced and growing field. + +Hard negative mining. While it has not been deeply explored in modern contrastive learning, negative mining has a rich line of research in the metric learning community. Deep metric learning utilizes triplet objectives of the form ${ \mathcal { L } } _ { \mathrm { t r i p l e t } } = d ( g _ { \theta } ( x _ { i } ) , g _ { \theta } ( x _ { + } ) ) - d ( g _ { \theta } ( x _ { i } ) , g _ { \theta } ( x _ { - } ) + \alpha )$ where $d$ is a distance function (e.g. $\mathrm { L _ { 2 } }$ distance), $x _ { + }$ and $x _ { - }$ are a positive and negative example, respectively, relative to $x _ { i }$ , the current instance, and $\alpha \in \mathbf { R } ^ { + }$ is a margin. In this context, several approaches pick semi-hard negatives: Schroff et al. $\textcircled { | 2 0 1 5 } )$ treats the furthest (in $\mathrm { L _ { 2 } }$ distance) example in the same minibatch as $x _ { i }$ as its negative, whereas Oh Song et al. $\underline { { \left( \bar { 2 } 0 1 6 \right) } }$ weight each example in the minibatch by its distance to $g _ { \boldsymbol { \theta } } ( x _ { i } )$ , thereby being a continuous version of $\boxed { \mathrm { S c h r o f f ~ e t ~ a l . } } \textcircled { 2 0 1 5 } )$ . More sophisticated negative sampling strategies developed over time. $\mathrm { I n } [ \mathrm { W u \ e t \ a l . } ] \mathbb { ( } 2 0 1 7 \mathrm { ) }$ , the authors pick negatives from a fixed normal distribution that is shown to approximate $\mathrm { L _ { 2 } }$ normalized embeddings in high dimensions. The authors show that weighting by this distribution samples more diverse negatives. Similarly, HDC $( \mathbf { Y } \mathbf { u } \mathbf { a } \mathbf { n } \operatorname { e t } \mathbf { a } \mathbf { l } . ) , [ 2 0 1 7 ]$ simulataneously optimizes a triplet loss using many levels of “hardness” in negatives, again improving the diversity. Although triplet objectives paved the way for modern NCE-based objectives, the focus on negative mining has largely been overlooked. Ring Discrimination, being inspired by the deep metric learning literature, reminds that negative sampling is still an effective way of learning stronger representations in the new NCE framework. As such, an important contribution was to do so while retaining the theoretical properties of NCE, namely in relation to mutual information. This, to the best of our knowledge, is novel as negative mining in metric learning literature was not characterized in terms of information theory. + +That being said, there are some cases of negative mining in contrastive literature. In CPC (Oord et al., 2018), the authors explore using negatives from the same speaker versus from mixed speakers in audio applications, the former of which can be interpreted as being more difficult. A recent paper, InterCLR (Xie et al., 2020), also finds that using “semi-hard negatives” is beneficial to contrastive learning whereas negatives that are too difficult or too easy produce worse representations. Where InterCLR uses a margin-based approach to sample negatives, we explore a wider family of negative distributions and show analysis that annealing offers a simple and easy solution to choosing between easy and hard negatives. Further, as InterCLR’s negative sampling procedure is a special case of CNCE, we provide theory grounding these approaches in information theory. Finally, a separate line of work in contrastive learning explores using neighboring examples (in embedding space) as “positive” views of the instance (Zhuang et al., 2019; Xie et al., 2020; Asano et al., 2019; Caron et al., $\boxed { 2 0 2 0 }$ Li et al., 2020). That is, finding a set $\overline { { \{ x _ { j } \} } }$ such that we consider ${ \overline { { x _ { j } \ } } } = { \overline { { t ( x _ { i } ) } } }$ for the current instance $x _ { i }$ . While this does not deal with negatives explicitly, it shares similarities to our approach by employing other examples in the contrastive objective to learn better representations. In the Appendix, we discuss how one of these algorithms, LA (Zhuang et al., 2019), implicitly uses hard negatives and expand the Ring family with ideas inspired by it. + +Contrastive learning. We focused primarily on comparing Ring Discrimination to three recent and highly performing contrastive algorithms, but the field contains much more. The basic idea of learning representations to be invariant under a family of transformations is an old one, having been explored with self-organizing maps $\scriptstyle ( \mathbf { B e c k e r } \ \& \ \mathbf { H i n t o n } , \mathbf { \left[ 1 9 9 2 \right] } )$ and dimensionality reduction (Hadsell et al., 2006). Before IR, the idea of instance discrimination was studied (Dosovitskiy et al., 2014; Wang & Gupta, $\underline { { 2 0 1 5 } } )$ among many pretext objectives such as position prediction (Doersch et al., 2015), color prediction (Zhang et al., $\boxed { 2 0 1 6 }$ , multi-task objectives (Doersch & Zisserman, 2017), rotation prediction (Gidaris et al., 2018; Chen et al., 2019), and many other “pretext” objectives (Pathak et al., 2017). As we have mentioned, one of the primary challenges to instance discrimination is making such a large softmax objective tractable. Moving from a parametric (Dosovitskiy et al., $\boxed { 2 0 1 4 }$ to a nonparametric softmax reduced issues with vanishing gradients, shifting the challenge to efficient negative sampling. The memory bank approach $\mathbb { W } \mathrm { u } \mathrm { ~ e t ~ a l . } \lVert \mathbb { Z } \mathrm { 0 . 8 } \rVert$ is a simple and memory-efficient solution, quickly being adopted by the research community (Zhuang et al., 2019; Tian et al., 2019; He et al., 2019; Chen et al., 2020b; Misra & Maaten, 2020). With enough computational resources, it is now also possible to reuse examples in a large minibatch and negatives of one another (Ye et al., 2019; Ji et al., 2019; Chen et al., 2020a). In our work, we focus on hard negative mining in the context of a memory bank or queue due to its computational efficiency. However, the same principles should be applicable to batch-based methods (e.g. SimCLR): assuming a large enough batch size, for each example, we only use a subset of the minibatch as negatives as in Ring. Finally, more recent work (Grill et al., 2020) removes negatives altogether, which is speculated to implicitly use negative samples via batch normalization (Ioffe & Szegedy, 2015); we leave a more thorough understanding of negatives in this setting to future work. + +# 7 DISCUSSION + +Computational cost of Ring. To measure the cost of CNCE, we compare the cost an epoch of training MoCo/IR versus MoCoRing/IRing on four image datasets. Table 5a reports the average cost over 200 epochs. We observe that Ring models cost no more than 1.5 times the cost of standard contrastive algorithms, amounting to a difference of 3 to 7 minutes in ImageNet and 10 to 60 seconds in three other datasets per epoch. In the context of deep learning, we do not find the cost increases to be substantial. In particular, since (1) the memory structure in IR and MoCo allow us to store and reuse embeddings and (2) gradients are not propagated through the memory structure, the additional compute of Ring amounts to one matrix multiplication, which is cheap on modern hardware. We used a single Titan X GPU with 8 CPU workers, and PyTorch Lightning (Falcon et al., 2019). + +
ModelCIFAR10 (sec.)ImageNet (min.)
IR136.0 ±443.9 ±1
IRing141.1 ± 5 (1.1x)51.0±1 (1.2x)
MoCo318.4±1661.1 ±1
MoCoRing383.4 ± 12 (1.2x)64.9 ±1 (1.1x)
+ +(b) Comparison with SOTA + +
DatasetArch.MoCo-v2MoCoRing-v2
CIFAR10ResNet-1890.191.9(+1.8)
CIFAR10ResNet-5092.494.1(+1.6)
CIFAR100ResNet-1865.167.3(+2.2)
STL10ResNet-1874.876.7(+1.9)
+ +(a) Average Epoch Cost + +
Transfer TaskMoCoMoCoRing
LibriSpeech Spk.ID (Panayotov et al.201595.596.6(+1.1)
AudioMNISTBecker et al. 2018)87.491.3(+3.9)
Google CommandsWarden 1201838.541.4(+2.9)
Fluent Actions Lugosch et al.] 201936.536.8(+0.3)
Fluent ObjectsLugosch et al. 1201941.944.1(+2.2)
Fluent Locations Lugoschet al. 1201960.963.9(+3.0)
+ +(c) Speech Extension + +Table 5: Generalizations of Ring to a new modality (a) and a batch-based algorithm (b). + +
DatasetSimCLRSimCLRing
CIFAR1088.989.3(+0.4)
CIFAR10063.564.1(+0.6)
STL1071.272.1(+0.9)
+ +(d) SimCLRing Extension + +Comparison with the state-of-the-art. Unlike the experiments in Sec. 5, we now choose the optimal hyperparameters for MoCo-v2 (Chen et al., 2020b) separately for CIFAR10, CIFAR100, and STL10. Table $5 \mathsf { b }$ compares MoCo-v2 and its CNCE equivalent, MoCoRing-v2 using linear evaluation. We observe comparable improvements as found in Table 1 even with optimal hyperparameters. Notably, the gains generalize to ResNet-50 encoders. Refer to Sec. F for hyperparameter choices. + +Generalization to other modalities. Thus far, we have focused on visual representation learning, although the same ideas apply to other domains. To exemplify the generality of CNCE, we apply MoCoRing to learning speech representations. Table $5 \mathrm { { c } }$ reports linear evaluation on six transfer datasets, ranging from predicting speaker identity to speech recognition to intent prediction. We find significant gains of 1 to 4 percent over 4 datasets and 6 transfer tasks with an average of 2.2 absolute percentage points. See Sec. $\mathrm { ~ E ~ }$ for experimental details. + +Batch-based negative sampling. In Ring, we assumed to have a memory structure that stores embeddings, which led to an efficient procedure of mining semi-hard negatives. However, another flavor of contrastive algorithms removes the memory structure entirely, using the examples in the minibatch as negatives of one another. Here, we motivate a possible extension of Ring to SimCLR, and leave more careful study to future work. In SimCLR, we are given a minibatch $M$ of examples. To sample hard negatives, as before, pick $\ell$ and $u$ as lower and upper percentiles. For every example $x _ { i }$ in the minibatch, only consider the subset of the minibatch $\{ x : x \subseteq M , \exp \{ g _ { \theta } ( t ( x _ { i } ) ) ^ { \cdot } g _ { \theta } ( t ^ { \prime } ( x ) ) \}$ in the $\ell \cdot$ -th and $u$ -th percentiles in $M \}$ as negative examples for $x _ { i }$ . This can be efficiently implemented as a matrix operation using an element-wise mask. Thus, we ignore gradient signal for examples too far or too close to $x _ { i }$ in representation. As before, we anneal $u$ from 100 to 10 and set $\ell = 1$ . Table $\textcircled { 5 } \textcircled { \scriptsize { \mathrm { d } } }$ report consistent but moderate gains over SimCLR, showing promise but room for improvement in future research. + +# 8 CONCLUDING REMARKS + +To conclude, we presented a family of mutual information estimators that approximate the partition function using samples from a class of conditional distributions. We proved several theoretical statements about this family, showing a bound on mutual information and a tradeoff between bias and variance. Then, we applied these estimators as objectives in contrastive representation learning. In doing so, we found that our representations outperform existing approaches consistently across a spectrum of contrastive objectives, data distributions, and transfer tasks. Overall, we hope our work to encourage more exploration of negative sampling in the recent growth of contrastive learning. + +# ACKNOWLEDGMENTS + +This research was supported by the Office of Naval Research grant ONR MURI N00014-16-1-2007. +MW is supported by the Stanford Interdisciplinary Graduate Fellowship as the Karr Family Fellow. + +# REFERENCES + +Yuki Markus Asano, Christian Rupprecht, and Andrea Vedaldi. Self-labelling via simultaneous clustering and representation learning. arXiv preprint arXiv:1911.05371, 2019. + +Philip Bachman, R Devon Hjelm, and William Buchwalter. 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In Proceedings of the IEEE International Conference on Computer Vision, pp. 6002–6012, 2019. \ No newline at end of file diff --git a/parse/train/v8b3e5jN66j/v8b3e5jN66j_content_list.json b/parse/train/v8b3e5jN66j/v8b3e5jN66j_content_list.json new file mode 100644 index 0000000000000000000000000000000000000000..64c500b06cf1dd6cbf1ea8c40a5488dd210fa0f6 --- /dev/null +++ b/parse/train/v8b3e5jN66j/v8b3e5jN66j_content_list.json @@ -0,0 +1,1536 @@ +[ + { + "type": "text", + "text": "CONDITIONAL NEGATIVE SAMPLING FOR CONTRASTIVE LEARNING OF VISUAL REPRESENTATIONS ", + "text_level": 1, + "bbox": [ + 176, + 98, + 821, + 146 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Mike $\\mathbf { W } \\mathbf { u } ^ { 1 }$ , Milan Mosse1,3, Chengxu Zhuang2, Daniel Yamins1,2, Noah Goodman $^ { 1 , 2 }$ Department of Computer Science1, Psychology2, and Philosophy3 Stanford University {wumike, chengxuz, mmosse19, yamins, ngoodman}@stanford.edu ", + "bbox": [ + 184, + 169, + 772, + 227 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "ABSTRACT ", + "text_level": 1, + "bbox": [ + 454, + 263, + 544, + 279 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Recent methods for learning unsupervised visual representations, dubbed contrastive learning, optimize the noise-contrastive estimation (NCE) bound on mutual information between two transformations of an image. NCE typically uses randomly sampled negative examples to normalize the objective, but this may often include many uninformative examples either because they are too easy or too hard to discriminate. Taking inspiration from metric learning, we show that choosing semi-hard negatives can yield stronger contrastive representations. To do this, we introduce a family of mutual information estimators that sample negatives conditionally – in a “ring” around each positive. We prove that these estimators remain lower-bounds of mutual information, with higher bias but lower variance than NCE. Experimentally, we find our approach, applied on top of existing models (IR, CMC, and MoCo) improves accuracy by $2 - 5 \\%$ absolute points in each case, measured by linear evaluation on four standard image benchmarks. Moreover, we find continued benefits when transferring features to a variety of new image distributions from the Meta-Dataset collection and to a variety of downstream tasks such as object detection, instance segmentation, and key-point detection. ", + "bbox": [ + 233, + 295, + 764, + 517 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "1 INTRODUCTION ", + "text_level": 1, + "bbox": [ + 176, + 545, + 336, + 560 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Supervised learning has given rise to human-level performance in several visual tasks (Russakovsky et al., 2015; He et al., 2017), relying heavily on large image datasets paired with semantic annotations. These annotations vary in difficulty and cost, spanning from simple class labels to more granular descriptions like bounding boxes and key-points. As it is impractical to scale high quality annotations, this reliance on supervision poses a barrier to widespread adoption. While supervised pretraining is still the dominant approach in computer vision, recent studies using unsupervised “contrastive” objectives, have achieved remarkable results in the last two years, closing the gap to supervised baselines (Wu et al., 2018; Oord et al., 2018; Hjelm et al., 2018; Zhuang et al., 2019; Henaff et al., 2019; Misra & Maaten, 2020; He et al., 2019; Chen et al.,´ 2020a;b; Grill et al., 2020) ", + "bbox": [ + 174, + 575, + 825, + 703 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Many contrastive algorithms are estimators of mutual information (Oord et al., 2018; Hjelm et al., 2018; Bachman et al., 2019), capturing the intuition that a good low-dimensional “representation” is one that linearizes the useful information embedded within a high-dimensional data point. In vision, these estimators maximize the similarity of encodings for two augmentations of the same image. This is trivial (e.g. assign all image pairs maximum similarity) unless the similarity function is normalized. This is typically done by comparing an image to “negative examples”, which a model must assign low similarity to. We hypothesize that how we choose these negatives greatly impacts the representation quality. With harder negatives, the encoder is encouraged to capture more granular information that may improve performance on downstream tasks. While research in contrastive learning has explored architectures, augmentations, and pretext tasks, there has been little attention given to the negative sampling procedure. Meanwhile, there is a rich body of work in deep metric learning showing semi-hard negative mining to improve the efficacy of triplet losses. Inspired by this, we hope to bring harder negative sampling to modern contrastive learning. ", + "bbox": [ + 174, + 708, + 825, + 888 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Naively choosing difficult negatives may yield an objective that no longer bounds mutual information, removing a theoretical connection that is core to contrastive learning and has been shown to be important for downstream performance (Tian et al., $\\boxed { 2 0 2 0 }$ . In this paper, we present a new estimator of mutual information based on the popular noise-contrastive estimator (NCE) that supports sampling negatives from conditional distributions. We summarize our contributions below: ", + "bbox": [ + 174, + 895, + 821, + 922 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "", + "bbox": [ + 176, + 102, + 821, + 146 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "1. We prove our Conditional-NCE (CNCE) objective to lower bound mutual information. Further, we show that although CNCE is a looser bound than NCE, it has lower variance. This motivates its value for representation learning. \n2. We use CNCE to generalize contrastive algorithms that utilize a memory structure like IR, CMC, and MoCo to sample semi-hard negatives in just a few lines of code and minimal compute overhead. \n3. We find that the naive strategy of sampling hard negatives throughout training can be detrimental. We then show that slowly introducing harder negatives yields good performance. \n4. On four image classification benchmarks, we find improvements of $2 - 5 \\%$ absolute points. We also find consistent improvements (1) when transferring features to new image datasets and (2) in object detection, instance segmentation, and key-point detection. ", + "bbox": [ + 210, + 156, + 825, + 325 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "2 BACKGROUND ", + "text_level": 1, + "bbox": [ + 174, + 344, + 326, + 361 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "We focus on exemplar-based contrastive objectives, where examples are compared to one another to learn a representation. Many of these objectives (Hjelm et al., 2018; Wu et al., 2018; Bachman et al., 2019; Tian et al., 2019; Chen et al., 2020a) are equivalent to NCE (Oord et al., 2018; Poole et al., $\\boxed { 2 0 1 9 }$ , a popular lower bound on the mutual information, denoted by $\\overline { { \\mathcal { T } } }$ , between two random variables. This connection is well-known and stated in several works (Chen et al., 2020a; Tschannen et al., 2019; Tian et al., 2020; Wu et al., 2020). To review, recall: ", + "bbox": [ + 173, + 376, + 825, + 462 + ], + "page_idx": 1 + }, + { + "type": "equation", + "img_path": "images/9b508a2100444b879d7728f2f489df02034bfc8ad23c00b00a06933c0c911422.jpg", + "text": "$$\n\\mathscr { Z } ( X ; Y ) \\geq \\mathscr { Z } _ { \\mathrm { N C E } } ( X ; Y ) = \\mathbf { E } _ { x _ { i } , y _ { i } \\sim p ( x , y ) } \\mathbf { E } _ { y _ { 1 : k } \\sim p ( y ) } \\left[ \\log \\frac { e ^ { f _ { \\theta } ( x _ { i } , y _ { i } ) } } { \\frac { 1 } { k + 1 } \\sum _ { j \\in \\{ i , 1 : k \\} } e ^ { f _ { \\theta } ( x _ { i } , y _ { j } ) } } \\right]\n$$", + "text_format": "latex", + "bbox": [ + 218, + 465, + 777, + 507 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "where $x , y$ are realizations of two random variables, $X$ and $Y$ , and $f _ { \\theta } : X \\times Y \\to \\mathbf { R }$ is a similarity function. We call $y _ { 1 : k } = \\{ y _ { 1 } , . . . y _ { k } \\}$ negative examples, being other realizations of $Y$ . ", + "bbox": [ + 174, + 512, + 826, + 542 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "Suppose the two random variables in Eq. $^ 1$ are both transformations of a common random variable $X$ . Let $\\tau$ be a family of transformations where each member $t$ is a composition of cropping, color jittering, gaussian blurring, among others $( | \\mathrm { W u \\ e t \\ a l . } | ) { | 2 0 1 8 } ; { | \\mathrm { B a c h m a n \\ e t \\ a l . } | 2 0 1 9 } ; { | \\mathrm { C h e n \\ e t \\ a l . } | } , | \\overline { { 2 0 2 0 \\mathrm { a } } } )$ We call a transformed input $t ( x )$ a “view” of $x$ . Let $\\overline { { p ( t ) } }$ denote a distribution over $\\tau$ , a common choice being uniform. Next, introduce an encoder $g _ { \\boldsymbol \\theta } : X \\to \\mathbf { S } ^ { n - 1 }$ that maps an example to a $\\mathrm { L _ { 2 } }$ -normalized representation. Suppose we have a dataset $\\mathcal { D } = \\{ x _ { i } \\} _ { i = 1 } ^ { n }$ of $n$ values for $X$ sampled from a distribution $p ( x )$ . Then, the contrastive objective for the $i$ -th example is: ", + "bbox": [ + 173, + 547, + 825, + 647 + ], + "page_idx": 1 + }, + { + "type": "equation", + "img_path": "images/d9c8572249da0f67a14912bbc18f05484c7cfa00f026649e8841c5a717b243bd.jpg", + "text": "$$\n\\mathcal { L } ( \\boldsymbol { x } _ { i } ) = \\mathbf { E } _ { t , t ^ { \\prime } , t _ { 1 : k } \\sim p ( t ) } \\mathbf { E } _ { \\boldsymbol { x } _ { 1 : k } \\sim p ( \\boldsymbol { x } ) } \\left[ \\log \\frac { e ^ { g _ { \\theta } ( t ( \\boldsymbol { x } _ { i } ) ) ^ { T } g _ { \\theta } ( t ^ { \\prime } ( \\boldsymbol { x } _ { i } ) ) / \\tau } } { \\frac { 1 } { k + 1 } \\sum _ { j \\in \\{ i , 1 : k \\} } e ^ { g _ { \\theta } ( t ( \\boldsymbol { x } _ { i } ) ) ^ { T } g _ { \\theta } ( t _ { j } ( \\boldsymbol { x } _ { j } ) ) / \\tau } } \\right]\n$$", + "text_format": "latex", + "bbox": [ + 240, + 651, + 756, + 694 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "where $\\tau$ is a temperature. The equivalence of Eq. $\\displaystyle { 2 }$ to NCE is immediate given $f _ { \\theta } ( x , y ) ~ =$ $g _ { \\theta } ( x ) ^ { T } g _ { \\theta } ( y ) / \\tau$ . Maximizing Eq. $\\bigstar$ chooses an embedding that pulls two views of the same example together while pushing two views of distinct examples apart. A drawback to this framework is that the number of negatives $k$ must be large to faithfully approximate the true partition. In practice, $k$ is limited by memory. Recent innovations have focused on tackling this challenge: ", + "bbox": [ + 173, + 699, + 825, + 771 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "Instance Discrimination $\\mathbb { W } \\mathrm { u } \\mathrm { ~ e t ~ a l . } \\backslash \\mathbb { Z } 0 1 8 \\}$ , or IR, introduces a memory bank of $n$ entries to cache embeddings of each example throughout training. Since every epoch we observe each example once, the memory bank will save the embedding of the view of the $i$ -th example observed last epoch in its $i$ - th entry. Representations stored in the memory bank are removed from the automatic differentiation tape, but in return, we can choose a large $k$ by querying $M$ . A follow up work, Contrastive Multiview Coding $( [ \\mathrm { T i a n ~ e t ~ a l . } ] [ 2 0 1 9 )$ , or CMC, decomposes an image into two color modalities. Then, CMC sums two IR losses where the memory banks for each modality are swapped. ", + "bbox": [ + 173, + 775, + 825, + 875 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "Momentum Contrast $( \\mathbb { H } { \\mathrm { e ~ e t ~ a l . } } ) , \\lfloor 2 0 1 9 \\rangle$ , or MoCo, observed that the representations stored in the memory bank grow stale, since possibly thousands of optimization steps pass before updating an entry again. So, MoCo makes two important changes. First, it replaces the memory bank with a first-in first-out (FIFO) queue of size $k$ . During each minibatch, representations are cached into the queue while the most stale ones are removed. Second, MoCo introduces a second (momentum) encoder $g _ { \\theta ^ { \\prime } } ^ { \\prime }$ as a copy of $g _ { \\theta }$ . The primary encoder $g _ { \\theta }$ is used to embed one view of $x _ { i }$ whereas the momentum encoder is used to embed the other. Again, gradients are not propagated to $g _ { \\theta ^ { \\prime } } ^ { \\prime }$ . ", + "bbox": [ + 174, + 881, + 825, + 924 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "", + "bbox": [ + 174, + 103, + 823, + 160 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "In this work, we focus on contrastive algorithms that utilize a memory structure that we repurpose in Sec. $\\boxed { 4 }$ to efficiently sample hard negatives from. In Sec. $\\textcircled { 7 }$ we briefly discuss generalizations to contrastive algorithms that do not use a memory structure. ", + "bbox": [ + 174, + 166, + 825, + 208 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "3 CONDITIONAL NOISE CONTRASTIVE ESTIMATION ", + "text_level": 1, + "bbox": [ + 174, + 229, + 622, + 246 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "In NCE, the negative examples are sampled i.i.d. from the marginal distribution, $p ( y )$ . Indeed, the existing proof that NCE lower bounds mutual information $\\pm { \\overline { { \\mathrm { ( P o o l e ~ e t ~ a l . } } } } , \\pm { \\overline { { 2 0 1 9 } } } )$ assumes this to be true. However, choosing negatives in this manner may not be the best choice for learning a good representation. For instance, prior work in metric learning has shown the effectiveness of semihard negative mining in optimizing triplet losses (Wu et al., 2017; Yuan et al., 2017; Schroff et al., $\\boxed { 2 0 1 5 }$ . We similarly wish to exploit choosing semi-hard negatives in NCE conditional on the current example but to do so in a manner that preserves the lower bound on mutual information. ", + "bbox": [ + 173, + 262, + 825, + 359 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "In presenting the theory, we assume two random variables $X$ and $Y$ , deriving a general bound; we will return to the contrastive learning setting in Sec. $\\boxed { 4 }$ To begin, in Eq. $\\bar { \\bigtriangledown }$ suppose we sample negatives from a distribution $q ( y | x )$ conditional on a value $x \\sim p ( x )$ rather than the marginal $p ( \\bar { y } )$ , which is independent of $X$ . Ideally, we would like to freely choose $q ( y | x )$ to be any distribution but not all choices preserve a bound on mutual information1. This does not, however, imply that we can only sample negatives from $\\upsilon ( y ) ~ ( { \\mathrm { P o o l e ~ e t ~ a l . } } ) ~ [ { \\underline { { { 2 0 1 9 } } } } \\vert ~ [ { \\mathrm { O o r d ~ e t ~ a l . } } ] [ { \\underline { { { 2 0 1 8 } } } } )$ . One of our contributions is to formally define a family of conditional distributions $\\mathcal { Q }$ such that for any $q ( y | x ) \\in \\mathcal { Q }$ , drawing negative examples from $q$ defines an estimator that lower bounds $\\mathcal { I } ( X ; Y )$ . We call this new bound the Conditional Noise Contrastive Estimator, or CNCE. We first prove CNCE to be a bound: ", + "bbox": [ + 173, + 366, + 825, + 492 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Theorem 3.1. (The Conditional NCE bound) Define $d$ -dimensional random variables $X$ and $Y$ by a joint distribution $p ( x , y )$ and let $Y _ { 1 } , . . . , Y _ { k }$ be i.i.d. copies of $Y$ with the marginal distribution $p ( y )$ . Fix any function $f : ( X , Y ) \\to \\mathbf { R }$ , any realization $x$ of $X$ , and let $c = \\mathbf { E } _ { y \\sim p ( y ) } [ e ^ { f ( x , y ) } ] ,$ , the expected exponentiated similarity. Pick a set $B \\subset \\mathbf { R }$ strictly lower-bounded by c. Assume the pulled back set $S _ { B } \\sp { \\bullet } = \\{ y | e \\sp { f ( x , y ) } \\in B \\}$ has non-zero probability (i.e. $p ( S _ { B } ) > 0 ,$ ). For $A _ { 1 } , \\ldots , A _ { k }$ in the Borel $\\sigma$ -algebra over $\\mathbf { R } ^ { d }$ , define $A = A _ { 1 } \\times \\ldots \\times A _ { k }$ and let ", + "bbox": [ + 173, + 496, + 825, + 583 + ], + "page_idx": 2 + }, + { + "type": "equation", + "img_path": "images/d6a9632fe22fac78ee179735a33e26f23bc90b27f9176ff009e648e566f18f2b.jpg", + "text": "$$\nq ( ( Y _ { 1 } , \\dots , Y _ { k } ) \\in A | X = x ) = \\prod _ { j = 1 } ^ { k } p ( A _ { j } | S _ { B } ) .\n$$", + "text_format": "latex", + "bbox": [ + 344, + 589, + 651, + 635 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Proof. To show $\\mathcal { T } _ { \\mathrm { C N C E } } \\leq \\mathcal { T } _ { \\mathrm { N C E } }$ , we show $\\begin{array} { r } { { \\bf E } _ { p } [ \\log \\sum _ { j = 1 } ^ { k } e ^ { f ( x , y _ { j } ) } ] < { \\bf E } _ { q } [ \\log \\sum _ { j = 1 } ^ { k } e ^ { f ( x , y _ { j } ) } ] } \\end{array}$ . To see this, apply Jensen’s to the left-hand side of $\\begin{array} { r } { \\log \\mathbf { E } _ { p } [ \\sum _ { j = 1 } ^ { k } e ^ { f ( x , y _ { j } ) } ] < \\log \\sum _ { j = 1 } ^ { k } e ^ { f ( x , y _ { j } ) } } \\end{array}$ , which holds if $y _ { j } \\in S _ { B }$ for $j = 1 , \\dotsc , k$ , and then take the expectation $\\mathbf { E } _ { q }$ of both sides. The last inequality holds by monoticity of log, linearity of expectation, and the fact that ${ \\bf E } _ { p } [ e ^ { f ( x , y _ { j } ) } ] \\le e ^ { f ( x , y _ { j } ) }$ . ", + "bbox": [ + 173, + 693, + 825, + 762 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Theorem Intuition. For intuition, although using arbitrary negative distributions in NCE does not bound mutual information, we have found a restricted class of distributions $\\mathcal { Q }$ where every member $q ( y | x )$ “subsets the support” of the distribution $p ( y )$ . That is, given some fixed value $x$ , we have defined $q ( y | x )$ to constrain the support of $p ( y )$ to a set $S _ { B }$ whose members are “close” to $x$ as measured by the similarity function $f$ . For every element $y \\in S _ { B }$ , the distribution $q ( y | x )$ wants to assign to it the same probability as $p ( y )$ . However, as $q ( y | x )$ is not defined outside of $S _ { B }$ , we must renormalize it to sum to one (hence $\\begin{array} { r } { p ( A _ { j } | S _ { B } ) = \\frac { p ( A _ { j } \\cap S _ { B } ) } { p ( S _ { B } ) } ) } \\end{array}$ . Intuitively, $q ( y | x )$ cannot change $p ( y )$ too much: it must redistribute mass proportionally. The primary distinction then, is the smaller support of $q ( y | x )$ , which forces samples from it to be harder for $f$ to distinguish from $x$ . Thm. 3.1 shows that substituting $q ( y | x )$ for $p ( y )$ in NCE still bounds mutual information. ", + "bbox": [ + 173, + 779, + 825, + 898 + ], + "page_idx": 2 + }, + { + "type": "image", + "img_path": "images/5cd68e2041e12a3bc85e25a2911d9fb4c19f8d06288b74bef3503bf2086665ed.jpg", + "image_caption": [ + "Figure 1: Visual illustration of Ring Discrimination. Black: view of example $x _ { i }$ ; gray: second view of $x _ { i }$ ; red: negative samples; gray area: distribution $q ( x | t ( x _ { i } ) )$ . In subfigure (c), the negative samples are annealed to be closer to $t ( x _ { i } )$ through training. In other words, the support of $q$ shrinks. " + ], + "image_footnote": [], + "bbox": [ + 199, + 97, + 792, + 183 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "", + "bbox": [ + 173, + 267, + 825, + 296 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Theorem Example 3.1. We give a concrete example for the choice $B$ that will be used in Sec. 4. For any realization $x$ , suppose we define two similarity thresholds $\\omega _ { \\ell } , \\omega _ { u } \\in \\mathbf { R }$ where $c < \\omega _ { \\ell } < \\omega _ { u }$ . Then, choose $B = [ w _ { \\ell } , w _ { u } ]$ . In this case, the set $S _ { B }$ , which defines the support of the distribution $q ( y | x )$ , contains values of $y$ that are not “too-close” to $x$ but not “too-far”. In contrastive learning, we might pick these similarity thresholds to vary the difficulty of negative samples. ", + "bbox": [ + 174, + 303, + 825, + 373 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Interestingly, Thm. $\\boxed { 3 . 1 }$ states that CNCE is looser than NCE, which raises the question: when is a looser bound useful? In reply, we show that while CNCE is a more biased estimator than NCE, in return it has lower variance. Intuitively, because $q ( y | x )$ is the result of restricting $p ( y )$ to a smaller support, samples from $q ( y | x )$ have less opportunity to deviate, hence lower variance. Formally: ", + "bbox": [ + 174, + 378, + 825, + 435 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Theorem 3.2. (Bias and Variance Tradeoff) Pick any $x , y \\sim p ( x , y )$ . Fix the distribution $q ( y _ { 1 : k } | x )$ as stated in Theorem 3.1 Define a new random variable $\\begin{array} { r } { Z ( y _ { 1 : k } ) = \\log \\left( \\frac { e ^ { f ( x , y ) } } { \\frac { 1 } { k } \\sum _ { j = 1 } ^ { k } e ^ { f ( x , y _ { j } ) } } \\right) } \\end{array}$ representing the normalized similarity. By Theorem $\\boxed { 3 . 1 }$ the expressions ${ \\bf E } _ { p ( y _ { 1 : k } ) } [ Z ]$ and ${ \\mathbf { E } } _ { q \\left( y _ { 1 : k } \\mid x \\right) } [ Z ]$ are estimators for $\\mathcal { I } ( X ; Y )$ . Suppose that the set $S _ { B }$ is chosen to ensure $W a r _ { q ( y _ { 1 : k } | x ) } [ Z ] \\ \\leq$ $V a r _ { \\tilde { q } ( y _ { 1 : k } | x ) } [ Z ]$ , where $\\tilde { q } ( A ) ~ = ~ p ( A |$ complement of $S _ { B }$ ). That is, we assume the variance of the normalized similarity when using $y _ { 1 : k } ~ \\in ~ S _ { B }$ is smaller than when using $y _ { 1 : k } ~ \\notin ~ S _ { B }$ . Then $\\mathrm { B i a s } _ { p ( y _ { 1 : k } ) } ( Z ) \\leq \\mathrm { B i a s } _ { q ( y _ { 1 : k } | x ) } ( Z )$ and $\\operatorname { V a r } _ { p ( y _ { 1 : k } ) } ( Z ) \\geq \\operatorname { V a r } _ { q ( y _ { 1 : k } | x ) } ( Z )$ . ", + "bbox": [ + 173, + 440, + 825, + 564 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "The proof can be found in $\\mathrm { S e c . } \\ \\mathrm { A . } 2 . \\ \\mathrm { T h m . } \\ 3 . 2$ provides one answer to our question of looseness. In stochastic optimization, a lower variance objective may lead to better local optima. For representation learning, using CNCE to sample more difficult negatives may (1) encourage the representation to distinguish fine-grained features useful in transfer tasks, and (2) provide less noisy gradients. ", + "bbox": [ + 174, + 574, + 825, + 632 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "4 RING DISCRIMINATION ", + "text_level": 1, + "bbox": [ + 176, + 656, + 400, + 672 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "We have shown CNCE to be a new bound on the mutual information that uses hard negative samples. Now we wish to apply CNCE to contrastive learning where the two random variables are again transformations of a single variable $X$ . In this setting, for a fixed $x _ { i } \\sim p ( x )$ , the CNCE distribution is written as $q ( x | t ( x _ { i } ) )$ for some transform $t \\in \\mathcal { T }$ . Samples from $x \\stackrel { \\cdot \\cdot } { \\sim } q ( x | t ( x _ { i } ) )$ will be such that the exponentiated distance, $\\exp \\{ g _ { \\theta } ( t ( x _ { i } ) ) ^ { T } g _ { \\theta } ( t ^ { \\prime } ( x ) ) \\}$ , is at least a minimum value $c$ . As in Example 3.1, we will choose $B = [ \\omega _ { \\ell } , \\omega _ { u } ]$ , a closed interval in $\\mathbf { R }$ defined by two thresholds. ", + "bbox": [ + 173, + 690, + 825, + 775 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Picking thresholds. We pick the thresholds conditioned on the $i$ -th example in the dataset, hence each example has a different set $B$ . We first describe how to pick the upper threshold $\\omega _ { u }$ . Given the $i$ -example $x _ { i }$ , we pick a number $u \\in [ 0 , 1 0 0 ]$ representing an upper “percentile”. We consider each example $x$ in the dataset to be in the support $S _ { B }$ if and only if the (exponentiated) distance between the embedding of $x _ { i }$ and $x$ , or $\\exp \\{ \\bar { g _ { \\boldsymbol { \\theta } } } ( t ( x _ { i } ) ) ^ { T } g _ { \\boldsymbol { \\theta } } ( t ^ { \\prime } ( x ) ) \\}$ , is below the $u$ -th percentile for all $x \\in \\mathcal { D }$ . Call this maximum distance $\\omega _ { u }$ . In other words, we construct $q ( x | t ( x _ { i } ) )$ such that we ignore examples from the dataset whose embedding dot producted with the embedding of $x _ { i }$ is above $\\omega _ { u }$ . (Note that $u = 1 0 0$ recovers NCE.) For a small enough choice of $u$ , the upper similarity threshold $\\omega _ { u }$ will be greater than $c$ (defined in Thm. $3 . 1$ as the expected distance with respect to $p ( x ) )$ ), and the samples from $q ( x | t ( x _ { i } ) )$ will be harder negatives to discriminate from $x _ { i }$ . ", + "bbox": [ + 173, + 781, + 825, + 921 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "In picking the lower threshold $\\omega _ { \\ell }$ , one could choose it to be 0, so $B = [ 0 , \\omega _ { u } )$ . However, picking the closest examples to $t ( x _ { i } )$ as its negative examples may be inappropriate, as these examples might be better suited as positive views rather than negatives (Zhuang et al., 2019; Xie et al., 2020). As an extreme case, if the same image is included in the dataset twice, we would not like to select it as a negative example for itself. Furthermore, choosing negatives “too close” to the current instance may result in representations that pick up on fine-grain details only, ignoring larger semantic concepts. This suggests removing examples from $q ( x | t ( x _ { i } ) )$ we consider “too close” to $x _ { i }$ . To do this, we pick a lower percentile $0 \\leq \\ell < u$ . For each example $x \\in \\mathcal { D }$ , we say it is in $S _ { B }$ if $\\exp \\{ g _ { \\theta } ( t ( x _ { i } ) ) ^ { T } g _ { \\theta } ( t ^ { \\prime } ( x ) ) \\}$ is below $\\omega _ { u }$ and also if it is above the $\\ell$ -th percentile of all distances with respect to $\\mathcal { D }$ . Call this minimum distance $\\omega _ { \\ell }$ . Fig. 2 visualizes this whole procedure. ", + "bbox": [ + 173, + 103, + 825, + 243 + ], + "page_idx": 4 + }, + { + "type": "image", + "img_path": "images/b9475398b61f033925ccb174c0553ebb227c591ac73f17496792590fbf7e7ce0.jpg", + "image_caption": [ + "Figure 2: Defining the CNCE distribution $q ( x | t ( x _ { i } ) )$ . By choosing a lower and upper percentile $\\ell$ and $u$ , we implicitly define similarity thresholds $\\omega _ { \\ell }$ and $\\omega _ { u }$ to construct a support of valid negative examples, $S _ { B }$ , which in turn, defines the distribution $q \\big ( \\boldsymbol { x } | t ( \\boldsymbol { x } _ { i } ) \\big )$ . " + ], + "image_footnote": [], + "bbox": [ + 173, + 253, + 821, + 328 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Algorithm 1: MoCoRing ", + "text_level": 1, + "bbox": [ + 174, + 421, + 343, + 435 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "# g q , g k : e n c o d e r n e t w o r k s \n# m: momentum ; t : t e m p e r a t u r e \n# u : r i n g u p p e r p e r c e n t i l e \n# l : r i n g l o w e r p e r c e n t i l e \n$\\operatorname { t x } 1 = \\operatorname { a u g } \\left( \\mathbf { x } \\right)$ # random a u g m e n t a t i o n \nt x 2 = aug ( x ) \nemb1=norm ( g q ( t x 1 ) ) \nemb2=norm ( g k ( t x 2 ) ) . d e t a c h ( ) \nd p s =sum ( t x 1 t x 2 ) / t # d o t p r o d u c t \n# s o r t f r o m c l o s e s t t o f a r t h e s t n e g \na l l $\\mathrm { | } \\mathbf { p } \\mathbf { s } = \\mathbf { s } \\mathbf { o } \\mathbf { r } \\mathbf { t }$ ( emb1@queue . T / t ) \n# f i n d i n d i c e s o f t h r e s h o l d s \ni x ${ \\bf \\Pi } _ { - } 1 = 1 * 1 \\mathbf { e } \\mathbf { n }$ ( q u e u e ) \ni x $\\mathbf { u = u * l e n }$ ( q u e u e ) \nr i n g d p $s =$ a l l d p s [ : , i x l : i x u ] \n# n o n p a r a m e t r i c s o f t m a x \nl o s s=\u0000dps + logsumexp ( r i n g d p s ) \nl o s s . b a c k w a r d ( ) \ns t e p ( g q . p a r a m s ) \n# moco u p d a t e s \ng k . params $\\mathbf { \\mu } = \\mathbf { \\mu } _ { \\mathbf { m } * \\mathbf { g } } \\mathbf { \\Lambda } _ { - } \\mathbf { k }$ . p a r a m s + $( 1 - \\mathrm { m } ) * .$ g q . p a r a m s \nenqueue ( queue , emb2 ) ; dequeue ( queue ) \n# t h r e s h o l d u p d a t e s \na n n e a l ( w l ) ; a n n e a l ( w u ) ", + "bbox": [ + 173, + 441, + 382, + 665 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Ring Discrimination. Having defined $\\omega _ { \\ell }$ and $\\omega _ { u }$ , we have a practical method of choosing $B$ , and thus $S _ { B }$ to define $q ( \\bar { x } | t ( x _ { i } ) )$ for $i$ -th example. Intuitively, we construct a conditional distribution for negative examples that are (1) not too easy since their representations are fairly similar to that of $x _ { i }$ and (2) not too hard since we remove the “closest” instances to $x _ { i }$ from $S _ { B }$ . We call this algorithm Ring Discrimination, or Ring, inspired by the shape of negative set (see Fig. 1). ", + "bbox": [ + 410, + 397, + 823, + 508 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Ring can be easily added to popular contrastive algorithms. For IR and CMC, this amounts to simply sampling entries in the memory bank that fall within the $\\ell$ -th to $u$ -th percentile of all distances to the current example view (in representation space). Similarly, for MoCo, we sample from a subset of the queue (chosen to be in the $\\ell$ -th to $u$ -th percentile), preserving the FIFO ordering. In our experiments, we refer to these as IRing, CMCRing, MoCoRing, respectively. Alg. $\\perp$ shows PyTorch-like pseudocode for MoCoRing. One of the strengths of this approach is the simplicity: the algorithm requires only a few lines of code on top of existing implementations. ", + "bbox": [ + 411, + 516, + 823, + 669 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Annealing Policy. Naively using hard negatives can collapse to a poor representation, especially if we choose the upper threshold, $\\omega _ { u }$ , to be very small early in training. At the start of training, the encoder $g _ { \\theta }$ is randomly initialized and cannot guarantee that elements in the $\\ell$ -th to $u$ -th percentile are properly calibrated: if the representations are near random, choosing negatives that are close in embedding distance may detrimentally exclude those examples that are “actually” close. This could lock in poor local minima. To avoid this, we propose to use an annealing policy that reduces $\\omega _ { u }$ (and thus the size of the support $S _ { B }$ ) throughout training. Early in training we choose $\\omega _ { u }$ to be large. Over many epochs, we slowly decrease $\\omega _ { u }$ closer to $\\omega _ { l }$ , thereby selecting more difficult negatives. We explored several annealing policies and found a linear schedule to be well-performing and simple (see Sec. $\\mathbf { G } )$ . In our experiments, annealing is shown to be crucial: being too aggressive with negatives early in training produced representations that performed poorly on downstream tasks. ", + "bbox": [ + 411, + 676, + 823, + 704 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "", + "bbox": [ + 173, + 704, + 825, + 842 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "5 EXPERIMENTS ", + "text_level": 1, + "bbox": [ + 176, + 863, + 326, + 878 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "We explore our method applied to IR, CMC, and MoCo in four commonly used visual datasets. As in prior work (Wu et al., 2018; Zhuang et al., 2019; He et al., 2019; Misra & Maaten, 2020; Henaff ´ et al., 2019; Kolesnikov et al., 2019; Donahue & Simonyan, 2019; Bachman et al., 2019; Tian et al., 2019; Chen et al., 2020a), we evaluate each method by linear classification on frozen embeddings. That is, we optimize a contrastive objective on a pretraining dataset to learn a representation; then, using a transfer dataset, we fit logistic regression on representations only. A better representation would contain more “object-centric” information, thereby achieving a higher classification score. ", + "bbox": [ + 174, + 895, + 823, + 926 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "", + "bbox": [ + 173, + 102, + 825, + 174 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Training Details. We pick the upper percentile $u = 1 0$ and the lower percentile $\\ell = 1$ although we anneal $u$ starting from 100. We resize input images to be 256 by 256 pixels, and normalize them using dataset mean and standard deviation. The temperature $\\tau$ is set to 0.07. We use a composition of a 224 by 224-pixel random crop, random color jittering, random horizontal flip, and random grayscale conversion as our augmentation family $\\tau$ . We use a ResNet-18 encoder with a output dimension of 128. For CMC, we use two ResNet-18 encoders, doubling the number of parameters. For linear classification, we treat the pre-pool output (size $5 1 2 \\times 7 \\times 7 )$ after the last convolutional layer as the input to the logistic regression. Note that this setup is equivalent to using a linear projection head $( \\mathbf { \\overline { { C h e n \\ e t \\ a l . } } } ) , \\mathbf { \\overline { { 2 0 2 0 a } } } , \\mathbf { \\overline { { b } } } )$ . In pretraining, we use SGD with learning rate 0.03, momentum 0.9 and weight decay 1e-4 for 300 epochs and batch size 256 (128 for CMC). We drop the learning rate twice by a factor of 10 on epochs 200 and 250. In transfer, we use SGD with learning rate 0.01, momentum 0.9, and no weight decay for 100 epochs without dropping learning rate. These hyperparameters were taken from $\\boxed { \\mathrm { W u \\ e t \\ a l . } } \\textcircled { 2 0 1 8 }$ and used in all of Table $\\bar { 1 }$ for a consistent comparison. We found normalizing hyperparameters to be important for a fair comparison as many competing algorithms use different hyperparameters. For a state-of-the-art comparison, see Table 5. ", + "bbox": [ + 173, + 181, + 825, + 388 + ], + "page_idx": 5 + }, + { + "type": "table", + "img_path": "images/222e2bb974c6c8538c42352d5300dcf98a33430b20398d186415163a606cc282.jpg", + "table_caption": [], + "table_footnote": [ + "(a) CIFAR10 (b) CIFAR100 (c) STL10 (d) ImageNet " + ], + "table_body": "
ModelLinear EvaluationModelLinear EvaluationModelLinear EvaluationModelLinear Evaluation
IR81.2IR60.4IR61.4IR43.2
IRing83.9 (+2.7)IRing62.3 (+1.9)IRing64.3 (+2.9)IRing48.4 (+5.2)
CMC*85.6CMC*56.0CMC*63.8CMC*48.2
CMCRing*87.6 (+2.0)CMCRing*56.0 (+0.0)CMCRing*66.4 (+2.6)CMCRing*50.4 (+2.2)
MoCo83.1MoCo59.1MoCo63.8MoCo52.8
MoCoRing86.1 (+3.0)MoCoRing61.5 (+2.4)MoCoRing65.2 (+1.4)MoCoRing54.6 (+1.8)
LA83.9LA61.4LA63.0LA48.0
", + "bbox": [ + 181, + 401, + 818, + 489 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Table 1: Comparison of contrastive algorithms on four image domains. Superscript $( ^ { * } )$ indicates models that use twice as many parameters as others e.g. CMC has “L” and “ab” encoders. ", + "bbox": [ + 174, + 518, + 823, + 546 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "The results for CIFAR10, CIFAR100, STL10, and ImageNet are in Table $\\bigstar$ Overall, IR, CMC, and MoCo all benefit from using more difficult negatives as shown by $2 - 5 \\%$ absolute points of improvement across the four datasets. While we find different contrastive objectives to perform best in each dataset, the improvements from Ring are consistent: the Ring variant outperforms the base for every model and every dataset. We also include as a baseline Local Aggregation, or LA (Zhuang et al., 2019), a popular contrastive algorithm (see ${ \\mathrm { S e c . } } \\mathrm { H } )$ that implicitly uses hard negatives without annealing. We find our methods to outperform LA by up to $4 \\%$ absolute. ", + "bbox": [ + 173, + 561, + 825, + 660 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Ablations: Annealing and Upper Boundary. Having found good performance with Ring Discrimination, we want to assess the importance of the individual components that comprise Ring. We focus on the annealing policy and the exclusion of very close negatives from $S _ { B }$ . Concretely, we measure the transfer accuracy of (1) IRing without annealing and (2) IRing with an lower percentile $\\ell = 0$ , thereby excluding no close negatives. That is, $S _ { B }$ contains all examples in the dataset with representation similarity less than the $\\omega _ { u }$ (a “ball” instead of a “ring”). Table $2$ compares these ablations to IR and full IRing on CIFAR10 and ImageNet classification transfer. We observe that both ablations result in worse transfer accuracy, with proper annealing being especially important to prevent convergence to bad minima. We also find even with $\\ell = 0$ , IRing outperforms IR, suggesting both removing negatives that are “too close” and “too far” contribute to the improved representation quality. ", + "bbox": [ + 400, + 666, + 823, + 888 + ], + "page_idx": 5 + }, + { + "type": "table", + "img_path": "images/3221e0554c498641ae6ea048bc24bd6833e7d80ab47d8140bbf45cbd2d4e3bde.jpg", + "table_caption": [ + "Table 2: Lesioning the effects of annealing and choice of $\\ell$ . " + ], + "table_footnote": [ + "(b) ImageNet " + ], + "table_body": "
ModelLinear Eval.
IR81.2
IRing83.9
IRing (No Anneal)81.4
IRing(l=0)82.1
(a) CIFAR10
ModelLinear Eval.
IR43.2
IRing48.4
IRing (No Anneal)41.3
IRing(e=0)47.3
", + "bbox": [ + 199, + 683, + 364, + 824 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Transferring Features. Thus far we have only evaluated the learned representations on unseen examples from the training distribution. As the goal of unsupervised learning is to capture general representations, we are also interested in their performance on new, unseen distributions. To gauge this, we use the same linear classification paradigm on a suite of image datasets from the “Meta Dataset” collection (Triantafillou et al., 2019) that have been used before in contrastive literature (Chen et al., 2020a). All representations were trained on CIFAR10. For each transfer dataset, we compute mean and variance from a training split to normalize input images, which we found important for generalization to new visual domains. ", + "bbox": [ + 179, + 896, + 825, + 924 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "", + "bbox": [ + 173, + 103, + 825, + 188 + ], + "page_idx": 6 + }, + { + "type": "table", + "img_path": "images/73a615917cc31eb85fca46833f539c2585e2e88596a343cbcfb1e23b89efbb93.jpg", + "table_caption": [ + "Table 3: Transferring CIFAR10 embeddings to various image distributions. " + ], + "table_footnote": [], + "table_body": "
ModelAircraftCUBirdsDTDFungiMNISTFashionMNISTTrafficSignVGGFlowerMSCOCO
IR40.917.939.22.796.991.797.168.152.4
IRing40.6 (-0.3)17.9 (+0.0)39.5 (+0.3)3.4(+0.7)97.8 (+0.9)91.6 (+0.1)98.8 (+1.7)68.5 (+0.4)52.5 (+0.1)
MoCo41.518.039.73.196.990.997.364.552.0
MoCoRing41.6(+0.1)18.6 (+0.6)39.5 (-0.2)3.6(+0.5)97.9 (+1.0)91.3 (+0.4)99.3 (+2.0)69.1 (+4.6)52.6 (+0.6)
CMC40.115.838.34.397.591.594.667.151.4
CMCRing40.8 (+0.7)16.8 (+1.0)40.6(+2.3)4.2 (-0.1)97.9 (+0.4)92.1 (+0.6)97.1( (+2.5)69.1 (+2.0)52.1 (+0.7)
LA41.317.839.02.397.292.398.266.952.3
", + "bbox": [ + 173, + 202, + 826, + 287 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "We find in Table $\\textcircled { 3 }$ that the Ring models are competitive with the non-Ring analogues, with increases in transfer accuracies of 0.5 to $2 \\%$ absolute. Most notable are the TrafficSign and VGGFlower datasets in which Ring models surpass others by a larger margin. We also observe that IRing largely outperforms LA. This suggests the features learned with more difficult negatives are not only useful for the training distribution but may also be transferrable to many visual datasets. ", + "bbox": [ + 174, + 337, + 825, + 406 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "More Downstream Tasks. Object classification is a popular transfer task, but we want our learned representations to capture holistic knowledge about the contents of an image. We must thus evaluate performance on transfer tasks such as detection and segmentation that require different kinds of visual information. We study four additional downstream tasks: object detection on COCO (Lin et al., $\\boxed { 2 0 1 4 }$ and Pascal VOC’07 (Everingham et al., 2010), instance segmentation on COCO, and keypoint detection on COCO. In all cases, we employ embeddings trained on ImageNet with a ResNet-18 encoder. We base these experiments after those found in $\\boxed { \\mathrm { H e ~ e t ~ a l . } } \\textcircled { 2 0 1 9 }$ with the same hyperparameters. However, we use a smaller backbone (ResNet-18 versus ResNet-50) and we freeze its parameters instead of finetuning them. We adapt code from Detectron2 (Wu et al., 2019). ", + "bbox": [ + 173, + 412, + 825, + 540 + ], + "page_idx": 6 + }, + { + "type": "table", + "img_path": "images/95fa676a48383ea8395bdbe317425f2d29d5d8c6b04fb6070efcdb6c5803b5c4.jpg", + "table_caption": [ + "Table 4: Evaluation of ImageNet representations using four visual transfer tasks. " + ], + "table_footnote": [], + "table_body": "
COCO: Object DetectionCOCO: Inst. SegmentationCOCO: Keypoint DetectionVOC: Object Detection
Arch.Mask R-CNN,R18-FPN,1x scheduleR-CNN,R18-FPNFaster R-CNN, R18-C4
ModelApbbAPAPApmkAPAPApkpAP5APApbbAPAP
IR8.619.06.68.517.47.434.663.032.95.514.53.3
IRing10.922.98.711.020.99.637.266.135.77.620.34.4
MoCo6.014.34.010.821.49.737.666.536.97.317.94.1
MoCoRing9.420.37.612.022.910.838.767.737.98.022.14.8
LA10.222.08.110.020.39.036.365.335.17.620.04.3
", + "bbox": [ + 205, + 553, + 794, + 655 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "We find IRing outperforms IR by around 2.3 points in COCO object detection, 2.5 points in COCO Instance Segmentation, 2.6 points in COCO keypoint detection, and 2.1 points in VOC object detection. Similarly, MoCoRing finds consistent improvements of 1-3 points over MoCo on the four tasks. Future work can investigate orthogonal directions of using larger encoders (e.g. ResNet-50) and finetuning ResNet parameters for these individual tasks. ", + "bbox": [ + 174, + 695, + 825, + 765 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "6 RELATED WORK ", + "text_level": 1, + "bbox": [ + 176, + 786, + 344, + 803 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "Several of the ideas in Ring Discrimination relate to existing work. Below, we explore these connections, and at the same time, place our work in a fast-paced and growing field. ", + "bbox": [ + 174, + 819, + 821, + 847 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "Hard negative mining. While it has not been deeply explored in modern contrastive learning, negative mining has a rich line of research in the metric learning community. Deep metric learning utilizes triplet objectives of the form ${ \\mathcal { L } } _ { \\mathrm { t r i p l e t } } = d ( g _ { \\theta } ( x _ { i } ) , g _ { \\theta } ( x _ { + } ) ) - d ( g _ { \\theta } ( x _ { i } ) , g _ { \\theta } ( x _ { - } ) + \\alpha )$ where $d$ is a distance function (e.g. $\\mathrm { L _ { 2 } }$ distance), $x _ { + }$ and $x _ { - }$ are a positive and negative example, respectively, relative to $x _ { i }$ , the current instance, and $\\alpha \\in \\mathbf { R } ^ { + }$ is a margin. In this context, several approaches pick semi-hard negatives: Schroff et al. $\\textcircled { | 2 0 1 5 } )$ treats the furthest (in $\\mathrm { L _ { 2 } }$ distance) example in the same minibatch as $x _ { i }$ as its negative, whereas Oh Song et al. $\\underline { { \\left( \\bar { 2 } 0 1 6 \\right) } }$ weight each example in the minibatch by its distance to $g _ { \\boldsymbol { \\theta } } ( x _ { i } )$ , thereby being a continuous version of $\\boxed { \\mathrm { S c h r o f f ~ e t ~ a l . } } \\textcircled { 2 0 1 5 } )$ . More sophisticated negative sampling strategies developed over time. $\\mathrm { I n } [ \\mathrm { W u \\ e t \\ a l . } ] \\mathbb { ( } 2 0 1 7 \\mathrm { ) }$ , the authors pick negatives from a fixed normal distribution that is shown to approximate $\\mathrm { L _ { 2 } }$ normalized embeddings in high dimensions. The authors show that weighting by this distribution samples more diverse negatives. Similarly, HDC $( \\mathbf { Y } \\mathbf { u } \\mathbf { a } \\mathbf { n } \\operatorname { e t } \\mathbf { a } \\mathbf { l } . ) , [ 2 0 1 7 ]$ simulataneously optimizes a triplet loss using many levels of “hardness” in negatives, again improving the diversity. Although triplet objectives paved the way for modern NCE-based objectives, the focus on negative mining has largely been overlooked. Ring Discrimination, being inspired by the deep metric learning literature, reminds that negative sampling is still an effective way of learning stronger representations in the new NCE framework. As such, an important contribution was to do so while retaining the theoretical properties of NCE, namely in relation to mutual information. This, to the best of our knowledge, is novel as negative mining in metric learning literature was not characterized in terms of information theory. ", + "bbox": [ + 174, + 853, + 823, + 924 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "", + "bbox": [ + 174, + 103, + 825, + 297 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "That being said, there are some cases of negative mining in contrastive literature. In CPC (Oord et al., 2018), the authors explore using negatives from the same speaker versus from mixed speakers in audio applications, the former of which can be interpreted as being more difficult. A recent paper, InterCLR (Xie et al., 2020), also finds that using “semi-hard negatives” is beneficial to contrastive learning whereas negatives that are too difficult or too easy produce worse representations. Where InterCLR uses a margin-based approach to sample negatives, we explore a wider family of negative distributions and show analysis that annealing offers a simple and easy solution to choosing between easy and hard negatives. Further, as InterCLR’s negative sampling procedure is a special case of CNCE, we provide theory grounding these approaches in information theory. Finally, a separate line of work in contrastive learning explores using neighboring examples (in embedding space) as “positive” views of the instance (Zhuang et al., 2019; Xie et al., 2020; Asano et al., 2019; Caron et al., $\\boxed { 2 0 2 0 }$ Li et al., 2020). That is, finding a set $\\overline { { \\{ x _ { j } \\} } }$ such that we consider ${ \\overline { { x _ { j } \\ } } } = { \\overline { { t ( x _ { i } ) } } }$ for the current instance $x _ { i }$ . While this does not deal with negatives explicitly, it shares similarities to our approach by employing other examples in the contrastive objective to learn better representations. In the Appendix, we discuss how one of these algorithms, LA (Zhuang et al., 2019), implicitly uses hard negatives and expand the Ring family with ideas inspired by it. ", + "bbox": [ + 173, + 305, + 825, + 526 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "Contrastive learning. We focused primarily on comparing Ring Discrimination to three recent and highly performing contrastive algorithms, but the field contains much more. The basic idea of learning representations to be invariant under a family of transformations is an old one, having been explored with self-organizing maps $\\scriptstyle ( \\mathbf { B e c k e r } \\ \\& \\ \\mathbf { H i n t o n } , \\mathbf { \\left[ 1 9 9 2 \\right] } )$ and dimensionality reduction (Hadsell et al., 2006). Before IR, the idea of instance discrimination was studied (Dosovitskiy et al., 2014; Wang & Gupta, $\\underline { { 2 0 1 5 } } )$ among many pretext objectives such as position prediction (Doersch et al., 2015), color prediction (Zhang et al., $\\boxed { 2 0 1 6 }$ , multi-task objectives (Doersch & Zisserman, 2017), rotation prediction (Gidaris et al., 2018; Chen et al., 2019), and many other “pretext” objectives (Pathak et al., 2017). As we have mentioned, one of the primary challenges to instance discrimination is making such a large softmax objective tractable. Moving from a parametric (Dosovitskiy et al., $\\boxed { 2 0 1 4 }$ to a nonparametric softmax reduced issues with vanishing gradients, shifting the challenge to efficient negative sampling. The memory bank approach $\\mathbb { W } \\mathrm { u } \\mathrm { ~ e t ~ a l . } \\lVert \\mathbb { Z } \\mathrm { 0 . 8 } \\rVert$ is a simple and memory-efficient solution, quickly being adopted by the research community (Zhuang et al., 2019; Tian et al., 2019; He et al., 2019; Chen et al., 2020b; Misra & Maaten, 2020). With enough computational resources, it is now also possible to reuse examples in a large minibatch and negatives of one another (Ye et al., 2019; Ji et al., 2019; Chen et al., 2020a). In our work, we focus on hard negative mining in the context of a memory bank or queue due to its computational efficiency. However, the same principles should be applicable to batch-based methods (e.g. SimCLR): assuming a large enough batch size, for each example, we only use a subset of the minibatch as negatives as in Ring. Finally, more recent work (Grill et al., 2020) removes negatives altogether, which is speculated to implicitly use negative samples via batch normalization (Ioffe & Szegedy, 2015); we leave a more thorough understanding of negatives in this setting to future work. ", + "bbox": [ + 173, + 534, + 825, + 839 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "7 DISCUSSION ", + "text_level": 1, + "bbox": [ + 176, + 862, + 310, + 877 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "Computational cost of Ring. To measure the cost of CNCE, we compare the cost an epoch of training MoCo/IR versus MoCoRing/IRing on four image datasets. Table 5a reports the average cost over 200 epochs. We observe that Ring models cost no more than 1.5 times the cost of standard contrastive algorithms, amounting to a difference of 3 to 7 minutes in ImageNet and 10 to 60 seconds in three other datasets per epoch. In the context of deep learning, we do not find the cost increases to be substantial. In particular, since (1) the memory structure in IR and MoCo allow us to store and reuse embeddings and (2) gradients are not propagated through the memory structure, the additional compute of Ring amounts to one matrix multiplication, which is cheap on modern hardware. We used a single Titan X GPU with 8 CPU workers, and PyTorch Lightning (Falcon et al., 2019). ", + "bbox": [ + 173, + 895, + 823, + 924 + ], + "page_idx": 7 + }, + { + "type": "table", + "img_path": "images/5cbfbc514efbb5768916ee08855043d4749ffe62f5255f92e599e51a8e317a0d.jpg", + "table_caption": [], + "table_footnote": [], + "table_body": "
ModelCIFAR10 (sec.)ImageNet (min.)
IR136.0 ±443.9 ±1
IRing141.1 ± 5 (1.1x)51.0±1 (1.2x)
MoCo318.4±1661.1 ±1
MoCoRing383.4 ± 12 (1.2x)64.9 ±1 (1.1x)
", + "bbox": [ + 227, + 101, + 482, + 167 + ], + "page_idx": 8 + }, + { + "type": "table", + "img_path": "images/59631d9f14bc25165deb85fa775e07545e34b054bce355042f8935d05c1d9164.jpg", + "table_caption": [ + "(b) Comparison with SOTA " + ], + "table_footnote": [], + "table_body": "
DatasetArch.MoCo-v2MoCoRing-v2
CIFAR10ResNet-1890.191.9(+1.8)
CIFAR10ResNet-5092.494.1(+1.6)
CIFAR100ResNet-1865.167.3(+2.2)
STL10ResNet-1874.876.7(+1.9)
", + "bbox": [ + 547, + 104, + 808, + 164 + ], + "page_idx": 8 + }, + { + "type": "table", + "img_path": "images/7c62f9cfe966ddea3bd2181593d0528f91803d4b4f4d6c61fe210fb97b3e50c7.jpg", + "table_caption": [ + "(a) Average Epoch Cost " + ], + "table_footnote": [ + "(c) Speech Extension " + ], + "table_body": "
Transfer TaskMoCoMoCoRing
LibriSpeech Spk.ID (Panayotov et al.201595.596.6(+1.1)
AudioMNISTBecker et al. 2018)87.491.3(+3.9)
Google CommandsWarden 1201838.541.4(+2.9)
Fluent Actions Lugosch et al.] 201936.536.8(+0.3)
Fluent ObjectsLugosch et al. 1201941.944.1(+2.2)
Fluent Locations Lugoschet al. 1201960.963.9(+3.0)
", + "bbox": [ + 210, + 188, + 532, + 263 + ], + "page_idx": 8 + }, + { + "type": "table", + "img_path": "images/1716cd92c30a2c00918d187ffa28e0c2ec8aa52a0cd43c43ae2e9cf5669d2555.jpg", + "table_caption": [ + "Table 5: Generalizations of Ring to a new modality (a) and a batch-based algorithm (b). " + ], + "table_footnote": [ + "(d) SimCLRing Extension " + ], + "table_body": "
DatasetSimCLRSimCLRing
CIFAR1088.989.3(+0.4)
CIFAR10063.564.1(+0.6)
STL1071.272.1(+0.9)
", + "bbox": [ + 594, + 200, + 792, + 251 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "", + "bbox": [ + 174, + 335, + 825, + 434 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "Comparison with the state-of-the-art. Unlike the experiments in Sec. 5, we now choose the optimal hyperparameters for MoCo-v2 (Chen et al., 2020b) separately for CIFAR10, CIFAR100, and STL10. Table $5 \\mathsf { b }$ compares MoCo-v2 and its CNCE equivalent, MoCoRing-v2 using linear evaluation. We observe comparable improvements as found in Table 1 even with optimal hyperparameters. Notably, the gains generalize to ResNet-50 encoders. Refer to Sec. F for hyperparameter choices. ", + "bbox": [ + 174, + 440, + 825, + 511 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "Generalization to other modalities. Thus far, we have focused on visual representation learning, although the same ideas apply to other domains. To exemplify the generality of CNCE, we apply MoCoRing to learning speech representations. Table $5 \\mathrm { { c } }$ reports linear evaluation on six transfer datasets, ranging from predicting speaker identity to speech recognition to intent prediction. We find significant gains of 1 to 4 percent over 4 datasets and 6 transfer tasks with an average of 2.2 absolute percentage points. See Sec. $\\mathrm { ~ E ~ }$ for experimental details. ", + "bbox": [ + 174, + 517, + 825, + 602 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "Batch-based negative sampling. In Ring, we assumed to have a memory structure that stores embeddings, which led to an efficient procedure of mining semi-hard negatives. However, another flavor of contrastive algorithms removes the memory structure entirely, using the examples in the minibatch as negatives of one another. Here, we motivate a possible extension of Ring to SimCLR, and leave more careful study to future work. In SimCLR, we are given a minibatch $M$ of examples. To sample hard negatives, as before, pick $\\ell$ and $u$ as lower and upper percentiles. For every example $x _ { i }$ in the minibatch, only consider the subset of the minibatch $\\{ x : x \\subseteq M , \\exp \\{ g _ { \\theta } ( t ( x _ { i } ) ) ^ { \\cdot } g _ { \\theta } ( t ^ { \\prime } ( x ) ) \\}$ in the $\\ell \\cdot$ -th and $u$ -th percentiles in $M \\}$ as negative examples for $x _ { i }$ . This can be efficiently implemented as a matrix operation using an element-wise mask. Thus, we ignore gradient signal for examples too far or too close to $x _ { i }$ in representation. As before, we anneal $u$ from 100 to 10 and set $\\ell = 1$ . Table $\\textcircled { 5 } \\textcircled { \\scriptsize { \\mathrm { d } } }$ report consistent but moderate gains over SimCLR, showing promise but room for improvement in future research. ", + "bbox": [ + 173, + 608, + 825, + 775 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "8 CONCLUDING REMARKS ", + "text_level": 1, + "bbox": [ + 176, + 795, + 411, + 810 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "To conclude, we presented a family of mutual information estimators that approximate the partition function using samples from a class of conditional distributions. We proved several theoretical statements about this family, showing a bound on mutual information and a tradeoff between bias and variance. Then, we applied these estimators as objectives in contrastive representation learning. In doing so, we found that our representations outperform existing approaches consistently across a spectrum of contrastive objectives, data distributions, and transfer tasks. Overall, we hope our work to encourage more exploration of negative sampling in the recent growth of contrastive learning. 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Taking inspiration from metric learning, we show that", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 141, + 299, + 470, + 313 + ], + "spans": [ + { + "bbox": [ + 141, + 299, + 470, + 313 + ], + "score": 1.0, + "content": "choosing semi-hard negatives can yield stronger contrastive representations. To do", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 142, + 312, + 469, + 322 + ], + "spans": [ + { + "bbox": [ + 142, + 312, + 469, + 322 + ], + "score": 1.0, + "content": "this, we introduce a family of mutual information estimators that sample negatives", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 142, + 323, + 470, + 334 + ], + "spans": [ + { + "bbox": [ + 142, + 323, + 470, + 334 + ], + "score": 1.0, + "content": "conditionally – in a “ring” around each positive. 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Experimentally, we find our approach, applied on top of existing mod-", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 141, + 355, + 469, + 367 + ], + "spans": [ + { + "bbox": [ + 141, + 355, + 347, + 367 + ], + "score": 1.0, + "content": "els (IR, CMC, and MoCo) improves accuracy by", + "type": "text" + }, + { + "bbox": [ + 347, + 355, + 370, + 365 + ], + "score": 0.86, + "content": "2 - 5 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 371, + 355, + 469, + 367 + ], + "score": 1.0, + "content": "absolute points in each", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 141, + 366, + 469, + 378 + ], + "spans": [ + { + "bbox": [ + 141, + 366, + 469, + 378 + ], + "score": 1.0, + "content": "case, measured by linear evaluation on four standard image benchmarks. More-", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 141, + 376, + 469, + 390 + ], + "spans": [ + { + "bbox": [ + 141, + 376, + 469, + 390 + ], + "score": 1.0, + "content": "over, we find continued benefits when transferring features to a variety of new im-", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 141, + 386, + 469, + 401 + ], + "spans": [ + { + "bbox": [ + 141, + 386, + 469, + 401 + ], + "score": 1.0, + "content": "age distributions from the Meta-Dataset collection and to a variety of downstream", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 142, + 399, + 456, + 411 + ], + "spans": [ + { + "bbox": [ + 142, + 399, + 456, + 411 + ], + "score": 1.0, + "content": "tasks such as object detection, instance segmentation, and key-point detection.", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 14.5, + "bbox_fs": [ + 141, + 234, + 470, + 411 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 432, + 206, + 444 + ], + "lines": [ + { + "bbox": [ + 105, + 430, + 208, + 447 + ], + "spans": [ + { + "bbox": [ + 105, + 430, + 208, + 447 + ], + "score": 1.0, + "content": "1 INTRODUCTION", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 23 + }, + { + "type": "text", + "bbox": [ + 107, + 456, + 505, + 557 + ], + "lines": [ + { + "bbox": [ + 106, + 456, + 506, + 468 + ], + "spans": [ + { + "bbox": [ + 106, + 456, + 506, + 468 + ], + "score": 1.0, + "content": "Supervised learning has given rise to human-level performance in several visual tasks (Russakovsky", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 466, + 505, + 479 + ], + "spans": [ + { + "bbox": [ + 105, + 466, + 505, + 479 + ], + "score": 1.0, + "content": "et al., 2015; He et al., 2017), relying heavily on large image datasets paired with semantic anno-", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 478, + 506, + 491 + ], + "spans": [ + { + "bbox": [ + 105, + 478, + 506, + 491 + ], + "score": 1.0, + "content": "tations. These annotations vary in difficulty and cost, spanning from simple class labels to more", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 488, + 505, + 502 + ], + "spans": [ + { + "bbox": [ + 105, + 488, + 505, + 502 + ], + "score": 1.0, + "content": "granular descriptions like bounding boxes and key-points. As it is impractical to scale high quality", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 500, + 505, + 512 + ], + "spans": [ + { + "bbox": [ + 105, + 500, + 505, + 512 + ], + "score": 1.0, + "content": "annotations, this reliance on supervision poses a barrier to widespread adoption. While supervised", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 511, + 505, + 523 + ], + "spans": [ + { + "bbox": [ + 106, + 511, + 505, + 523 + ], + "score": 1.0, + "content": "pretraining is still the dominant approach in computer vision, recent studies using unsupervised", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 521, + 506, + 535 + ], + "spans": [ + { + "bbox": [ + 106, + 521, + 506, + 535 + ], + "score": 1.0, + "content": "“contrastive” objectives, have achieved remarkable results in the last two years, closing the gap to", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 532, + 504, + 546 + ], + "spans": [ + { + "bbox": [ + 105, + 532, + 504, + 546 + ], + "score": 1.0, + "content": "supervised baselines (Wu et al., 2018; Oord et al., 2018; Hjelm et al., 2018; Zhuang et al., 2019;", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 543, + 501, + 557 + ], + "spans": [ + { + "bbox": [ + 105, + 543, + 392, + 557 + ], + "score": 1.0, + "content": "Henaff et al., 2019; Misra & Maaten, 2020; He et al., 2019; Chen et al.,´", + "type": "text" + }, + { + "bbox": [ + 392, + 543, + 501, + 556 + ], + "score": 1.0, + "content": "2020a;b; Grill et al., 2020)", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 28, + "bbox_fs": [ + 105, + 456, + 506, + 557 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 561, + 505, + 704 + ], + "lines": [ + { + "bbox": [ + 105, + 560, + 502, + 573 + ], + "spans": [ + { + "bbox": [ + 105, + 560, + 502, + 573 + ], + "score": 1.0, + "content": "Many contrastive algorithms are estimators of mutual information (Oord et al., 2018; Hjelm et al.,", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 572, + 502, + 584 + ], + "spans": [ + { + "bbox": [ + 106, + 572, + 502, + 584 + ], + "score": 1.0, + "content": "2018; Bachman et al., 2019), capturing the intuition that a good low-dimensional “representation”", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 582, + 505, + 595 + ], + "spans": [ + { + "bbox": [ + 105, + 582, + 505, + 595 + ], + "score": 1.0, + "content": "is one that linearizes the useful information embedded within a high-dimensional data point. In", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 594, + 504, + 606 + ], + "spans": [ + { + "bbox": [ + 105, + 594, + 504, + 606 + ], + "score": 1.0, + "content": "vision, these estimators maximize the similarity of encodings for two augmentations of the same", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 605, + 505, + 618 + ], + "spans": [ + { + "bbox": [ + 105, + 605, + 505, + 618 + ], + "score": 1.0, + "content": "image. This is trivial (e.g. assign all image pairs maximum similarity) unless the similarity function", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 615, + 506, + 628 + ], + "spans": [ + { + "bbox": [ + 105, + 615, + 506, + 628 + ], + "score": 1.0, + "content": "is normalized. This is typically done by comparing an image to “negative examples”, which a", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 626, + 505, + 640 + ], + "spans": [ + { + "bbox": [ + 105, + 626, + 505, + 640 + ], + "score": 1.0, + "content": "model must assign low similarity to. We hypothesize that how we choose these negatives greatly", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 637, + 506, + 650 + ], + "spans": [ + { + "bbox": [ + 105, + 637, + 506, + 650 + ], + "score": 1.0, + "content": "impacts the representation quality. With harder negatives, the encoder is encouraged to capture", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 649, + 505, + 660 + ], + "spans": [ + { + "bbox": [ + 105, + 649, + 505, + 660 + ], + "score": 1.0, + "content": "more granular information that may improve performance on downstream tasks. While research", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 106, + 660, + 505, + 672 + ], + "spans": [ + { + "bbox": [ + 106, + 660, + 505, + 672 + ], + "score": 1.0, + "content": "in contrastive learning has explored architectures, augmentations, and pretext tasks, there has been", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 671, + 505, + 684 + ], + "spans": [ + { + "bbox": [ + 105, + 671, + 505, + 684 + ], + "score": 1.0, + "content": "little attention given to the negative sampling procedure. Meanwhile, there is a rich body of work", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 681, + 505, + 695 + ], + "spans": [ + { + "bbox": [ + 105, + 681, + 505, + 695 + ], + "score": 1.0, + "content": "in deep metric learning showing semi-hard negative mining to improve the efficacy of triplet losses.", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 106, + 691, + 472, + 707 + ], + "spans": [ + { + "bbox": [ + 106, + 691, + 472, + 707 + ], + "score": 1.0, + "content": "Inspired by this, we hope to bring harder negative sampling to modern contrastive learning.", + "type": "text" + } + ], + "index": 45 + } + ], + "index": 39, + "bbox_fs": [ + 105, + 560, + 506, + 707 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 709, + 503, + 731 + ], + "lines": [ + { + "bbox": [ + 105, + 709, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 105, + 709, + 505, + 722 + ], + "score": 1.0, + "content": "Naively choosing difficult negatives may yield an objective that no longer bounds mutual informa-", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 720, + 505, + 732 + ], + "spans": [ + { + "bbox": [ + 105, + 720, + 505, + 732 + ], + "score": 1.0, + "content": "tion, removing a theoretical connection that is core to contrastive learning and has been shown to", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 106, + 82, + 504, + 96 + ], + "spans": [ + { + "bbox": [ + 106, + 82, + 326, + 96 + ], + "score": 1.0, + "content": "be important for downstream performance (Tian et al.,", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 326, + 82, + 352, + 95 + ], + "score": 0.36, + "content": "\\boxed { 2 0 2 0 }", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 352, + 82, + 504, + 96 + ], + "score": 1.0, + "content": ". In this paper, we present a new esti-", + "type": "text", + "cross_page": true + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 93, + 506, + 106 + ], + "spans": [ + { + "bbox": [ + 106, + 93, + 506, + 106 + ], + "score": 1.0, + "content": "mator of mutual information based on the popular noise-contrastive estimator (NCE) that supports", + "type": "text", + "cross_page": true + } + ], + "index": 1 + }, + { + "bbox": [ + 106, + 104, + 472, + 117 + ], + "spans": [ + { + "bbox": [ + 106, + 104, + 472, + 117 + ], + "score": 1.0, + "content": "sampling negatives from conditional distributions. 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We prove our Conditional-NCE (CNCE) objective to lower bound mutual information.", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 141, + 136, + 505, + 149 + ], + "spans": [ + { + "bbox": [ + 141, + 136, + 505, + 149 + ], + "score": 1.0, + "content": "Further, we show that although CNCE is a looser bound than NCE, it has lower variance.", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 141, + 145, + 350, + 161 + ], + "spans": [ + { + "bbox": [ + 141, + 145, + 350, + 161 + ], + "score": 1.0, + "content": "This motivates its value for representation learning.", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 129, + 162, + 505, + 174 + ], + "spans": [ + { + "bbox": [ + 129, + 162, + 505, + 174 + ], + "score": 1.0, + "content": "2. We use CNCE to generalize contrastive algorithms that utilize a memory structure like IR,", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 142, + 173, + 505, + 186 + ], + "spans": [ + { + "bbox": [ + 142, + 173, + 505, + 186 + ], + "score": 1.0, + "content": "CMC, and MoCo to sample semi-hard negatives in just a few lines of code and minimal", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 141, + 185, + 220, + 196 + ], + "spans": [ + { + "bbox": [ + 141, + 185, + 220, + 196 + ], + "score": 1.0, + "content": "compute overhead.", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 128, + 198, + 505, + 212 + ], + "spans": [ + { + "bbox": [ + 128, + 198, + 505, + 212 + ], + "score": 1.0, + "content": "3. We find that the naive strategy of sampling hard negatives throughout training can be detri-", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 141, + 209, + 500, + 223 + ], + "spans": [ + { + "bbox": [ + 141, + 209, + 500, + 223 + ], + "score": 1.0, + "content": "mental. We then show that slowly introducing harder negatives yields good performance.", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 129, + 224, + 505, + 237 + ], + "spans": [ + { + "bbox": [ + 129, + 224, + 415, + 237 + ], + "score": 1.0, + "content": "4. 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Many of these objectives (Hjelm et al., 2018; Wu et al., 2018; Bachman", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 320, + 506, + 334 + ], + "spans": [ + { + "bbox": [ + 105, + 320, + 155, + 333 + ], + "score": 1.0, + "content": "et al., 2019;", + "type": "text" + }, + { + "bbox": [ + 158, + 320, + 228, + 333 + ], + "score": 1.0, + "content": "Tian et al., 2019;", + "type": "text" + }, + { + "bbox": [ + 229, + 321, + 277, + 332 + ], + "score": 1.0, + "content": "Chen et al.,", + "type": "text" + }, + { + "bbox": [ + 277, + 320, + 506, + 334 + ], + "score": 1.0, + "content": "2020a) are equivalent to NCE (Oord et al., 2018; Poole", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 331, + 506, + 344 + ], + "spans": [ + { + "bbox": [ + 105, + 331, + 129, + 344 + ], + "score": 1.0, + "content": "et al.,", + "type": "text" + }, + { + "bbox": [ + 129, + 332, + 155, + 344 + ], + "score": 0.37, + "content": "\\boxed { 2 0 1 9 }", + "type": "inline_equation" + }, + { + "bbox": [ + 155, + 331, + 407, + 344 + ], + "score": 1.0, + "content": ", a popular lower bound on the mutual information, denoted by", + "type": "text" + }, + { + "bbox": [ + 408, + 332, + 415, + 342 + ], + "score": 0.75, + "content": "\\overline { { \\mathcal { T } } }", + "type": "inline_equation" + }, + { + "bbox": [ + 415, + 331, + 506, + 344 + ], + "score": 1.0, + "content": ", between two random", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 342, + 506, + 355 + ], + "spans": [ + { + "bbox": [ + 105, + 342, + 506, + 355 + ], + "score": 1.0, + "content": "variables. 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We prove our Conditional-NCE (CNCE) objective to lower bound mutual information.", + "type": "text" + } + ], + "index": 3, + "is_list_start_line": true + }, + { + "bbox": [ + 141, + 136, + 505, + 149 + ], + "spans": [ + { + "bbox": [ + 141, + 136, + 505, + 149 + ], + "score": 1.0, + "content": "Further, we show that although CNCE is a looser bound than NCE, it has lower variance.", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 141, + 145, + 350, + 161 + ], + "spans": [ + { + "bbox": [ + 141, + 145, + 350, + 161 + ], + "score": 1.0, + "content": "This motivates its value for representation learning.", + "type": "text" + } + ], + "index": 5, + "is_list_end_line": true + }, + { + "bbox": [ + 129, + 162, + 505, + 174 + ], + "spans": [ + { + "bbox": [ + 129, + 162, + 505, + 174 + ], + "score": 1.0, + "content": "2. We use CNCE to generalize contrastive algorithms that utilize a memory structure like IR,", + "type": "text" + } + ], + "index": 6, + "is_list_start_line": true + }, + { + "bbox": [ + 142, + 173, + 505, + 186 + ], + "spans": [ + { + "bbox": [ + 142, + 173, + 505, + 186 + ], + "score": 1.0, + "content": "CMC, and MoCo to sample semi-hard negatives in just a few lines of code and minimal", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 141, + 185, + 220, + 196 + ], + "spans": [ + { + "bbox": [ + 141, + 185, + 220, + 196 + ], + "score": 1.0, + "content": "compute overhead.", + "type": "text" + } + ], + "index": 8, + "is_list_end_line": true + }, + { + "bbox": [ + 128, + 198, + 505, + 212 + ], + "spans": [ + { + "bbox": [ + 128, + 198, + 505, + 212 + ], + "score": 1.0, + "content": "3. We find that the naive strategy of sampling hard negatives throughout training can be detri-", + "type": "text" + } + ], + "index": 9, + "is_list_start_line": true + }, + { + "bbox": [ + 141, + 209, + 500, + 223 + ], + "spans": [ + { + "bbox": [ + 141, + 209, + 500, + 223 + ], + "score": 1.0, + "content": "mental. We then show that slowly introducing harder negatives yields good performance.", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 129, + 224, + 505, + 237 + ], + "spans": [ + { + "bbox": [ + 129, + 224, + 415, + 237 + ], + "score": 1.0, + "content": "4. 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One of our contributions", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 104, + 355, + 506, + 370 + ], + "spans": [ + { + "bbox": [ + 104, + 355, + 337, + 370 + ], + "score": 1.0, + "content": "is to formally define a family of conditional distributions", + "type": "text" + }, + { + "bbox": [ + 337, + 358, + 347, + 367 + ], + "score": 0.7, + "content": "\\mathcal { Q }", + "type": "inline_equation" + }, + { + "bbox": [ + 347, + 355, + 417, + 370 + ], + "score": 1.0, + "content": "such that for any", + "type": "text" + }, + { + "bbox": [ + 418, + 356, + 466, + 369 + ], + "score": 0.92, + "content": "q ( y | x ) \\in \\mathcal { Q }", + "type": "inline_equation" + }, + { + "bbox": [ + 466, + 355, + 506, + 370 + ], + "score": 1.0, + "content": ", drawing", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 367, + 505, + 379 + ], + "spans": [ + { + "bbox": [ + 105, + 367, + 204, + 379 + ], + "score": 1.0, + "content": "negative examples from", + "type": "text" + }, + { + "bbox": [ + 204, + 369, + 210, + 379 + ], + "score": 0.72, + "content": "q", + "type": "inline_equation" + }, + { + "bbox": [ + 211, + 367, + 367, + 379 + ], + "score": 1.0, + "content": "defines an estimator that lower bounds", + "type": "text" + }, + { + "bbox": [ + 368, + 367, + 404, + 379 + ], + "score": 0.92, + "content": "\\mathcal { I } ( X ; Y )", + "type": "inline_equation" + }, + { + "bbox": [ + 405, + 367, + 505, + 379 + ], + "score": 1.0, + "content": ". We call this new bound", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 378, + 477, + 390 + ], + "spans": [ + { + "bbox": [ + 106, + 378, + 477, + 390 + ], + "score": 1.0, + "content": "the Conditional Noise Contrastive Estimator, or CNCE. We first prove CNCE to be a bound:", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 19 + }, + { + "type": "text", + "bbox": [ + 106, + 393, + 505, + 462 + ], + "lines": [ + { + "bbox": [ + 106, + 393, + 504, + 405 + ], + "spans": [ + { + "bbox": [ + 106, + 393, + 329, + 405 + ], + "score": 1.0, + "content": "Theorem 3.1. (The Conditional NCE bound) Define", + "type": "text" + }, + { + "bbox": [ + 329, + 394, + 335, + 403 + ], + "score": 0.65, + "content": "d", + "type": "inline_equation" + }, + { + "bbox": [ + 336, + 393, + 464, + 405 + ], + "score": 1.0, + "content": "-dimensional random variables", + "type": "text" + }, + { + "bbox": [ + 464, + 394, + 474, + 403 + ], + "score": 0.78, + "content": "X", + "type": "inline_equation" + }, + { + "bbox": [ + 475, + 393, + 494, + 405 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 495, + 394, + 504, + 403 + ], + "score": 0.73, + "content": "Y", + "type": "inline_equation" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 404, + 505, + 417 + ], + "spans": [ + { + "bbox": [ + 106, + 404, + 196, + 417 + ], + "score": 1.0, + "content": "by a joint distribution", + "type": "text" + }, + { + "bbox": [ + 197, + 405, + 226, + 417 + ], + "score": 0.92, + "content": "p ( x , y )", + "type": "inline_equation" + }, + { + "bbox": [ + 226, + 404, + 258, + 417 + ], + "score": 1.0, + "content": "and let", + "type": "text" + }, + { + "bbox": [ + 258, + 405, + 297, + 416 + ], + "score": 0.91, + "content": "Y _ { 1 } , . . . , Y _ { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 297, + 404, + 371, + 417 + ], + "score": 1.0, + "content": "be i.i.d. copies of", + "type": "text" + }, + { + "bbox": [ + 372, + 405, + 381, + 415 + ], + "score": 0.74, + "content": "Y", + "type": "inline_equation" + }, + { + "bbox": [ + 381, + 404, + 505, + 417 + ], + "score": 1.0, + "content": "with the marginal distribution", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 415, + 507, + 432 + ], + "spans": [ + { + "bbox": [ + 106, + 417, + 125, + 429 + ], + "score": 0.9, + "content": "p ( y )", + "type": "inline_equation" + }, + { + "bbox": [ + 126, + 415, + 199, + 432 + ], + "score": 1.0, + "content": ". Fix any function", + "type": "text" + }, + { + "bbox": [ + 200, + 417, + 269, + 429 + ], + "score": 0.91, + "content": "f : ( X , Y ) \\to \\mathbf { R }", + "type": "inline_equation" + }, + { + "bbox": [ + 270, + 415, + 336, + 432 + ], + "score": 1.0, + "content": ", any realization", + "type": "text" + }, + { + "bbox": [ + 336, + 419, + 343, + 427 + ], + "score": 0.61, + "content": "x", + "type": "inline_equation" + }, + { + "bbox": [ + 344, + 415, + 356, + 432 + ], + "score": 1.0, + "content": "of", + "type": "text" + }, + { + "bbox": [ + 356, + 417, + 365, + 427 + ], + "score": 0.79, + "content": "X", + "type": "inline_equation" + }, + { + "bbox": [ + 366, + 415, + 400, + 432 + ], + "score": 1.0, + "content": ", and let", + "type": "text" + }, + { + "bbox": [ + 401, + 416, + 486, + 430 + ], + "score": 0.92, + "content": "c = \\mathbf { E } _ { y \\sim p ( y ) } [ e ^ { f ( x , y ) } ] ,", + "type": "inline_equation" + }, + { + "bbox": [ + 487, + 415, + 507, + 432 + ], + "score": 1.0, + "content": ", the", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 427, + 506, + 440 + ], + "spans": [ + { + "bbox": [ + 105, + 427, + 285, + 440 + ], + "score": 1.0, + "content": "expected exponentiated similarity. Pick a set", + "type": "text" + }, + { + "bbox": [ + 285, + 428, + 316, + 438 + ], + "score": 0.89, + "content": "B \\subset \\mathbf { R }", + "type": "inline_equation" + }, + { + "bbox": [ + 316, + 427, + 506, + 440 + ], + "score": 1.0, + "content": "strictly lower-bounded by c. Assume the pulled", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 104, + 437, + 507, + 452 + ], + "spans": [ + { + "bbox": [ + 104, + 437, + 142, + 452 + ], + "score": 1.0, + "content": "back set", + "type": "text" + }, + { + "bbox": [ + 142, + 438, + 237, + 451 + ], + "score": 0.91, + "content": "S _ { B } \\sp { \\bullet } = \\{ y | e \\sp { f ( x , y ) } \\in B \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 237, + 437, + 358, + 452 + ], + "score": 1.0, + "content": "has non-zero probability (i.e.", + "type": "text" + }, + { + "bbox": [ + 359, + 439, + 406, + 451 + ], + "score": 0.92, + "content": "p ( S _ { B } ) > 0 ,", + "type": "inline_equation" + }, + { + "bbox": [ + 407, + 437, + 430, + 452 + ], + "score": 1.0, + "content": "). For", + "type": "text" + }, + { + "bbox": [ + 431, + 439, + 478, + 450 + ], + "score": 0.93, + "content": "A _ { 1 } , \\ldots , A _ { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 478, + 437, + 507, + 452 + ], + "score": 1.0, + "content": "in the", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 449, + 352, + 462 + ], + "spans": [ + { + "bbox": [ + 106, + 449, + 131, + 462 + ], + "score": 1.0, + "content": "Borel", + "type": "text" + }, + { + "bbox": [ + 131, + 451, + 138, + 460 + ], + "score": 0.72, + "content": "\\sigma", + "type": "inline_equation" + }, + { + "bbox": [ + 138, + 449, + 193, + 462 + ], + "score": 1.0, + "content": "-algebra over", + "type": "text" + }, + { + "bbox": [ + 194, + 450, + 207, + 460 + ], + "score": 0.79, + "content": "\\mathbf { R } ^ { d }", + "type": "inline_equation" + }, + { + "bbox": [ + 208, + 449, + 238, + 462 + ], + "score": 1.0, + "content": ", define", + "type": "text" + }, + { + "bbox": [ + 239, + 450, + 320, + 461 + ], + "score": 0.92, + "content": "A = A _ { 1 } \\times \\ldots \\times A _ { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 320, + 449, + 352, + 462 + ], + "score": 1.0, + "content": "and let", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 26.5 + }, + { + "type": "interline_equation", + "bbox": [ + 211, + 467, + 399, + 503 + ], + "lines": [ + { + "bbox": [ + 211, + 467, + 399, + 503 + ], + "spans": [ + { + "bbox": [ + 211, + 467, + 399, + 503 + ], + "score": 0.93, + "content": "q ( ( Y _ { 1 } , \\dots , Y _ { k } ) \\in A | X = x ) = \\prod _ { j = 1 } ^ { k } p ( A _ { j } | S _ { B } ) .", + "type": "interline_equation", + "image_path": "d6a9632fe22fac78ee179735a33e26f23bc90b27f9176ff009e648e566f18f2b.jpg" + } + ] + } + ], + "index": 30.5, + "virtual_lines": [ + { + "bbox": [ + 211, + 467, + 399, + 485.0 + ], + "spans": [], + "index": 30 + }, + { + "bbox": [ + 211, + 485.0, + 399, + 503.0 + ], + "spans": [], + "index": 31 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 549, + 505, + 604 + ], + "lines": [ + { + "bbox": [ + 101, + 541, + 506, + 573 + ], + "spans": [ + { + "bbox": [ + 101, + 541, + 173, + 573 + ], + "score": 1.0, + "content": "Proof. To show", + "type": "text" + }, + { + "bbox": [ + 173, + 552, + 234, + 563 + ], + "score": 0.9, + "content": "\\mathcal { T } _ { \\mathrm { C N C E } } \\leq \\mathcal { T } _ { \\mathrm { N C E } }", + "type": "inline_equation" + }, + { + "bbox": [ + 234, + 541, + 277, + 573 + ], + "score": 1.0, + "content": ", we show", + "type": "text" + }, + { + "bbox": [ + 277, + 550, + 470, + 564 + ], + "score": 0.91, + "content": "\\begin{array} { r } { { \\bf E } _ { p } [ \\log \\sum _ { j = 1 } ^ { k } e ^ { f ( x , y _ { j } ) } ] < { \\bf E } _ { q } [ \\log \\sum _ { j = 1 } ^ { k } e ^ { f ( x , y _ { j } ) } ] } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 470, + 541, + 506, + 573 + ], + "score": 1.0, + "content": ". To see", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 102, + 561, + 509, + 586 + ], + "spans": [ + { + "bbox": [ + 102, + 561, + 276, + 586 + ], + "score": 1.0, + "content": "this, apply Jensen’s to the left-hand side of", + "type": "text" + }, + { + "bbox": [ + 276, + 565, + 451, + 581 + ], + "score": 0.9, + "content": "\\begin{array} { r } { \\log \\mathbf { E } _ { p } [ \\sum _ { j = 1 } ^ { k } e ^ { f ( x , y _ { j } ) } ] < \\log \\sum _ { j = 1 } ^ { k } e ^ { f ( x , y _ { j } ) } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 451, + 561, + 509, + 586 + ], + "score": 1.0, + "content": ", which holds", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 578, + 505, + 592 + ], + "spans": [ + { + "bbox": [ + 105, + 578, + 115, + 592 + ], + "score": 1.0, + "content": "if", + "type": "text" + }, + { + "bbox": [ + 116, + 579, + 154, + 591 + ], + "score": 0.92, + "content": "y _ { j } \\in S _ { B }", + "type": "inline_equation" + }, + { + "bbox": [ + 154, + 578, + 171, + 592 + ], + "score": 1.0, + "content": "for", + "type": "text" + }, + { + "bbox": [ + 171, + 579, + 225, + 591 + ], + "score": 0.91, + "content": "j = 1 , \\dotsc , k", + "type": "inline_equation" + }, + { + "bbox": [ + 225, + 578, + 352, + 592 + ], + "score": 1.0, + "content": ", and then take the expectation", + "type": "text" + }, + { + "bbox": [ + 352, + 579, + 365, + 591 + ], + "score": 0.86, + "content": "\\mathbf { E } _ { q }", + "type": "inline_equation" + }, + { + "bbox": [ + 365, + 578, + 505, + 592 + ], + "score": 1.0, + "content": "of both sides. The last inequality", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 104, + 589, + 504, + 605 + ], + "spans": [ + { + "bbox": [ + 104, + 589, + 381, + 605 + ], + "score": 1.0, + "content": "holds by monoticity of log, linearity of expectation, and the fact that", + "type": "text" + }, + { + "bbox": [ + 381, + 590, + 474, + 604 + ], + "score": 0.92, + "content": "{ \\bf E } _ { p } [ e ^ { f ( x , y _ { j } ) } ] \\le e ^ { f ( x , y _ { j } ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 475, + 589, + 504, + 605 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 33.5 + }, + { + "type": "text", + "bbox": [ + 106, + 617, + 505, + 712 + ], + "lines": [ + { + "bbox": [ + 106, + 617, + 505, + 630 + ], + "spans": [ + { + "bbox": [ + 106, + 617, + 505, + 630 + ], + "score": 1.0, + "content": "Theorem Intuition. For intuition, although using arbitrary negative distributions in NCE does not", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 628, + 505, + 641 + ], + "spans": [ + { + "bbox": [ + 105, + 628, + 408, + 641 + ], + "score": 1.0, + "content": "bound mutual information, we have found a restricted class of distributions", + "type": "text" + }, + { + "bbox": [ + 408, + 629, + 418, + 639 + ], + "score": 0.82, + "content": "\\mathcal { Q }", + "type": "inline_equation" + }, + { + "bbox": [ + 419, + 628, + 505, + 641 + ], + "score": 1.0, + "content": "where every member", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 107, + 639, + 506, + 652 + ], + "spans": [ + { + "bbox": [ + 107, + 639, + 134, + 651 + ], + "score": 0.92, + "content": "q ( y | x )", + "type": "inline_equation" + }, + { + "bbox": [ + 134, + 639, + 301, + 652 + ], + "score": 1.0, + "content": "“subsets the support” of the distribution", + "type": "text" + }, + { + "bbox": [ + 301, + 639, + 320, + 651 + ], + "score": 0.91, + "content": "p ( y )", + "type": "inline_equation" + }, + { + "bbox": [ + 320, + 639, + 457, + 652 + ], + "score": 1.0, + "content": ". That is, given some fixed value", + "type": "text" + }, + { + "bbox": [ + 458, + 642, + 465, + 650 + ], + "score": 0.72, + "content": "x", + "type": "inline_equation" + }, + { + "bbox": [ + 465, + 639, + 506, + 652 + ], + "score": 1.0, + "content": ", we have", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 106, + 650, + 506, + 663 + ], + "spans": [ + { + "bbox": [ + 106, + 650, + 139, + 663 + ], + "score": 1.0, + "content": "defined", + "type": "text" + }, + { + "bbox": [ + 140, + 650, + 167, + 662 + ], + "score": 0.93, + "content": "q ( y | x )", + "type": "inline_equation" + }, + { + "bbox": [ + 167, + 650, + 282, + 663 + ], + "score": 1.0, + "content": "to constrain the support of", + "type": "text" + }, + { + "bbox": [ + 282, + 651, + 302, + 663 + ], + "score": 0.91, + "content": "p ( y )", + "type": "inline_equation" + }, + { + "bbox": [ + 302, + 650, + 338, + 663 + ], + "score": 1.0, + "content": "to a set", + "type": "text" + }, + { + "bbox": [ + 339, + 651, + 353, + 662 + ], + "score": 0.88, + "content": "S _ { B }", + "type": "inline_equation" + }, + { + "bbox": [ + 353, + 650, + 484, + 663 + ], + "score": 1.0, + "content": "whose members are “close” to", + "type": "text" + }, + { + "bbox": [ + 485, + 653, + 492, + 660 + ], + "score": 0.76, + "content": "x", + "type": "inline_equation" + }, + { + "bbox": [ + 492, + 650, + 506, + 663 + ], + "score": 1.0, + "content": "as", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 660, + 506, + 675 + ], + "spans": [ + { + "bbox": [ + 105, + 660, + 254, + 675 + ], + "score": 1.0, + "content": "measured by the similarity function", + "type": "text" + }, + { + "bbox": [ + 255, + 662, + 262, + 673 + ], + "score": 0.83, + "content": "f", + "type": "inline_equation" + }, + { + "bbox": [ + 262, + 660, + 345, + 675 + ], + "score": 1.0, + "content": ". For every element", + "type": "text" + }, + { + "bbox": [ + 346, + 662, + 380, + 673 + ], + "score": 0.91, + "content": "y \\in S _ { B }", + "type": "inline_equation" + }, + { + "bbox": [ + 380, + 660, + 450, + 675 + ], + "score": 1.0, + "content": ", the distribution", + "type": "text" + }, + { + "bbox": [ + 450, + 661, + 477, + 673 + ], + "score": 0.92, + "content": "q ( y | x )", + "type": "inline_equation" + }, + { + "bbox": [ + 477, + 660, + 506, + 675 + ], + "score": 1.0, + "content": "wants", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 104, + 672, + 506, + 686 + ], + "spans": [ + { + "bbox": [ + 104, + 672, + 262, + 686 + ], + "score": 1.0, + "content": "to assign to it the same probability as", + "type": "text" + }, + { + "bbox": [ + 262, + 673, + 281, + 685 + ], + "score": 0.91, + "content": "p ( y )", + "type": "inline_equation" + }, + { + "bbox": [ + 282, + 672, + 341, + 686 + ], + "score": 1.0, + "content": ". However, as", + "type": "text" + }, + { + "bbox": [ + 341, + 673, + 368, + 685 + ], + "score": 0.91, + "content": "q ( y | x )", + "type": "inline_equation" + }, + { + "bbox": [ + 369, + 672, + 472, + 686 + ], + "score": 1.0, + "content": "is not defined outside of", + "type": "text" + }, + { + "bbox": [ + 472, + 673, + 486, + 684 + ], + "score": 0.89, + "content": "S _ { B }", + "type": "inline_equation" + }, + { + "bbox": [ + 486, + 672, + 506, + 686 + ], + "score": 1.0, + "content": ", we", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 102, + 680, + 507, + 703 + ], + "spans": [ + { + "bbox": [ + 102, + 680, + 271, + 703 + ], + "score": 1.0, + "content": "must renormalize it to sum to one (hence", + "type": "text" + }, + { + "bbox": [ + 271, + 684, + 367, + 701 + ], + "score": 0.93, + "content": "\\begin{array} { r } { p ( A _ { j } | S _ { B } ) = \\frac { p ( A _ { j } \\cap S _ { B } ) } { p ( S _ { B } ) } ) } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 367, + 680, + 417, + 703 + ], + "score": 1.0, + "content": ". 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The primary distinction then, is the smaller", + "type": "text" + } + ], + "index": 43 + } + ], + "index": 39.5 + } + ], + "page_idx": 2, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 108, + 27, + 292, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 25, + 293, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 25, + 293, + 38 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2021", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 120, + 720, + 275, + 733 + ], + "lines": [ + { + "bbox": [ + 119, + 719, + 275, + 735 + ], + "spans": [ + { + "bbox": [ + 119, + 719, + 275, + 735 + ], + "score": 1.0, + "content": "1 We provide a counterexample in Sec. A.1.", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 302, + 751, + 309, + 760 + ], + "lines": [ + { + "bbox": [ + 301, + 750, + 310, + 762 + ], + "spans": [ + { + "bbox": [ + 301, + 750, + 310, + 762 + ], + "score": 1.0, + "content": "3", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 82, + 504, + 127 + ], + "lines": [], + "index": 1.5, + "bbox_fs": [ + 105, + 82, + 506, + 129 + ], + "lines_deleted": true + }, + { + "type": "text", + "bbox": [ + 107, + 132, + 505, + 165 + ], + "lines": [ + { + "bbox": [ + 105, + 131, + 505, + 146 + ], + "spans": [ + { + "bbox": [ + 105, + 131, + 505, + 146 + ], + "score": 1.0, + "content": "In this work, we focus on contrastive algorithms that utilize a memory structure that we repurpose", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 106, + 142, + 506, + 157 + ], + "spans": [ + { + "bbox": [ + 106, + 142, + 135, + 157 + ], + "score": 1.0, + "content": "in Sec.", + "type": "text" + }, + { + "bbox": [ + 135, + 143, + 145, + 156 + ], + "score": 0.62, + "content": "\\boxed { 4 }", + "type": "inline_equation" + }, + { + "bbox": [ + 145, + 142, + 345, + 157 + ], + "score": 1.0, + "content": "to efficiently sample hard negatives from. In Sec.", + "type": "text" + }, + { + "bbox": [ + 345, + 143, + 355, + 156 + ], + "score": 0.53, + "content": "\\textcircled { 7 }", + "type": "inline_equation" + }, + { + "bbox": [ + 356, + 142, + 506, + 157 + ], + "score": 1.0, + "content": "we briefly discuss generalizations to", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 154, + 342, + 167 + ], + "spans": [ + { + "bbox": [ + 105, + 154, + 342, + 167 + ], + "score": 1.0, + "content": "contrastive algorithms that do not use a memory structure.", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 5, + "bbox_fs": [ + 105, + 131, + 506, + 167 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 182, + 381, + 195 + ], + "lines": [ + { + "bbox": [ + 104, + 181, + 383, + 198 + ], + "spans": [ + { + "bbox": [ + 104, + 181, + 383, + 198 + ], + "score": 1.0, + "content": "3 CONDITIONAL NOISE CONTRASTIVE ESTIMATION", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 7 + }, + { + "type": "text", + "bbox": [ + 106, + 208, + 505, + 285 + ], + "lines": [ + { + "bbox": [ + 105, + 207, + 506, + 221 + ], + "spans": [ + { + "bbox": [ + 105, + 207, + 433, + 221 + ], + "score": 1.0, + "content": "In NCE, the negative examples are sampled i.i.d. from the marginal distribution,", + "type": "text" + }, + { + "bbox": [ + 433, + 208, + 452, + 220 + ], + "score": 0.9, + "content": "p ( y )", + "type": "inline_equation" + }, + { + "bbox": [ + 452, + 207, + 506, + 221 + ], + "score": 1.0, + "content": ". Indeed, the", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 218, + 505, + 232 + ], + "spans": [ + { + "bbox": [ + 105, + 218, + 347, + 232 + ], + "score": 1.0, + "content": "existing proof that NCE lower bounds mutual information", + "type": "text" + }, + { + "bbox": [ + 347, + 219, + 426, + 232 + ], + "score": 0.6, + "content": "\\pm { \\overline { { \\mathrm { ( P o o l e ~ e t ~ a l . } } } } , \\pm { \\overline { { 2 0 1 9 } } } )", + "type": "inline_equation" + }, + { + "bbox": [ + 427, + 218, + 505, + 232 + ], + "score": 1.0, + "content": "assumes this to be", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 106, + 230, + 505, + 242 + ], + "spans": [ + { + "bbox": [ + 106, + 230, + 505, + 242 + ], + "score": 1.0, + "content": "true. However, choosing negatives in this manner may not be the best choice for learning a good", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 241, + 505, + 253 + ], + "spans": [ + { + "bbox": [ + 105, + 241, + 505, + 253 + ], + "score": 1.0, + "content": "representation. For instance, prior work in metric learning has shown the effectiveness of semi-", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 252, + 503, + 263 + ], + "spans": [ + { + "bbox": [ + 106, + 252, + 503, + 263 + ], + "score": 1.0, + "content": "hard negative mining in optimizing triplet losses (Wu et al., 2017; Yuan et al., 2017; Schroff et al.,", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 262, + 505, + 275 + ], + "spans": [ + { + "bbox": [ + 105, + 262, + 131, + 275 + ], + "score": 0.32, + "content": "\\boxed { 2 0 1 5 }", + "type": "inline_equation" + }, + { + "bbox": [ + 131, + 262, + 505, + 275 + ], + "score": 1.0, + "content": ". 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This does not, however, imply that we can", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 344, + 505, + 359 + ], + "spans": [ + { + "bbox": [ + 105, + 344, + 221, + 359 + ], + "score": 1.0, + "content": "only sample negatives from", + "type": "text" + }, + { + "bbox": [ + 224, + 345, + 396, + 358 + ], + "score": 0.68, + "content": "\\upsilon ( y ) ~ ( { \\mathrm { P o o l e ~ e t ~ a l . } } ) ~ [ { \\underline { { { 2 0 1 9 } } } } \\vert ~ [ { \\mathrm { O o r d ~ e t ~ a l . } } ] [ { \\underline { { { 2 0 1 8 } } } } )", + "type": "inline_equation" + }, + { + "bbox": [ + 396, + 344, + 505, + 359 + ], + "score": 1.0, + "content": ". One of our contributions", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 104, + 355, + 506, + 370 + ], + "spans": [ + { + "bbox": [ + 104, + 355, + 337, + 370 + ], + "score": 1.0, + "content": "is to formally define a family of conditional distributions", + "type": "text" + }, + { + "bbox": [ + 337, + 358, + 347, + 367 + ], + "score": 0.7, + "content": "\\mathcal { Q }", + "type": "inline_equation" + }, + { + "bbox": [ + 347, + 355, + 417, + 370 + ], + "score": 1.0, + "content": "such that for any", + "type": "text" + }, + { + "bbox": [ + 418, + 356, + 466, + 369 + ], + "score": 0.92, + "content": "q ( y | x ) \\in \\mathcal { Q }", + "type": "inline_equation" + }, + { + "bbox": [ + 466, + 355, + 506, + 370 + ], + "score": 1.0, + "content": ", drawing", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 367, + 505, + 379 + ], + "spans": [ + { + "bbox": [ + 105, + 367, + 204, + 379 + ], + "score": 1.0, + "content": "negative examples from", + "type": "text" + }, + { + "bbox": [ + 204, + 369, + 210, + 379 + ], + "score": 0.72, + "content": "q", + "type": "inline_equation" + }, + { + "bbox": [ + 211, + 367, + 367, + 379 + ], + "score": 1.0, + "content": "defines an estimator that lower bounds", + "type": "text" + }, + { + "bbox": [ + 368, + 367, + 404, + 379 + ], + "score": 0.92, + "content": "\\mathcal { I } ( X ; Y )", + "type": "inline_equation" + }, + { + "bbox": [ + 405, + 367, + 505, + 379 + ], + "score": 1.0, + "content": ". We call this new bound", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 378, + 477, + 390 + ], + "spans": [ + { + "bbox": [ + 106, + 378, + 477, + 390 + ], + "score": 1.0, + "content": "the Conditional Noise Contrastive Estimator, or CNCE. We first prove CNCE to be a bound:", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 19, + "bbox_fs": [ + 104, + 290, + 506, + 390 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 393, + 505, + 462 + ], + "lines": [ + { + "bbox": [ + 106, + 393, + 504, + 405 + ], + "spans": [ + { + "bbox": [ + 106, + 393, + 329, + 405 + ], + "score": 1.0, + "content": "Theorem 3.1. (The Conditional NCE bound) Define", + "type": "text" + }, + { + "bbox": [ + 329, + 394, + 335, + 403 + ], + "score": 0.65, + "content": "d", + "type": "inline_equation" + }, + { + "bbox": [ + 336, + 393, + 464, + 405 + ], + "score": 1.0, + "content": "-dimensional random variables", + "type": "text" + }, + { + "bbox": [ + 464, + 394, + 474, + 403 + ], + "score": 0.78, + "content": "X", + "type": "inline_equation" + }, + { + "bbox": [ + 475, + 393, + 494, + 405 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 495, + 394, + 504, + 403 + ], + "score": 0.73, + "content": "Y", + "type": "inline_equation" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 404, + 505, + 417 + ], + "spans": [ + { + "bbox": [ + 106, + 404, + 196, + 417 + ], + "score": 1.0, + "content": "by a joint distribution", + "type": "text" + }, + { + "bbox": [ + 197, + 405, + 226, + 417 + ], + "score": 0.92, + "content": "p ( x , y )", + "type": "inline_equation" + }, + { + "bbox": [ + 226, + 404, + 258, + 417 + ], + "score": 1.0, + "content": "and let", + "type": "text" + }, + { + "bbox": [ + 258, + 405, + 297, + 416 + ], + "score": 0.91, + "content": "Y _ { 1 } , . . . , Y _ { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 297, + 404, + 371, + 417 + ], + "score": 1.0, + "content": "be i.i.d. copies of", + "type": "text" + }, + { + "bbox": [ + 372, + 405, + 381, + 415 + ], + "score": 0.74, + "content": "Y", + "type": "inline_equation" + }, + { + "bbox": [ + 381, + 404, + 505, + 417 + ], + "score": 1.0, + "content": "with the marginal distribution", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 415, + 507, + 432 + ], + "spans": [ + { + "bbox": [ + 106, + 417, + 125, + 429 + ], + "score": 0.9, + "content": "p ( y )", + "type": "inline_equation" + }, + { + "bbox": [ + 126, + 415, + 199, + 432 + ], + "score": 1.0, + "content": ". Fix any function", + "type": "text" + }, + { + "bbox": [ + 200, + 417, + 269, + 429 + ], + "score": 0.91, + "content": "f : ( X , Y ) \\to \\mathbf { R }", + "type": "inline_equation" + }, + { + "bbox": [ + 270, + 415, + 336, + 432 + ], + "score": 1.0, + "content": ", any realization", + "type": "text" + }, + { + "bbox": [ + 336, + 419, + 343, + 427 + ], + "score": 0.61, + "content": "x", + "type": "inline_equation" + }, + { + "bbox": [ + 344, + 415, + 356, + 432 + ], + "score": 1.0, + "content": "of", + "type": "text" + }, + { + "bbox": [ + 356, + 417, + 365, + 427 + ], + "score": 0.79, + "content": "X", + "type": "inline_equation" + }, + { + "bbox": [ + 366, + 415, + 400, + 432 + ], + "score": 1.0, + "content": ", and let", + "type": "text" + }, + { + "bbox": [ + 401, + 416, + 486, + 430 + ], + "score": 0.92, + "content": "c = \\mathbf { E } _ { y \\sim p ( y ) } [ e ^ { f ( x , y ) } ] ,", + "type": "inline_equation" + }, + { + "bbox": [ + 487, + 415, + 507, + 432 + ], + "score": 1.0, + "content": ", the", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 427, + 506, + 440 + ], + "spans": [ + { + "bbox": [ + 105, + 427, + 285, + 440 + ], + "score": 1.0, + "content": "expected exponentiated similarity. Pick a set", + "type": "text" + }, + { + "bbox": [ + 285, + 428, + 316, + 438 + ], + "score": 0.89, + "content": "B \\subset \\mathbf { R }", + "type": "inline_equation" + }, + { + "bbox": [ + 316, + 427, + 506, + 440 + ], + "score": 1.0, + "content": "strictly lower-bounded by c. Assume the pulled", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 104, + 437, + 507, + 452 + ], + "spans": [ + { + "bbox": [ + 104, + 437, + 142, + 452 + ], + "score": 1.0, + "content": "back set", + "type": "text" + }, + { + "bbox": [ + 142, + 438, + 237, + 451 + ], + "score": 0.91, + "content": "S _ { B } \\sp { \\bullet } = \\{ y | e \\sp { f ( x , y ) } \\in B \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 237, + 437, + 358, + 452 + ], + "score": 1.0, + "content": "has non-zero probability (i.e.", + "type": "text" + }, + { + "bbox": [ + 359, + 439, + 406, + 451 + ], + "score": 0.92, + "content": "p ( S _ { B } ) > 0 ,", + "type": "inline_equation" + }, + { + "bbox": [ + 407, + 437, + 430, + 452 + ], + "score": 1.0, + "content": "). 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To show", + "type": "text" + }, + { + "bbox": [ + 173, + 552, + 234, + 563 + ], + "score": 0.9, + "content": "\\mathcal { T } _ { \\mathrm { C N C E } } \\leq \\mathcal { T } _ { \\mathrm { N C E } }", + "type": "inline_equation" + }, + { + "bbox": [ + 234, + 541, + 277, + 573 + ], + "score": 1.0, + "content": ", we show", + "type": "text" + }, + { + "bbox": [ + 277, + 550, + 470, + 564 + ], + "score": 0.91, + "content": "\\begin{array} { r } { { \\bf E } _ { p } [ \\log \\sum _ { j = 1 } ^ { k } e ^ { f ( x , y _ { j } ) } ] < { \\bf E } _ { q } [ \\log \\sum _ { j = 1 } ^ { k } e ^ { f ( x , y _ { j } ) } ] } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 470, + 541, + 506, + 573 + ], + "score": 1.0, + "content": ". To see", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 102, + 561, + 509, + 586 + ], + "spans": [ + { + "bbox": [ + 102, + 561, + 276, + 586 + ], + "score": 1.0, + "content": "this, apply Jensen’s to the left-hand side of", + "type": "text" + }, + { + "bbox": [ + 276, + 565, + 451, + 581 + ], + "score": 0.9, + "content": "\\begin{array} { r } { \\log \\mathbf { E } _ { p } [ \\sum _ { j = 1 } ^ { k } e ^ { f ( x , y _ { j } ) } ] < \\log \\sum _ { j = 1 } ^ { k } e ^ { f ( x , y _ { j } ) } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 451, + 561, + 509, + 586 + ], + "score": 1.0, + "content": ", which holds", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 578, + 505, + 592 + ], + "spans": [ + { + "bbox": [ + 105, + 578, + 115, + 592 + ], + "score": 1.0, + "content": "if", + "type": "text" + }, + { + "bbox": [ + 116, + 579, + 154, + 591 + ], + "score": 0.92, + "content": "y _ { j } \\in S _ { B }", + "type": "inline_equation" + }, + { + "bbox": [ + 154, + 578, + 171, + 592 + ], + "score": 1.0, + "content": "for", + "type": "text" + }, + { + "bbox": [ + 171, + 579, + 225, + 591 + ], + "score": 0.91, + "content": "j = 1 , \\dotsc , k", + "type": "inline_equation" + }, + { + "bbox": [ + 225, + 578, + 352, + 592 + ], + "score": 1.0, + "content": ", and then take the expectation", + "type": "text" + }, + { + "bbox": [ + 352, + 579, + 365, + 591 + ], + "score": 0.86, + "content": "\\mathbf { E } _ { q }", + "type": "inline_equation" + }, + { + "bbox": [ + 365, + 578, + 505, + 592 + ], + "score": 1.0, + "content": "of both sides. The last inequality", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 104, + 589, + 504, + 605 + ], + "spans": [ + { + "bbox": [ + 104, + 589, + 381, + 605 + ], + "score": 1.0, + "content": "holds by monoticity of log, linearity of expectation, and the fact that", + "type": "text" + }, + { + "bbox": [ + 381, + 590, + 474, + 604 + ], + "score": 0.92, + "content": "{ \\bf E } _ { p } [ e ^ { f ( x , y _ { j } ) } ] \\le e ^ { f ( x , y _ { j } ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 475, + 589, + 504, + 605 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 33.5, + "bbox_fs": [ + 101, + 541, + 509, + 605 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 617, + 505, + 712 + ], + "lines": [ + { + "bbox": [ + 106, + 617, + 505, + 630 + ], + "spans": [ + { + "bbox": [ + 106, + 617, + 505, + 630 + ], + "score": 1.0, + "content": "Theorem Intuition. For intuition, although using arbitrary negative distributions in NCE does not", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 628, + 505, + 641 + ], + "spans": [ + { + "bbox": [ + 105, + 628, + 408, + 641 + ], + "score": 1.0, + "content": "bound mutual information, we have found a restricted class of distributions", + "type": "text" + }, + { + "bbox": [ + 408, + 629, + 418, + 639 + ], + "score": 0.82, + "content": "\\mathcal { Q }", + "type": "inline_equation" + }, + { + "bbox": [ + 419, + 628, + 505, + 641 + ], + "score": 1.0, + "content": "where every member", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 107, + 639, + 506, + 652 + ], + "spans": [ + { + "bbox": [ + 107, + 639, + 134, + 651 + ], + "score": 0.92, + "content": "q ( y | x )", + "type": "inline_equation" + }, + { + "bbox": [ + 134, + 639, + 301, + 652 + ], + "score": 1.0, + "content": "“subsets the support” of the distribution", + "type": "text" + }, + { + "bbox": [ + 301, + 639, + 320, + 651 + ], + "score": 0.91, + "content": "p ( y )", + "type": "inline_equation" + }, + { + "bbox": [ + 320, + 639, + 457, + 652 + ], + "score": 1.0, + "content": ". 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Thm. 3.1", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 223, + 428, + 236 + ], + "spans": [ + { + "bbox": [ + 106, + 223, + 200, + 236 + ], + "score": 1.0, + "content": "shows that substituting", + "type": "text" + }, + { + "bbox": [ + 200, + 223, + 227, + 235 + ], + "score": 0.92, + "content": "q ( y | x )", + "type": "inline_equation" + }, + { + "bbox": [ + 228, + 223, + 243, + 236 + ], + "score": 1.0, + "content": "for", + "type": "text" + }, + { + "bbox": [ + 243, + 223, + 262, + 235 + ], + "score": 0.91, + "content": "p ( y )", + "type": "inline_equation" + }, + { + "bbox": [ + 262, + 223, + 428, + 236 + ], + "score": 1.0, + "content": "in NCE still bounds mutual information.", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 6.5 + }, + { + "type": "text", + "bbox": [ + 107, + 240, + 505, + 296 + ], + "lines": [ + { + "bbox": [ + 105, + 238, + 505, + 254 + ], + "spans": [ + { + "bbox": [ + 105, + 238, + 383, + 254 + ], + "score": 1.0, + "content": "Theorem Example 3.1. We give a concrete example for the choice", + "type": "text" + }, + { + "bbox": [ + 384, + 241, + 393, + 250 + ], + "score": 0.79, + "content": "B", + "type": "inline_equation" + }, + { + "bbox": [ + 393, + 238, + 505, + 254 + ], + "score": 1.0, + "content": "that will be used in Sec. 4.", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 249, + 506, + 265 + ], + "spans": [ + { + "bbox": [ + 105, + 249, + 184, + 265 + ], + "score": 1.0, + "content": "For any realization", + "type": "text" + }, + { + "bbox": [ + 184, + 253, + 191, + 261 + ], + "score": 0.73, + "content": "x", + "type": "inline_equation" + }, + { + "bbox": [ + 191, + 249, + 371, + 265 + ], + "score": 1.0, + "content": ", suppose we define two similarity thresholds", + "type": "text" + }, + { + "bbox": [ + 372, + 252, + 420, + 262 + ], + "score": 0.91, + "content": "\\omega _ { \\ell } , \\omega _ { u } \\in \\mathbf { R }", + "type": "inline_equation" + }, + { + "bbox": [ + 420, + 249, + 447, + 265 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 448, + 252, + 501, + 262 + ], + "score": 0.9, + "content": "c < \\omega _ { \\ell } < \\omega _ { u }", + "type": "inline_equation" + }, + { + "bbox": [ + 501, + 249, + 506, + 265 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 261, + 506, + 275 + ], + "spans": [ + { + "bbox": [ + 105, + 261, + 163, + 275 + ], + "score": 1.0, + "content": "Then, choose", + "type": "text" + }, + { + "bbox": [ + 163, + 262, + 220, + 274 + ], + "score": 0.92, + "content": "B = [ w _ { \\ell } , w _ { u } ]", + "type": "inline_equation" + }, + { + "bbox": [ + 220, + 261, + 306, + 275 + ], + "score": 1.0, + "content": ". In this case, the set", + "type": "text" + }, + { + "bbox": [ + 306, + 263, + 320, + 273 + ], + "score": 0.89, + "content": "S _ { B }", + "type": "inline_equation" + }, + { + "bbox": [ + 320, + 261, + 506, + 275 + ], + "score": 1.0, + "content": ", which defines the support of the distribution", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 107, + 271, + 506, + 288 + ], + "spans": [ + { + "bbox": [ + 107, + 273, + 134, + 285 + ], + "score": 0.94, + "content": "q ( y | x )", + "type": "inline_equation" + }, + { + "bbox": [ + 134, + 271, + 213, + 288 + ], + "score": 1.0, + "content": ", contains values of", + "type": "text" + }, + { + "bbox": [ + 213, + 275, + 220, + 284 + ], + "score": 0.83, + "content": "y", + "type": "inline_equation" + }, + { + "bbox": [ + 220, + 271, + 328, + 288 + ], + "score": 1.0, + "content": "that are not “too-close” to", + "type": "text" + }, + { + "bbox": [ + 328, + 275, + 335, + 283 + ], + "score": 0.76, + "content": "x", + "type": "inline_equation" + }, + { + "bbox": [ + 336, + 271, + 506, + 288 + ], + "score": 1.0, + "content": "but not “too-far”. In contrastive learning,", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 284, + 440, + 297 + ], + "spans": [ + { + "bbox": [ + 105, + 284, + 440, + 297 + ], + "score": 1.0, + "content": "we might pick these similarity thresholds to vary the difficulty of negative samples.", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 10 + }, + { + "type": "text", + "bbox": [ + 107, + 300, + 505, + 345 + ], + "lines": [ + { + "bbox": [ + 106, + 300, + 506, + 313 + ], + "spans": [ + { + "bbox": [ + 106, + 300, + 184, + 313 + ], + "score": 1.0, + "content": "Interestingly, Thm.", + "type": "text" + }, + { + "bbox": [ + 184, + 300, + 201, + 313 + ], + "score": 0.81, + "content": "\\boxed { 3 . 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 201, + 300, + 506, + 313 + ], + "score": 1.0, + "content": "states that CNCE is looser than NCE, which raises the question: when is a", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 311, + 505, + 324 + ], + "spans": [ + { + "bbox": [ + 105, + 311, + 505, + 324 + ], + "score": 1.0, + "content": "looser bound useful? In reply, we show that while CNCE is a more biased estimator than NCE, in", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 323, + 505, + 335 + ], + "spans": [ + { + "bbox": [ + 105, + 323, + 302, + 335 + ], + "score": 1.0, + "content": "return it has lower variance. Intuitively, because", + "type": "text" + }, + { + "bbox": [ + 303, + 323, + 330, + 335 + ], + "score": 0.93, + "content": "q ( y | x )", + "type": "inline_equation" + }, + { + "bbox": [ + 330, + 323, + 434, + 335 + ], + "score": 1.0, + "content": "is the result of restricting", + "type": "text" + }, + { + "bbox": [ + 435, + 323, + 454, + 335 + ], + "score": 0.91, + "content": "p ( y )", + "type": "inline_equation" + }, + { + "bbox": [ + 454, + 323, + 505, + 335 + ], + "score": 1.0, + "content": "to a smaller", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 334, + 489, + 347 + ], + "spans": [ + { + "bbox": [ + 105, + 334, + 198, + 347 + ], + "score": 1.0, + "content": "support, samples from", + "type": "text" + }, + { + "bbox": [ + 198, + 334, + 225, + 346 + ], + "score": 0.92, + "content": "q ( y | x )", + "type": "inline_equation" + }, + { + "bbox": [ + 225, + 334, + 489, + 347 + ], + "score": 1.0, + "content": "have less opportunity to deviate, hence lower variance. Formally:", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 14.5 + }, + { + "type": "text", + "bbox": [ + 106, + 349, + 505, + 447 + ], + "lines": [ + { + "bbox": [ + 106, + 348, + 504, + 363 + ], + "spans": [ + { + "bbox": [ + 106, + 348, + 325, + 363 + ], + "score": 1.0, + "content": "Theorem 3.2. (Bias and Variance Tradeoff) Pick any", + "type": "text" + }, + { + "bbox": [ + 325, + 350, + 383, + 362 + ], + "score": 0.92, + "content": "x , y \\sim p ( x , y )", + "type": "inline_equation" + }, + { + "bbox": [ + 383, + 348, + 466, + 363 + ], + "score": 1.0, + "content": ". Fix the distribution", + "type": "text" + }, + { + "bbox": [ + 466, + 350, + 504, + 362 + ], + "score": 0.9, + "content": "q ( y _ { 1 : k } | x )", + "type": "inline_equation" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 362, + 505, + 387 + ], + "spans": [ + { + "bbox": [ + 106, + 368, + 194, + 379 + ], + "score": 1.0, + "content": "as stated in Theorem", + "type": "text" + }, + { + "bbox": [ + 195, + 368, + 208, + 379 + ], + "score": 1.0, + "content": "3.1", + "type": "text" + }, + { + "bbox": [ + 212, + 365, + 340, + 383 + ], + "score": 1.0, + "content": "Define a new random variable", + "type": "text" + }, + { + "bbox": [ + 340, + 362, + 476, + 387 + ], + "score": 0.94, + "content": "\\begin{array} { r } { Z ( y _ { 1 : k } ) = \\log \\left( \\frac { e ^ { f ( x , y ) } } { \\frac { 1 } { k } \\sum _ { j = 1 } ^ { k } e ^ { f ( x , y _ { j } ) } } \\right) } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 477, + 369, + 505, + 381 + ], + "score": 1.0, + "content": "repre-", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 104, + 383, + 504, + 401 + ], + "spans": [ + { + "bbox": [ + 104, + 383, + 299, + 401 + ], + "score": 1.0, + "content": "senting the normalized similarity. By Theorem", + "type": "text" + }, + { + "bbox": [ + 299, + 385, + 317, + 400 + ], + "score": 0.7, + "content": "\\boxed { 3 . 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 317, + 383, + 384, + 401 + ], + "score": 1.0, + "content": "the expressions", + "type": "text" + }, + { + "bbox": [ + 384, + 387, + 430, + 399 + ], + "score": 0.91, + "content": "{ \\bf E } _ { p ( y _ { 1 : k } ) } [ Z ]", + "type": "inline_equation" + }, + { + "bbox": [ + 430, + 383, + 451, + 401 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 451, + 387, + 504, + 399 + ], + "score": 0.85, + "content": "{ \\mathbf { E } } _ { q \\left( y _ { 1 : k } \\mid x \\right) } [ Z ]", + "type": "inline_equation" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 397, + 505, + 413 + ], + "spans": [ + { + "bbox": [ + 105, + 397, + 186, + 413 + ], + "score": 1.0, + "content": "are estimators for", + "type": "text" + }, + { + "bbox": [ + 186, + 398, + 223, + 410 + ], + "score": 0.93, + "content": "\\mathcal { I } ( X ; Y )", + "type": "inline_equation" + }, + { + "bbox": [ + 223, + 397, + 325, + 413 + ], + "score": 1.0, + "content": ". Suppose that the set", + "type": "text" + }, + { + "bbox": [ + 325, + 398, + 339, + 410 + ], + "score": 0.87, + "content": "S _ { B }", + "type": "inline_equation" + }, + { + "bbox": [ + 339, + 397, + 430, + 413 + ], + "score": 1.0, + "content": "is chosen to ensure", + "type": "text" + }, + { + "bbox": [ + 430, + 399, + 505, + 411 + ], + "score": 0.89, + "content": "W a r _ { q ( y _ { 1 : k } | x ) } [ Z ] \\ \\leq", + "type": "inline_equation" + } + ], + "index": 20 + }, + { + "bbox": [ + 107, + 408, + 507, + 424 + ], + "spans": [ + { + "bbox": [ + 107, + 410, + 165, + 424 + ], + "score": 0.92, + "content": "V a r _ { \\tilde { q } ( y _ { 1 : k } | x ) } [ Z ]", + "type": "inline_equation" + }, + { + "bbox": [ + 166, + 408, + 200, + 424 + ], + "score": 1.0, + "content": ", where", + "type": "text" + }, + { + "bbox": [ + 201, + 410, + 262, + 423 + ], + "score": 0.92, + "content": "\\tilde { q } ( A ) ~ = ~ p ( A |", + "type": "inline_equation" + }, + { + "bbox": [ + 262, + 408, + 324, + 424 + ], + "score": 1.0, + "content": "complement of", + "type": "text" + }, + { + "bbox": [ + 324, + 411, + 338, + 422 + ], + "score": 0.82, + "content": "S _ { B }", + "type": "inline_equation" + }, + { + "bbox": [ + 338, + 408, + 507, + 424 + ], + "score": 1.0, + "content": "). That is, we assume the variance of", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 421, + 506, + 435 + ], + "spans": [ + { + "bbox": [ + 105, + 421, + 262, + 435 + ], + "score": 1.0, + "content": "the normalized similarity when using", + "type": "text" + }, + { + "bbox": [ + 263, + 423, + 309, + 434 + ], + "score": 0.89, + "content": "y _ { 1 : k } ~ \\in ~ S _ { B }", + "type": "inline_equation" + }, + { + "bbox": [ + 310, + 421, + 427, + 435 + ], + "score": 1.0, + "content": "is smaller than when using", + "type": "text" + }, + { + "bbox": [ + 428, + 422, + 474, + 434 + ], + "score": 0.91, + "content": "y _ { 1 : k } ~ \\notin ~ S _ { B }", + "type": "inline_equation" + }, + { + "bbox": [ + 474, + 421, + 506, + 435 + ], + "score": 1.0, + "content": ". Then", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 432, + 394, + 448 + ], + "spans": [ + { + "bbox": [ + 106, + 433, + 243, + 447 + ], + "score": 0.92, + "content": "\\mathrm { B i a s } _ { p ( y _ { 1 : k } ) } ( Z ) \\leq \\mathrm { B i a s } _ { q ( y _ { 1 : k } | x ) } ( Z )", + "type": "inline_equation" + }, + { + "bbox": [ + 243, + 432, + 262, + 448 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 262, + 434, + 390, + 447 + ], + "score": 0.93, + "content": "\\operatorname { V a r } _ { p ( y _ { 1 : k } ) } ( Z ) \\geq \\operatorname { V a r } _ { q ( y _ { 1 : k } | x ) } ( Z )", + "type": "inline_equation" + }, + { + "bbox": [ + 390, + 432, + 394, + 448 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 20 + }, + { + "type": "text", + "bbox": [ + 107, + 455, + 505, + 501 + ], + "lines": [ + { + "bbox": [ + 106, + 455, + 505, + 469 + ], + "spans": [ + { + "bbox": [ + 106, + 456, + 213, + 468 + ], + "score": 1.0, + "content": "The proof can be found in", + "type": "text" + }, + { + "bbox": [ + 214, + 455, + 293, + 469 + ], + "score": 0.27, + "content": "\\mathrm { S e c . } \\ \\mathrm { A . } 2 . \\ \\mathrm { T h m . } \\ 3 . 2", + "type": "inline_equation" + }, + { + "bbox": [ + 293, + 456, + 505, + 468 + ], + "score": 1.0, + "content": "provides one answer to our question of looseness. In", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 467, + 505, + 479 + ], + "spans": [ + { + "bbox": [ + 106, + 467, + 505, + 479 + ], + "score": 1.0, + "content": "stochastic optimization, a lower variance objective may lead to better local optima. For representa-", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 478, + 504, + 490 + ], + "spans": [ + { + "bbox": [ + 105, + 478, + 504, + 490 + ], + "score": 1.0, + "content": "tion learning, using CNCE to sample more difficult negatives may (1) encourage the representation", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 489, + 489, + 501 + ], + "spans": [ + { + "bbox": [ + 105, + 489, + 489, + 501 + ], + "score": 1.0, + "content": "to distinguish fine-grained features useful in transfer tasks, and (2) provide less noisy gradients.", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 25.5 + }, + { + "type": "title", + "bbox": [ + 108, + 520, + 245, + 533 + ], + "lines": [ + { + "bbox": [ + 105, + 519, + 246, + 535 + ], + "spans": [ + { + "bbox": [ + 105, + 519, + 246, + 535 + ], + "score": 1.0, + "content": "4 RING DISCRIMINATION", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 28 + }, + { + "type": "text", + "bbox": [ + 106, + 547, + 505, + 614 + ], + "lines": [ + { + "bbox": [ + 106, + 547, + 504, + 559 + ], + "spans": [ + { + "bbox": [ + 106, + 547, + 504, + 559 + ], + "score": 1.0, + "content": "We have shown CNCE to be a new bound on the mutual information that uses hard negative samples.", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 558, + 505, + 571 + ], + "spans": [ + { + "bbox": [ + 105, + 558, + 505, + 571 + ], + "score": 1.0, + "content": "Now we wish to apply CNCE to contrastive learning where the two random variables are again", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 569, + 505, + 582 + ], + "spans": [ + { + "bbox": [ + 106, + 569, + 250, + 582 + ], + "score": 1.0, + "content": "transformations of a single variable", + "type": "text" + }, + { + "bbox": [ + 250, + 570, + 260, + 579 + ], + "score": 0.75, + "content": "X", + "type": "inline_equation" + }, + { + "bbox": [ + 261, + 569, + 367, + 582 + ], + "score": 1.0, + "content": ". In this setting, for a fixed", + "type": "text" + }, + { + "bbox": [ + 367, + 570, + 409, + 581 + ], + "score": 0.93, + "content": "x _ { i } \\sim p ( x )", + "type": "inline_equation" + }, + { + "bbox": [ + 409, + 569, + 505, + 582 + ], + "score": 1.0, + "content": ", the CNCE distribution", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 580, + 505, + 594 + ], + "spans": [ + { + "bbox": [ + 105, + 580, + 160, + 594 + ], + "score": 1.0, + "content": "is written as", + "type": "text" + }, + { + "bbox": [ + 161, + 581, + 203, + 593 + ], + "score": 0.93, + "content": "q ( x | t ( x _ { i } ) )", + "type": "inline_equation" + }, + { + "bbox": [ + 203, + 580, + 287, + 594 + ], + "score": 1.0, + "content": "for some transform", + "type": "text" + }, + { + "bbox": [ + 288, + 581, + 316, + 591 + ], + "score": 0.88, + "content": "t \\in \\mathcal { T }", + "type": "inline_equation" + }, + { + "bbox": [ + 316, + 580, + 384, + 594 + ], + "score": 1.0, + "content": ". Samples from", + "type": "text" + }, + { + "bbox": [ + 385, + 581, + 450, + 593 + ], + "score": 0.93, + "content": "x \\stackrel { \\cdot \\cdot } { \\sim } q ( x | t ( x _ { i } ) )", + "type": "inline_equation" + }, + { + "bbox": [ + 450, + 580, + 505, + 594 + ], + "score": 1.0, + "content": "will be such", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 591, + 505, + 604 + ], + "spans": [ + { + "bbox": [ + 105, + 591, + 239, + 604 + ], + "score": 1.0, + "content": "that the exponentiated distance,", + "type": "text" + }, + { + "bbox": [ + 239, + 591, + 345, + 604 + ], + "score": 0.9, + "content": "\\exp \\{ g _ { \\theta } ( t ( x _ { i } ) ) ^ { T } g _ { \\theta } ( t ^ { \\prime } ( x ) ) \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 346, + 591, + 467, + 604 + ], + "score": 1.0, + "content": ", is at least a minimum value", + "type": "text" + }, + { + "bbox": [ + 468, + 594, + 473, + 601 + ], + "score": 0.64, + "content": "c", + "type": "inline_equation" + }, + { + "bbox": [ + 474, + 591, + 505, + 604 + ], + "score": 1.0, + "content": ". As in", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 602, + 479, + 615 + ], + "spans": [ + { + "bbox": [ + 106, + 602, + 224, + 615 + ], + "score": 1.0, + "content": "Example 3.1, we will choose", + "type": "text" + }, + { + "bbox": [ + 224, + 603, + 278, + 614 + ], + "score": 0.92, + "content": "B = [ \\omega _ { \\ell } , \\omega _ { u } ]", + "type": "inline_equation" + }, + { + "bbox": [ + 278, + 602, + 360, + 615 + ], + "score": 1.0, + "content": ", a closed interval in", + "type": "text" + }, + { + "bbox": [ + 360, + 603, + 370, + 612 + ], + "score": 0.55, + "content": "\\mathbf { R }", + "type": "inline_equation" + }, + { + "bbox": [ + 370, + 602, + 479, + 615 + ], + "score": 1.0, + "content": "defined by two thresholds.", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 31.5 + }, + { + "type": "text", + "bbox": [ + 106, + 619, + 505, + 730 + ], + "lines": [ + { + "bbox": [ + 105, + 618, + 506, + 632 + ], + "spans": [ + { + "bbox": [ + 105, + 618, + 366, + 632 + ], + "score": 1.0, + "content": "Picking thresholds. We pick the thresholds conditioned on the", + "type": "text" + }, + { + "bbox": [ + 366, + 621, + 371, + 630 + ], + "score": 0.74, + "content": "i", + "type": "inline_equation" + }, + { + "bbox": [ + 371, + 618, + 506, + 632 + ], + "score": 1.0, + "content": "-th example in the dataset, hence", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 630, + 506, + 643 + ], + "spans": [ + { + "bbox": [ + 106, + 630, + 240, + 643 + ], + "score": 1.0, + "content": "each example has a different set", + "type": "text" + }, + { + "bbox": [ + 240, + 631, + 249, + 640 + ], + "score": 0.77, + "content": "B", + "type": "inline_equation" + }, + { + "bbox": [ + 250, + 630, + 459, + 643 + ], + "score": 1.0, + "content": ". 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We consider", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 106, + 652, + 505, + 665 + ], + "spans": [ + { + "bbox": [ + 106, + 652, + 164, + 665 + ], + "score": 1.0, + "content": "each example", + "type": "text" + }, + { + "bbox": [ + 165, + 654, + 172, + 662 + ], + "score": 0.76, + "content": "x", + "type": "inline_equation" + }, + { + "bbox": [ + 172, + 652, + 315, + 665 + ], + "score": 1.0, + "content": "in the dataset to be in the support", + "type": "text" + }, + { + "bbox": [ + 315, + 653, + 329, + 663 + ], + "score": 0.89, + "content": "S _ { B }", + "type": "inline_equation" + }, + { + "bbox": [ + 329, + 652, + 505, + 665 + ], + "score": 1.0, + "content": "if and only if the (exponentiated) distance", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 663, + 505, + 676 + ], + "spans": [ + { + "bbox": [ + 105, + 663, + 215, + 676 + ], + "score": 1.0, + "content": "between the embedding of", + "type": "text" + }, + { + "bbox": [ + 216, + 664, + 226, + 674 + ], + "score": 0.86, + "content": "x _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 226, + 663, + 244, + 676 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 245, + 665, + 252, + 673 + ], + "score": 0.74, + "content": "x", + "type": "inline_equation" + }, + { + "bbox": [ + 252, + 663, + 267, + 676 + ], + "score": 1.0, + "content": ", or", + "type": "text" + }, + { + "bbox": [ + 267, + 663, + 374, + 675 + ], + "score": 0.91, + "content": "\\exp \\{ \\bar { g _ { \\boldsymbol { \\theta } } } ( t ( x _ { i } ) ) ^ { T } g _ { \\boldsymbol { \\theta } } ( t ^ { \\prime } ( x ) ) \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 374, + 663, + 429, + 676 + ], + "score": 1.0, + "content": ", is below the", + "type": "text" + }, + { + "bbox": [ + 429, + 665, + 436, + 673 + ], + "score": 0.77, + "content": "u", + "type": "inline_equation" + }, + { + "bbox": [ + 437, + 663, + 505, + 676 + ], + "score": 1.0, + "content": "-th percentile for", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 674, + 506, + 687 + ], + "spans": [ + { + "bbox": [ + 105, + 674, + 120, + 687 + ], + "score": 1.0, + "content": "all", + "type": "text" + }, + { + "bbox": [ + 120, + 675, + 151, + 685 + ], + "score": 0.91, + "content": "x \\in \\mathcal { D }", + "type": "inline_equation" + }, + { + "bbox": [ + 152, + 674, + 277, + 687 + ], + "score": 1.0, + "content": ". Call this maximum distance", + "type": "text" + }, + { + "bbox": [ + 278, + 676, + 290, + 685 + ], + "score": 0.87, + "content": "\\omega _ { u }", + "type": "inline_equation" + }, + { + "bbox": [ + 290, + 674, + 420, + 687 + ], + "score": 1.0, + "content": ". 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We give a concrete example for the choice", + "type": "text" + }, + { + "bbox": [ + 384, + 241, + 393, + 250 + ], + "score": 0.79, + "content": "B", + "type": "inline_equation" + }, + { + "bbox": [ + 393, + 238, + 505, + 254 + ], + "score": 1.0, + "content": "that will be used in Sec. 4.", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 249, + 506, + 265 + ], + "spans": [ + { + "bbox": [ + 105, + 249, + 184, + 265 + ], + "score": 1.0, + "content": "For any realization", + "type": "text" + }, + { + "bbox": [ + 184, + 253, + 191, + 261 + ], + "score": 0.73, + "content": "x", + "type": "inline_equation" + }, + { + "bbox": [ + 191, + 249, + 371, + 265 + ], + "score": 1.0, + "content": ", suppose we define two similarity thresholds", + "type": "text" + }, + { + "bbox": [ + 372, + 252, + 420, + 262 + ], + "score": 0.91, + "content": "\\omega _ { \\ell } , \\omega _ { u } \\in \\mathbf { R }", + "type": "inline_equation" + }, + { + "bbox": [ + 420, + 249, + 447, + 265 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 448, + 252, + 501, + 262 + ], + "score": 0.9, + "content": "c < \\omega _ { \\ell } < \\omega _ { u }", + "type": "inline_equation" + }, + { + "bbox": [ + 501, + 249, + 506, + 265 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 261, + 506, + 275 + ], + "spans": [ + { + "bbox": [ + 105, + 261, + 163, + 275 + ], + "score": 1.0, + "content": "Then, choose", + "type": "text" + }, + { + "bbox": [ + 163, + 262, + 220, + 274 + ], + "score": 0.92, + "content": "B = [ w _ { \\ell } , w _ { u } ]", + "type": "inline_equation" + }, + { + "bbox": [ + 220, + 261, + 306, + 275 + ], + "score": 1.0, + "content": ". In this case, the set", + "type": "text" + }, + { + "bbox": [ + 306, + 263, + 320, + 273 + ], + "score": 0.89, + "content": "S _ { B }", + "type": "inline_equation" + }, + { + "bbox": [ + 320, + 261, + 506, + 275 + ], + "score": 1.0, + "content": ", which defines the support of the distribution", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 107, + 271, + 506, + 288 + ], + "spans": [ + { + "bbox": [ + 107, + 273, + 134, + 285 + ], + "score": 0.94, + "content": "q ( y | x )", + "type": "inline_equation" + }, + { + "bbox": [ + 134, + 271, + 213, + 288 + ], + "score": 1.0, + "content": ", contains values of", + "type": "text" + }, + { + "bbox": [ + 213, + 275, + 220, + 284 + ], + "score": 0.83, + "content": "y", + "type": "inline_equation" + }, + { + "bbox": [ + 220, + 271, + 328, + 288 + ], + "score": 1.0, + "content": "that are not “too-close” to", + "type": "text" + }, + { + "bbox": [ + 328, + 275, + 335, + 283 + ], + "score": 0.76, + "content": "x", + "type": "inline_equation" + }, + { + "bbox": [ + 336, + 271, + 506, + 288 + ], + "score": 1.0, + "content": "but not “too-far”. In contrastive learning,", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 284, + 440, + 297 + ], + "spans": [ + { + "bbox": [ + 105, + 284, + 440, + 297 + ], + "score": 1.0, + "content": "we might pick these similarity thresholds to vary the difficulty of negative samples.", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 10, + "bbox_fs": [ + 105, + 238, + 506, + 297 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 300, + 505, + 345 + ], + "lines": [ + { + "bbox": [ + 106, + 300, + 506, + 313 + ], + "spans": [ + { + "bbox": [ + 106, + 300, + 184, + 313 + ], + "score": 1.0, + "content": "Interestingly, Thm.", + "type": "text" + }, + { + "bbox": [ + 184, + 300, + 201, + 313 + ], + "score": 0.81, + "content": "\\boxed { 3 . 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 201, + 300, + 506, + 313 + ], + "score": 1.0, + "content": "states that CNCE is looser than NCE, which raises the question: when is a", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 311, + 505, + 324 + ], + "spans": [ + { + "bbox": [ + 105, + 311, + 505, + 324 + ], + "score": 1.0, + "content": "looser bound useful? In reply, we show that while CNCE is a more biased estimator than NCE, in", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 323, + 505, + 335 + ], + "spans": [ + { + "bbox": [ + 105, + 323, + 302, + 335 + ], + "score": 1.0, + "content": "return it has lower variance. 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Formally:", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 14.5, + "bbox_fs": [ + 105, + 300, + 506, + 347 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 349, + 505, + 447 + ], + "lines": [ + { + "bbox": [ + 106, + 348, + 504, + 363 + ], + "spans": [ + { + "bbox": [ + 106, + 348, + 325, + 363 + ], + "score": 1.0, + "content": "Theorem 3.2. (Bias and Variance Tradeoff) Pick any", + "type": "text" + }, + { + "bbox": [ + 325, + 350, + 383, + 362 + ], + "score": 0.92, + "content": "x , y \\sim p ( x , y )", + "type": "inline_equation" + }, + { + "bbox": [ + 383, + 348, + 466, + 363 + ], + "score": 1.0, + "content": ". 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Suppose that the set", + "type": "text" + }, + { + "bbox": [ + 325, + 398, + 339, + 410 + ], + "score": 0.87, + "content": "S _ { B }", + "type": "inline_equation" + }, + { + "bbox": [ + 339, + 397, + 430, + 413 + ], + "score": 1.0, + "content": "is chosen to ensure", + "type": "text" + }, + { + "bbox": [ + 430, + 399, + 505, + 411 + ], + "score": 0.89, + "content": "W a r _ { q ( y _ { 1 : k } | x ) } [ Z ] \\ \\leq", + "type": "inline_equation" + } + ], + "index": 20 + }, + { + "bbox": [ + 107, + 408, + 507, + 424 + ], + "spans": [ + { + "bbox": [ + 107, + 410, + 165, + 424 + ], + "score": 0.92, + "content": "V a r _ { \\tilde { q } ( y _ { 1 : k } | x ) } [ Z ]", + "type": "inline_equation" + }, + { + "bbox": [ + 166, + 408, + 200, + 424 + ], + "score": 1.0, + "content": ", where", + "type": "text" + }, + { + "bbox": [ + 201, + 410, + 262, + 423 + ], + "score": 0.92, + "content": "\\tilde { q } ( A ) ~ = ~ p ( A |", + "type": "inline_equation" + }, + { + "bbox": [ + 262, + 408, + 324, + 424 + ], + "score": 1.0, + "content": "complement of", + "type": "text" + }, + { + "bbox": [ + 324, + 411, + 338, + 422 + ], + "score": 0.82, + "content": "S _ { B }", + "type": "inline_equation" + }, + { + "bbox": [ + 338, + 408, + 507, + 424 + ], + "score": 1.0, + "content": "). 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In", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 467, + 505, + 479 + ], + "spans": [ + { + "bbox": [ + 106, + 467, + 505, + 479 + ], + "score": 1.0, + "content": "stochastic optimization, a lower variance objective may lead to better local optima. 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We consider", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 106, + 652, + 505, + 665 + ], + "spans": [ + { + "bbox": [ + 106, + 652, + 164, + 665 + ], + "score": 1.0, + "content": "each example", + "type": "text" + }, + { + "bbox": [ + 165, + 654, + 172, + 662 + ], + "score": 0.76, + "content": "x", + "type": "inline_equation" + }, + { + "bbox": [ + 172, + 652, + 315, + 665 + ], + "score": 1.0, + "content": "in the dataset to be in the support", + "type": "text" + }, + { + "bbox": [ + 315, + 653, + 329, + 663 + ], + "score": 0.89, + "content": "S _ { B }", + "type": "inline_equation" + }, + { + "bbox": [ + 329, + 652, + 505, + 665 + ], + "score": 1.0, + "content": "if and only if the (exponentiated) distance", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 663, + 505, + 676 + ], + "spans": [ + { + "bbox": [ + 105, + 663, + 215, + 676 + ], + "score": 1.0, + "content": "between the embedding of", + "type": "text" + }, + { + "bbox": [ + 216, + 664, + 226, + 674 + ], + "score": 0.86, + "content": "x _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 226, + 663, + 244, + 676 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 245, + 665, + 252, + 673 + ], + "score": 0.74, + "content": "x", + "type": "inline_equation" + }, + { + "bbox": [ + 252, + 663, + 267, + 676 + ], + "score": 1.0, + "content": ", or", + "type": "text" + }, + { + "bbox": [ + 267, + 663, + 374, + 675 + ], + "score": 0.91, + "content": "\\exp \\{ \\bar { g _ { \\boldsymbol { \\theta } } } ( t ( x _ { i } ) ) ^ { T } g _ { \\boldsymbol { \\theta } } ( t ^ { \\prime } ( x ) ) \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 374, + 663, + 429, + 676 + ], + "score": 1.0, + "content": ", is below the", + "type": "text" + }, + { + "bbox": [ + 429, + 665, + 436, + 673 + ], + "score": 0.77, + "content": "u", + "type": "inline_equation" + }, + { + "bbox": [ + 437, + 663, + 505, + 676 + ], + "score": 1.0, + "content": "-th percentile for", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 674, + 506, + 687 + ], + "spans": [ + { + "bbox": [ + 105, + 674, + 120, + 687 + ], + "score": 1.0, + "content": "all", + "type": "text" + }, + { + "bbox": [ + 120, + 675, + 151, + 685 + ], + "score": 0.91, + "content": "x \\in \\mathcal { D }", + "type": "inline_equation" + }, + { + "bbox": [ + 152, + 674, + 277, + 687 + ], + "score": 1.0, + "content": ". Call this maximum distance", + "type": "text" + }, + { + "bbox": [ + 278, + 676, + 290, + 685 + ], + "score": 0.87, + "content": "\\omega _ { u }", + "type": "inline_equation" + }, + { + "bbox": [ + 290, + 674, + 420, + 687 + ], + "score": 1.0, + "content": ". In other words, we construct", + "type": "text" + }, + { + "bbox": [ + 421, + 674, + 463, + 686 + ], + "score": 0.92, + "content": "q ( x | t ( x _ { i } ) )", + "type": "inline_equation" + }, + { + "bbox": [ + 463, + 674, + 506, + 687 + ], + "score": 1.0, + "content": "such that", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 106, + 685, + 506, + 698 + ], + "spans": [ + { + "bbox": [ + 106, + 685, + 484, + 698 + ], + "score": 1.0, + "content": "we ignore examples from the dataset whose embedding dot producted with the embedding of", + "type": "text" + }, + { + "bbox": [ + 484, + 686, + 494, + 696 + ], + "score": 0.84, + "content": "x _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 495, + 685, + 506, + 698 + ], + "score": 1.0, + "content": "is", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 106, + 696, + 505, + 709 + ], + "spans": [ + { + "bbox": [ + 106, + 696, + 132, + 709 + ], + "score": 1.0, + "content": "above", + "type": "text" + }, + { + "bbox": [ + 132, + 698, + 145, + 707 + ], + "score": 0.86, + "content": "\\omega _ { u }", + "type": "inline_equation" + }, + { + "bbox": [ + 145, + 696, + 192, + 709 + ], + "score": 1.0, + "content": ". 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However, picking the", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 93, + 506, + 106 + ], + "spans": [ + { + "bbox": [ + 106, + 94, + 187, + 105 + ], + "score": 1.0, + "content": "closest examples to", + "type": "text" + }, + { + "bbox": [ + 188, + 93, + 209, + 106 + ], + "score": 0.92, + "content": "t ( x _ { i } )", + "type": "inline_equation" + }, + { + "bbox": [ + 210, + 94, + 506, + 105 + ], + "score": 1.0, + "content": "as its negative examples may be inappropriate, as these examples might", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 104, + 506, + 117 + ], + "spans": [ + { + "bbox": [ + 105, + 104, + 486, + 117 + ], + "score": 1.0, + "content": "be better suited as positive views rather than negatives (Zhuang et al., 2019; Xie et al., 2020).", + "type": "text" + }, + { + "bbox": [ + 486, + 104, + 506, + 117 + ], + "score": 1.0, + "content": "As", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 106, + 115, + 506, + 127 + ], + "spans": [ + { + "bbox": [ + 106, + 115, + 506, + 127 + ], + "score": 1.0, + "content": "an extreme case, if the same image is included in the dataset twice, we would not like to select", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 126, + 506, + 139 + ], + "spans": [ + { + "bbox": [ + 105, + 126, + 506, + 139 + ], + "score": 1.0, + "content": "it as a negative example for itself. 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This could lock in poor local min-", + "type": "text" + } + ], + "index": 66 + }, + { + "bbox": [ + 105, + 601, + 505, + 614 + ], + "spans": [ + { + "bbox": [ + 105, + 601, + 392, + 614 + ], + "score": 1.0, + "content": "ima. To avoid this, we propose to use an annealing policy that reduces", + "type": "text" + }, + { + "bbox": [ + 392, + 603, + 405, + 612 + ], + "score": 0.87, + "content": "\\omega _ { u }", + "type": "inline_equation" + }, + { + "bbox": [ + 405, + 601, + 505, + 614 + ], + "score": 1.0, + "content": "(and thus the size of the", + "type": "text" + } + ], + "index": 67 + }, + { + "bbox": [ + 105, + 612, + 505, + 626 + ], + "spans": [ + { + "bbox": [ + 105, + 612, + 139, + 626 + ], + "score": 1.0, + "content": "support", + "type": "text" + }, + { + "bbox": [ + 140, + 613, + 154, + 623 + ], + "score": 0.85, + "content": "S _ { B }", + "type": "inline_equation" + }, + { + "bbox": [ + 154, + 612, + 359, + 626 + ], + "score": 1.0, + "content": ") throughout training. 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We explored several", + "type": "text" + } + ], + "index": 69 + }, + { + "bbox": [ + 106, + 633, + 504, + 647 + ], + "spans": [ + { + "bbox": [ + 106, + 634, + 462, + 647 + ], + "score": 1.0, + "content": "annealing policies and found a linear schedule to be well-performing and simple (see Sec.", + "type": "text" + }, + { + "bbox": [ + 462, + 633, + 475, + 646 + ], + "score": 0.38, + "content": "\\mathbf { G } )", + "type": "inline_equation" + }, + { + "bbox": [ + 475, + 634, + 504, + 647 + ], + "score": 1.0, + "content": ". In our", + "type": "text" + } + ], + "index": 70 + }, + { + "bbox": [ + 105, + 644, + 505, + 659 + ], + "spans": [ + { + "bbox": [ + 105, + 644, + 505, + 659 + ], + "score": 1.0, + "content": "experiments, annealing is shown to be crucial: being too aggressive with negatives early in training", + "type": "text" + } + ], + "index": 71 + }, + { + "bbox": [ + 105, + 656, + 387, + 668 + ], + "spans": [ + { + "bbox": [ + 105, + 656, + 387, + 668 + ], + "score": 1.0, + "content": "produced representations that performed poorly on downstream tasks.", + "type": "text" + } + ], + "index": 72 + } + ], + "index": 67.5 + }, + { + "type": "title", + "bbox": [ + 108, + 684, + 200, + 696 + ], + "lines": [ + { + "bbox": [ + 105, + 683, + 201, + 699 + ], + "spans": [ + { + "bbox": [ + 105, + 683, + 201, + 699 + ], + "score": 1.0, + "content": "5 EXPERIMENTS", + "type": "text" + } + ], + "index": 73 + } + ], + "index": 73 + }, + { + "type": "text", + "bbox": [ + 107, + 709, + 504, + 734 + ], + "lines": [ + { + "bbox": [ + 106, + 708, + 505, + 721 + ], + "spans": [ + { + "bbox": [ + 106, + 708, + 505, + 721 + ], + "score": 1.0, + "content": "We explore our method applied to IR, CMC, and MoCo in four commonly used visual datasets. As", + "type": "text" + } + ], + "index": 74 + }, + { + "bbox": [ + 105, + 719, + 506, + 733 + ], + "spans": [ + { + "bbox": [ + 105, + 719, + 506, + 733 + ], + "score": 1.0, + "content": "in prior work (Wu et al., 2018; Zhuang et al., 2019; He et al., 2019; Misra & Maaten, 2020; Henaff ´", + "type": "text" + } + ], + "index": 75 + } + ], + "index": 74.5 + } + ], + "page_idx": 4, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 107, + 27, + 292, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 26, + 293, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 26, + 293, + 38 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2021", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 302, + 751, + 309, + 760 + ], + "lines": [ + { + "bbox": [ + 301, + 750, + 310, + 762 + ], + "spans": [ + { + "bbox": [ + 301, + 750, + 310, + 762 + ], + "score": 1.0, + "content": "5", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 106, + 82, + 505, + 193 + ], + "lines": [ + { + "bbox": [ + 105, + 82, + 505, + 95 + ], + "spans": [ + { + "bbox": [ + 105, + 82, + 227, + 95 + ], + "score": 1.0, + "content": "In picking the lower threshold", + "type": "text" + }, + { + "bbox": [ + 228, + 84, + 239, + 93 + ], + "score": 0.84, + "content": "\\omega _ { \\ell }", + "type": "inline_equation" + }, + { + "bbox": [ + 239, + 82, + 363, + 95 + ], + "score": 1.0, + "content": ", one could choose it to be 0, so", + "type": "text" + }, + { + "bbox": [ + 364, + 83, + 414, + 95 + ], + "score": 0.92, + "content": "B = [ 0 , \\omega _ { u } )", + "type": "inline_equation" + }, + { + "bbox": [ + 414, + 82, + 505, + 95 + ], + "score": 1.0, + "content": ". 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We explored several", + "type": "text" + } + ], + "index": 69 + }, + { + "bbox": [ + 106, + 633, + 504, + 647 + ], + "spans": [ + { + "bbox": [ + 106, + 634, + 462, + 647 + ], + "score": 1.0, + "content": "annealing policies and found a linear schedule to be well-performing and simple (see Sec.", + "type": "text" + }, + { + "bbox": [ + 462, + 633, + 475, + 646 + ], + "score": 0.38, + "content": "\\mathbf { G } )", + "type": "inline_equation" + }, + { + "bbox": [ + 475, + 634, + 504, + 647 + ], + "score": 1.0, + "content": ". In our", + "type": "text" + } + ], + "index": 70 + }, + { + "bbox": [ + 105, + 644, + 505, + 659 + ], + "spans": [ + { + "bbox": [ + 105, + 644, + 505, + 659 + ], + "score": 1.0, + "content": "experiments, annealing is shown to be crucial: being too aggressive with negatives early in training", + "type": "text" + } + ], + "index": 71 + }, + { + "bbox": [ + 105, + 656, + 387, + 668 + ], + "spans": [ + { + "bbox": [ + 105, + 656, + 387, + 668 + ], + "score": 1.0, + "content": "produced representations that performed poorly on downstream tasks.", + "type": "text" + } + ], + "index": 72 + } + ], + "index": 61.5, + "bbox_fs": [ + 250, + 535, + 505, + 559 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 558, + 505, + 667 + ], + "lines": [], + "index": 67.5, + "bbox_fs": [ + 104, + 558, + 505, + 668 + ], + "lines_deleted": true + }, + { + "type": "title", + "bbox": [ + 108, + 684, + 200, + 696 + ], + "lines": [ + { + "bbox": [ + 105, + 683, + 201, + 699 + ], + "spans": [ + { + "bbox": [ + 105, + 683, + 201, + 699 + ], + "score": 1.0, + "content": "5 EXPERIMENTS", + "type": "text" + } + ], + "index": 73 + } + ], + "index": 73 + }, + { + "type": "text", + "bbox": [ + 107, + 709, + 504, + 734 + ], + "lines": [ + { + "bbox": [ + 106, + 708, + 505, + 721 + ], + "spans": [ + { + "bbox": [ + 106, + 708, + 505, + 721 + ], + "score": 1.0, + "content": "We explore our method applied to IR, CMC, and MoCo in four commonly used visual datasets. As", + "type": "text" + } + ], + "index": 74 + }, + { + "bbox": [ + 105, + 719, + 506, + 733 + ], + "spans": [ + { + "bbox": [ + 105, + 719, + 506, + 733 + ], + "score": 1.0, + "content": "in prior work (Wu et al., 2018; Zhuang et al., 2019; He et al., 2019; Misra & Maaten, 2020; Henaff ´", + "type": "text" + } + ], + "index": 75 + }, + { + "bbox": [ + 106, + 82, + 503, + 96 + ], + "spans": [ + { + "bbox": [ + 106, + 82, + 503, + 96 + ], + "score": 1.0, + "content": "et al., 2019; Kolesnikov et al., 2019; Donahue & Simonyan, 2019; Bachman et al., 2019; Tian et al.,", + "type": "text", + "cross_page": true + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 93, + 505, + 106 + ], + "spans": [ + { + "bbox": [ + 106, + 93, + 131, + 106 + ], + "score": 1.0, + "content": "2019;", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 131, + 93, + 505, + 106 + ], + "score": 1.0, + "content": "Chen et al., 2020a), we evaluate each method by linear classification on frozen embeddings.", + "type": "text", + "cross_page": true + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 103, + 505, + 118 + ], + "spans": [ + { + "bbox": [ + 105, + 103, + 505, + 118 + ], + "score": 1.0, + "content": "That is, we optimize a contrastive objective on a pretraining dataset to learn a representation; then,", + "type": "text", + "cross_page": true + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 115, + 505, + 128 + ], + "spans": [ + { + "bbox": [ + 105, + 115, + 505, + 128 + ], + "score": 1.0, + "content": "using a transfer dataset, we fit logistic regression on representations only. A better representation", + "type": "text", + "cross_page": true + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 126, + 496, + 140 + ], + "spans": [ + { + "bbox": [ + 105, + 126, + 496, + 140 + ], + "score": 1.0, + "content": "would contain more “object-centric” information, thereby achieving a higher classification score.", + "type": "text", + "cross_page": true + } + ], + "index": 4 + } + ], + "index": 74.5, + "bbox_fs": [ + 105, + 708, + 506, + 733 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 106, + 81, + 505, + 138 + ], + "lines": [ + { + "bbox": [ + 106, + 82, + 503, + 96 + ], + "spans": [ + { + "bbox": [ + 106, + 82, + 503, + 96 + ], + "score": 1.0, + "content": "et al., 2019; Kolesnikov et al., 2019; Donahue & Simonyan, 2019; Bachman et al., 2019; Tian et al.,", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 93, + 505, + 106 + ], + "spans": [ + { + "bbox": [ + 106, + 93, + 131, + 106 + ], + "score": 1.0, + "content": "2019;", + "type": "text" + }, + { + "bbox": [ + 131, + 93, + 505, + 106 + ], + "score": 1.0, + "content": "Chen et al., 2020a), we evaluate each method by linear classification on frozen embeddings.", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 103, + 505, + 118 + ], + "spans": [ + { + "bbox": [ + 105, + 103, + 505, + 118 + ], + "score": 1.0, + "content": "That is, we optimize a contrastive objective on a pretraining dataset to learn a representation; then,", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 115, + 505, + 128 + ], + "spans": [ + { + "bbox": [ + 105, + 115, + 505, + 128 + ], + "score": 1.0, + "content": "using a transfer dataset, we fit logistic regression on representations only. A better representation", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 126, + 496, + 140 + ], + "spans": [ + { + "bbox": [ + 105, + 126, + 496, + 140 + ], + "score": 1.0, + "content": "would contain more “object-centric” information, thereby achieving a higher classification score.", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 2 + }, + { + "type": "text", + "bbox": [ + 106, + 144, + 505, + 308 + ], + "lines": [ + { + "bbox": [ + 106, + 142, + 505, + 155 + ], + "spans": [ + { + "bbox": [ + 106, + 142, + 298, + 155 + ], + "score": 1.0, + "content": "Training Details. We pick the upper percentile", + "type": "text" + }, + { + "bbox": [ + 298, + 144, + 329, + 154 + ], + "score": 0.9, + "content": "u = 1 0", + "type": "inline_equation" + }, + { + "bbox": [ + 329, + 142, + 428, + 155 + ], + "score": 1.0, + "content": "and the lower percentile", + "type": "text" + }, + { + "bbox": [ + 428, + 144, + 452, + 154 + ], + "score": 0.9, + "content": "\\ell = 1", + "type": "inline_equation" + }, + { + "bbox": [ + 452, + 142, + 505, + 155 + ], + "score": 1.0, + "content": "although we", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 154, + 505, + 167 + ], + "spans": [ + { + "bbox": [ + 106, + 154, + 135, + 167 + ], + "score": 1.0, + "content": "anneal", + "type": "text" + }, + { + "bbox": [ + 135, + 156, + 142, + 164 + ], + "score": 0.71, + "content": "u", + "type": "inline_equation" + }, + { + "bbox": [ + 142, + 154, + 505, + 167 + ], + "score": 1.0, + "content": "starting from 100. 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ModelLinear EvaluationModelLinear EvaluationModelLinear EvaluationModelLinear Evaluation
IR81.2IR60.4IR61.4IR43.2
IRing83.9 (+2.7)IRing62.3 (+1.9)IRing64.3 (+2.9)IRing48.4 (+5.2)
CMC*85.6CMC*56.0CMC*63.8CMC*48.2
CMCRing*87.6 (+2.0)CMCRing*56.0 (+0.0)CMCRing*66.4 (+2.6)CMCRing*50.4 (+2.2)
MoCo83.1MoCo59.1MoCo63.8MoCo52.8
MoCoRing86.1 (+3.0)MoCoRing61.5 (+2.4)MoCoRing65.2 (+1.4)MoCoRing54.6 (+1.8)
LA83.9LA61.4LA63.0LA48.0
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Superscript", + "type": "text" + }, + { + "bbox": [ + 453, + 411, + 465, + 422 + ], + "score": 0.54, + "content": "( ^ { * } )", + "type": "inline_equation" + }, + { + "bbox": [ + 465, + 411, + 505, + 423 + ], + "score": 1.0, + "content": "indicates", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 421, + 466, + 434 + ], + "spans": [ + { + "bbox": [ + 106, + 421, + 466, + 434 + ], + "score": 1.0, + "content": "models that use twice as many parameters as others e.g. CMC has “L” and “ab” encoders.", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 24.5 + }, + { + "type": "text", + "bbox": [ + 106, + 445, + 505, + 523 + ], + "lines": [ + { + "bbox": [ + 106, + 445, + 505, + 459 + ], + "spans": [ + { + "bbox": [ + 106, + 445, + 410, + 459 + ], + "score": 1.0, + "content": "The results for CIFAR10, CIFAR100, STL10, and ImageNet are in Table", + "type": "text" + }, + { + "bbox": [ + 411, + 445, + 421, + 457 + ], + "score": 0.6, + "content": "\\bigstar", + "type": "inline_equation" + }, + { + "bbox": [ + 422, + 445, + 505, + 459 + ], + "score": 1.0, + "content": "Overall, IR, CMC,", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 456, + 506, + 469 + ], + "spans": [ + { + "bbox": [ + 106, + 456, + 403, + 469 + ], + "score": 1.0, + "content": "and MoCo all benefit from using more difficult negatives as shown by", + "type": "text" + }, + { + "bbox": [ + 403, + 457, + 426, + 467 + ], + "score": 0.84, + "content": "2 - 5 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 427, + 456, + 506, + 469 + ], + "score": 1.0, + "content": "absolute points of", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 467, + 505, + 479 + ], + "spans": [ + { + "bbox": [ + 106, + 467, + 505, + 479 + ], + "score": 1.0, + "content": "improvement across the four datasets. While we find different contrastive objectives to perform best", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 479, + 506, + 491 + ], + "spans": [ + { + "bbox": [ + 105, + 479, + 506, + 491 + ], + "score": 1.0, + "content": "in each dataset, the improvements from Ring are consistent: the Ring variant outperforms the base", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 487, + 506, + 503 + ], + "spans": [ + { + "bbox": [ + 105, + 487, + 506, + 503 + ], + "score": 1.0, + "content": "for every model and every dataset. We also include as a baseline Local Aggregation, or LA (Zhuang", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 500, + 506, + 513 + ], + "spans": [ + { + "bbox": [ + 106, + 500, + 303, + 513 + ], + "score": 1.0, + "content": "et al., 2019), a popular contrastive algorithm (see", + "type": "text" + }, + { + "bbox": [ + 303, + 500, + 334, + 513 + ], + "score": 0.28, + "content": "{ \\mathrm { S e c . } } \\mathrm { H } )", + "type": "inline_equation" + }, + { + "bbox": [ + 334, + 500, + 506, + 513 + ], + "score": 1.0, + "content": "that implicitly uses hard negatives without", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 511, + 399, + 524 + ], + "spans": [ + { + "bbox": [ + 106, + 511, + 344, + 524 + ], + "score": 1.0, + "content": "annealing. 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Having found", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 243, + 539, + 506, + 552 + ], + "spans": [ + { + "bbox": [ + 243, + 539, + 506, + 552 + ], + "score": 1.0, + "content": "good performance with Ring Discrimination, we want to assess", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 243, + 549, + 505, + 564 + ], + "spans": [ + { + "bbox": [ + 243, + 549, + 505, + 564 + ], + "score": 1.0, + "content": "the importance of the individual components that comprise Ring.", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 243, + 560, + 506, + 574 + ], + "spans": [ + { + "bbox": [ + 243, + 560, + 506, + 574 + ], + "score": 1.0, + "content": "We focus on the annealing policy and the exclusion of very close", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 243, + 572, + 505, + 585 + ], + "spans": [ + { + "bbox": [ + 243, + 572, + 305, + 585 + ], + "score": 1.0, + "content": "negatives from", + "type": "text" + }, + { + "bbox": [ + 305, + 572, + 319, + 583 + ], + "score": 0.87, + "content": "S _ { B }", + "type": "inline_equation" + }, + { + "bbox": [ + 319, + 572, + 505, + 585 + ], + "score": 1.0, + "content": ". 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ModelLinear Eval.
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IRing (No Anneal)81.4
IRing(l=0)82.1
(a) CIFAR10
ModelLinear Eval.
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IRing (No Anneal)41.3
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For linear", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 208, + 505, + 222 + ], + "spans": [ + { + "bbox": [ + 105, + 208, + 302, + 222 + ], + "score": 1.0, + "content": "classification, we treat the pre-pool output (size", + "type": "text" + }, + { + "bbox": [ + 303, + 209, + 357, + 220 + ], + "score": 0.9, + "content": "5 1 2 \\times 7 \\times 7 )", + "type": "inline_equation" + }, + { + "bbox": [ + 357, + 208, + 505, + 222 + ], + "score": 1.0, + "content": "after the last convolutional layer as", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 220, + 505, + 233 + ], + "spans": [ + { + "bbox": [ + 105, + 220, + 505, + 233 + ], + "score": 1.0, + "content": "the input to the logistic regression. 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We drop the learning rate", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 253, + 505, + 266 + ], + "spans": [ + { + "bbox": [ + 106, + 253, + 505, + 266 + ], + "score": 1.0, + "content": "twice by a factor of 10 on epochs 200 and 250. In transfer, we use SGD with learning rate 0.01,", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 264, + 505, + 277 + ], + "spans": [ + { + "bbox": [ + 105, + 264, + 505, + 277 + ], + "score": 1.0, + "content": "momentum 0.9, and no weight decay for 100 epochs without dropping learning rate. These hyper-", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 274, + 504, + 288 + ], + "spans": [ + { + "bbox": [ + 106, + 275, + 219, + 287 + ], + "score": 1.0, + "content": "parameters were taken from", + "type": "text" + }, + { + "bbox": [ + 219, + 274, + 286, + 287 + ], + "score": 0.81, + "content": "\\boxed { \\mathrm { W u \\ e t \\ a l . } } \\textcircled { 2 0 1 8 }", + "type": "inline_equation" + }, + { + "bbox": [ + 286, + 275, + 381, + 287 + ], + "score": 1.0, + "content": "and used in all of Table", + "type": "text" + }, + { + "bbox": [ + 381, + 275, + 391, + 288 + ], + "score": 0.48, + "content": "\\bar { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 391, + 275, + 504, + 287 + ], + "score": 1.0, + "content": "for a consistent comparison.", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 285, + 505, + 299 + ], + "spans": [ + { + "bbox": [ + 106, + 285, + 505, + 299 + ], + "score": 1.0, + "content": "We found normalizing hyperparameters to be important for a fair comparison as many competing", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 297, + 460, + 309 + ], + "spans": [ + { + "bbox": [ + 106, + 297, + 460, + 309 + ], + "score": 1.0, + "content": "algorithms use different hyperparameters. 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ModelLinear EvaluationModelLinear EvaluationModelLinear EvaluationModelLinear Evaluation
IR81.2IR60.4IR61.4IR43.2
IRing83.9 (+2.7)IRing62.3 (+1.9)IRing64.3 (+2.9)IRing48.4 (+5.2)
CMC*85.6CMC*56.0CMC*63.8CMC*48.2
CMCRing*87.6 (+2.0)CMCRing*56.0 (+0.0)CMCRing*66.4 (+2.6)CMCRing*50.4 (+2.2)
MoCo83.1MoCo59.1MoCo63.8MoCo52.8
MoCoRing86.1 (+3.0)MoCoRing61.5 (+2.4)MoCoRing65.2 (+1.4)MoCoRing54.6 (+1.8)
LA83.9LA61.4LA63.0LA48.0
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Superscript", + "type": "text" + }, + { + "bbox": [ + 453, + 411, + 465, + 422 + ], + "score": 0.54, + "content": "( ^ { * } )", + "type": "inline_equation" + }, + { + "bbox": [ + 465, + 411, + 505, + 423 + ], + "score": 1.0, + "content": "indicates", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 421, + 466, + 434 + ], + "spans": [ + { + "bbox": [ + 106, + 421, + 466, + 434 + ], + "score": 1.0, + "content": "models that use twice as many parameters as others e.g. CMC has “L” and “ab” encoders.", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 24.5, + "bbox_fs": [ + 106, + 411, + 505, + 434 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 445, + 505, + 523 + ], + "lines": [ + { + "bbox": [ + 106, + 445, + 505, + 459 + ], + "spans": [ + { + "bbox": [ + 106, + 445, + 410, + 459 + ], + "score": 1.0, + "content": "The results for CIFAR10, CIFAR100, STL10, and ImageNet are in Table", + "type": "text" + }, + { + "bbox": [ + 411, + 445, + 421, + 457 + ], + "score": 0.6, + "content": "\\bigstar", + "type": "inline_equation" + }, + { + "bbox": [ + 422, + 445, + 505, + 459 + ], + "score": 1.0, + "content": "Overall, IR, CMC,", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 456, + 506, + 469 + ], + "spans": [ + { + "bbox": [ + 106, + 456, + 403, + 469 + ], + "score": 1.0, + "content": "and MoCo all benefit from using more difficult negatives as shown by", + "type": "text" + }, + { + "bbox": [ + 403, + 457, + 426, + 467 + ], + "score": 0.84, + "content": "2 - 5 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 427, + 456, + 506, + 469 + ], + "score": 1.0, + "content": "absolute points of", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 467, + 505, + 479 + ], + "spans": [ + { + "bbox": [ + 106, + 467, + 505, + 479 + ], + "score": 1.0, + "content": "improvement across the four datasets. While we find different contrastive objectives to perform best", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 479, + 506, + 491 + ], + "spans": [ + { + "bbox": [ + 105, + 479, + 506, + 491 + ], + "score": 1.0, + "content": "in each dataset, the improvements from Ring are consistent: the Ring variant outperforms the base", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 487, + 506, + 503 + ], + "spans": [ + { + "bbox": [ + 105, + 487, + 506, + 503 + ], + "score": 1.0, + "content": "for every model and every dataset. We also include as a baseline Local Aggregation, or LA (Zhuang", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 500, + 506, + 513 + ], + "spans": [ + { + "bbox": [ + 106, + 500, + 303, + 513 + ], + "score": 1.0, + "content": "et al., 2019), a popular contrastive algorithm (see", + "type": "text" + }, + { + "bbox": [ + 303, + 500, + 334, + 513 + ], + "score": 0.28, + "content": "{ \\mathrm { S e c . } } \\mathrm { H } )", + "type": "inline_equation" + }, + { + "bbox": [ + 334, + 500, + 506, + 513 + ], + "score": 1.0, + "content": "that implicitly uses hard negatives without", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 511, + 399, + 524 + ], + "spans": [ + { + "bbox": [ + 106, + 511, + 344, + 524 + ], + "score": 1.0, + "content": "annealing. 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Having found", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 243, + 539, + 506, + 552 + ], + "spans": [ + { + "bbox": [ + 243, + 539, + 506, + 552 + ], + "score": 1.0, + "content": "good performance with Ring Discrimination, we want to assess", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 243, + 549, + 505, + 564 + ], + "spans": [ + { + "bbox": [ + 243, + 549, + 505, + 564 + ], + "score": 1.0, + "content": "the importance of the individual components that comprise Ring.", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 243, + 560, + 506, + 574 + ], + "spans": [ + { + "bbox": [ + 243, + 560, + 506, + 574 + ], + "score": 1.0, + "content": "We focus on the annealing policy and the exclusion of very close", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 243, + 572, + 505, + 585 + ], + "spans": [ + { + "bbox": [ + 243, + 572, + 305, + 585 + ], + "score": 1.0, + "content": "negatives from", + "type": "text" + }, + { + "bbox": [ + 305, + 572, + 319, + 583 + ], + "score": 0.87, + "content": "S _ { B }", + "type": "inline_equation" + }, + { + "bbox": [ + 319, + 572, + 505, + 585 + ], + "score": 1.0, + "content": ". Concretely, we measure the transfer accuracy", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 243, + 583, + 505, + 595 + ], + "spans": [ + { + "bbox": [ + 243, + 583, + 505, + 595 + ], + "score": 1.0, + "content": "of (1) IRing without annealing and (2) IRing with an lower per-", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 243, + 594, + 504, + 607 + ], + "spans": [ + { + "bbox": [ + 243, + 594, + 273, + 607 + ], + "score": 1.0, + "content": "centile", + "type": "text" + }, + { + "bbox": [ + 273, + 594, + 298, + 604 + ], + "score": 0.9, + "content": "\\ell = 0", + "type": "inline_equation" + }, + { + "bbox": [ + 298, + 594, + 489, + 607 + ], + "score": 1.0, + "content": ", thereby excluding no close negatives. That is,", + "type": "text" + }, + { + "bbox": [ + 489, + 594, + 504, + 605 + ], + "score": 0.87, + "content": "S _ { B }", + "type": "inline_equation" + } + ], + "index": 40 + }, + { + "bbox": [ + 243, + 605, + 504, + 617 + ], + "spans": [ + { + "bbox": [ + 243, + 605, + 504, + 617 + ], + "score": 1.0, + "content": "contains all examples in the dataset with representation similarity", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 242, + 615, + 506, + 629 + ], + "spans": [ + { + "bbox": [ + 242, + 615, + 296, + 629 + ], + "score": 1.0, + "content": "less than the", + "type": "text" + }, + { + "bbox": [ + 296, + 617, + 309, + 627 + ], + "score": 0.88, + "content": "\\omega _ { u }", + "type": "inline_equation" + }, + { + "bbox": [ + 310, + 615, + 455, + 629 + ], + "score": 1.0, + "content": "(a “ball” instead of a “ring”). Table", + "type": "text" + }, + { + "bbox": [ + 455, + 615, + 465, + 628 + ], + "score": 0.37, + "content": "2", + "type": "inline_equation" + }, + { + "bbox": [ + 465, + 615, + 506, + 629 + ], + "score": 1.0, + "content": "compares", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 243, + 627, + 505, + 640 + ], + "spans": [ + { + "bbox": [ + 243, + 627, + 505, + 640 + ], + "score": 1.0, + "content": "these ablations to IR and full IRing on CIFAR10 and ImageNet", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 244, + 638, + 505, + 650 + ], + "spans": [ + { + "bbox": [ + 244, + 638, + 505, + 650 + ], + "score": 1.0, + "content": "classification transfer. We observe that both ablations result in", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 243, + 649, + 505, + 662 + ], + "spans": [ + { + "bbox": [ + 243, + 649, + 505, + 662 + ], + "score": 1.0, + "content": "worse transfer accuracy, with proper annealing being especially", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 243, + 660, + 505, + 672 + ], + "spans": [ + { + "bbox": [ + 243, + 660, + 505, + 672 + ], + "score": 1.0, + "content": "important to prevent convergence to bad minima. 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ModelLinear Eval.
IR81.2
IRing83.9
IRing (No Anneal)81.4
IRing(l=0)82.1
(a) CIFAR10
ModelLinear Eval.
IR43.2
IRing48.4
IRing (No Anneal)41.3
IRing(e=0)47.3
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Thus far we have only evaluated the", + "type": "text" + } + ], + "index": 54 + }, + { + "bbox": [ + 109, + 721, + 506, + 733 + ], + "spans": [ + { + "bbox": [ + 109, + 721, + 506, + 733 + ], + "score": 1.0, + "content": "learned representations on unseen examples from the training distribution. As the goal of unsu-", + "type": "text" + } + ], + "index": 55 + }, + { + "bbox": [ + 105, + 82, + 505, + 95 + ], + "spans": [ + { + "bbox": [ + 105, + 82, + 505, + 95 + ], + "score": 1.0, + "content": "pervised learning is to capture general representations, we are also interested in their performance", + "type": "text", + "cross_page": true + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 94, + 505, + 105 + ], + "spans": [ + { + "bbox": [ + 106, + 94, + 505, + 105 + ], + "score": 1.0, + "content": "on new, unseen distributions. 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All representations were trained on CIFAR10.", + "type": "text", + "cross_page": true + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 126, + 506, + 139 + ], + "spans": [ + { + "bbox": [ + 105, + 126, + 506, + 139 + ], + "score": 1.0, + "content": "For each transfer dataset, we compute mean and variance from a training split to normalize input", + "type": "text", + "cross_page": true + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 137, + 416, + 149 + ], + "spans": [ + { + "bbox": [ + 105, + 137, + 416, + 149 + ], + "score": 1.0, + "content": "images, which we found important for generalization to new visual domains.", + "type": "text", + "cross_page": true + } + ], + "index": 5 + } + ], + "index": 54.5, + "bbox_fs": [ + 109, + 710, + 506, + 733 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 106, + 82, + 505, + 149 + ], + "lines": [ + { + "bbox": [ + 105, + 82, + 505, + 95 + ], + "spans": [ + { + "bbox": [ + 105, + 82, + 505, + 95 + ], + "score": 1.0, + "content": "pervised learning is to capture general representations, we are also interested in their performance", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 94, + 505, + 105 + ], + "spans": [ + { + "bbox": [ + 106, + 94, + 505, + 105 + ], + "score": 1.0, + "content": "on new, unseen distributions. 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ModelAircraftCUBirdsDTDFungiMNISTFashionMNISTTrafficSignVGGFlowerMSCOCO
IR40.917.939.22.796.991.797.168.152.4
IRing40.6 (-0.3)17.9 (+0.0)39.5 (+0.3)3.4(+0.7)97.8 (+0.9)91.6 (+0.1)98.8 (+1.7)68.5 (+0.4)52.5 (+0.1)
MoCo41.518.039.73.196.990.997.364.552.0
MoCoRing41.6(+0.1)18.6 (+0.6)39.5 (-0.2)3.6(+0.5)97.9 (+1.0)91.3 (+0.4)99.3 (+2.0)69.1 (+4.6)52.6 (+0.6)
CMC40.115.838.34.397.591.594.667.151.4
CMCRing40.8 (+0.7)16.8 (+1.0)40.6(+2.3)4.2 (-0.1)97.9 (+0.4)92.1 (+0.6)97.1( (+2.5)69.1 (+2.0)52.1 (+0.7)
LA41.317.839.02.397.292.398.266.952.3
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COCO: Object DetectionCOCO: Inst. SegmentationCOCO: Keypoint DetectionVOC: Object Detection
Arch.Mask R-CNN,R18-FPN,1x scheduleR-CNN,R18-FPNFaster R-CNN, R18-C4
ModelApbbAPAPApmkAPAPApkpAP5APApbbAPAP
IR8.619.06.68.517.47.434.663.032.95.514.53.3
IRing10.922.98.711.020.99.637.266.135.77.620.34.4
MoCo6.014.34.010.821.49.737.666.536.97.317.94.1
MoCoRing9.420.37.612.022.910.838.767.737.98.022.14.8
LA10.222.08.110.020.39.036.365.335.17.620.04.3
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ModelAircraftCUBirdsDTDFungiMNISTFashionMNISTTrafficSignVGGFlowerMSCOCO
IR40.917.939.22.796.991.797.168.152.4
IRing40.6 (-0.3)17.9 (+0.0)39.5 (+0.3)3.4(+0.7)97.8 (+0.9)91.6 (+0.1)98.8 (+1.7)68.5 (+0.4)52.5 (+0.1)
MoCo41.518.039.73.196.990.997.364.552.0
MoCoRing41.6(+0.1)18.6 (+0.6)39.5 (-0.2)3.6(+0.5)97.9 (+1.0)91.3 (+0.4)99.3 (+2.0)69.1 (+4.6)52.6 (+0.6)
CMC40.115.838.34.397.591.594.667.151.4
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LA41.317.839.02.397.292.398.266.952.3
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Object classification is a popular transfer task, but we want our learned", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 338, + 506, + 351 + ], + "spans": [ + { + "bbox": [ + 105, + 338, + 506, + 351 + ], + "score": 1.0, + "content": "representations to capture holistic knowledge about the contents of an image. We must thus evaluate", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 350, + 506, + 362 + ], + "spans": [ + { + "bbox": [ + 105, + 350, + 506, + 362 + ], + "score": 1.0, + "content": "performance on transfer tasks such as detection and segmentation that require different kinds of", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 361, + 506, + 374 + ], + "spans": [ + { + "bbox": [ + 106, + 361, + 506, + 374 + ], + "score": 1.0, + "content": "visual information. We study four additional downstream tasks: object detection on COCO (Lin", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 371, + 506, + 385 + ], + "spans": [ + { + "bbox": [ + 105, + 371, + 130, + 385 + ], + "score": 1.0, + "content": "et al.,", + "type": "text" + }, + { + "bbox": [ + 131, + 371, + 156, + 384 + ], + "score": 0.45, + "content": "\\boxed { 2 0 1 4 }", + "type": "inline_equation" + }, + { + "bbox": [ + 157, + 371, + 506, + 385 + ], + "score": 1.0, + "content": "and Pascal VOC’07 (Everingham et al., 2010), instance segmentation on COCO, and", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 382, + 506, + 396 + ], + "spans": [ + { + "bbox": [ + 105, + 382, + 506, + 396 + ], + "score": 1.0, + "content": "keypoint detection on COCO. In all cases, we employ embeddings trained on ImageNet with a", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 392, + 505, + 406 + ], + "spans": [ + { + "bbox": [ + 105, + 392, + 380, + 406 + ], + "score": 1.0, + "content": "ResNet-18 encoder. We base these experiments after those found in", + "type": "text" + }, + { + "bbox": [ + 380, + 393, + 445, + 406 + ], + "score": 0.48, + "content": "\\boxed { \\mathrm { H e ~ e t ~ a l . } } \\textcircled { 2 0 1 9 }", + "type": "inline_equation" + }, + { + "bbox": [ + 445, + 392, + 505, + 406 + ], + "score": 1.0, + "content": "with the same", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 405, + 506, + 416 + ], + "spans": [ + { + "bbox": [ + 106, + 405, + 506, + 416 + ], + "score": 1.0, + "content": "hyperparameters. However, we use a smaller backbone (ResNet-18 versus ResNet-50) and we freeze", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 414, + 474, + 429 + ], + "spans": [ + { + "bbox": [ + 106, + 414, + 474, + 429 + ], + "score": 1.0, + "content": "its parameters instead of finetuning them. We adapt code from Detectron2 (Wu et al., 2019).", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 19, + "bbox_fs": [ + 105, + 328, + 506, + 429 + ] + }, + { + "type": "table", + "bbox": [ + 126, + 438, + 486, + 519 + ], + "blocks": [ + { + "type": "table_body", + "bbox": [ + 126, + 438, + 486, + 519 + ], + "group_id": 1, + "lines": [ + { + "bbox": [ + 126, + 438, + 486, + 519 + ], + "spans": [ + { + "bbox": [ + 126, + 438, + 486, + 519 + ], + "score": 0.976, + "html": "
COCO: Object DetectionCOCO: Inst. SegmentationCOCO: Keypoint DetectionVOC: Object Detection
Arch.Mask R-CNN,R18-FPN,1x scheduleR-CNN,R18-FPNFaster R-CNN, R18-C4
ModelApbbAPAPApmkAPAPApkpAP5APApbbAPAP
IR8.619.06.68.517.47.434.663.032.95.514.53.3
IRing10.922.98.711.020.99.637.266.135.77.620.34.4
MoCo6.014.34.010.821.49.737.666.536.97.317.94.1
MoCoRing9.420.37.612.022.910.838.767.737.98.022.14.8
LA10.222.08.110.020.39.036.365.335.17.620.04.3
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Similarly, MoCoRing finds consistent improvements of 1-3 points over MoCo on the four", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 584, + 505, + 597 + ], + "spans": [ + { + "bbox": [ + 105, + 584, + 505, + 597 + ], + "score": 1.0, + "content": "tasks. Future work can investigate orthogonal directions of using larger encoders (e.g. ResNet-50)", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 596, + 348, + 608 + ], + "spans": [ + { + "bbox": [ + 106, + 596, + 348, + 608 + ], + "score": 1.0, + "content": "and finetuning ResNet parameters for these individual tasks.", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 30, + "bbox_fs": [ + 105, + 551, + 505, + 608 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 623, + 211, + 636 + ], + "lines": [ + { + "bbox": [ + 105, + 622, + 213, + 637 + ], + "spans": [ + { + "bbox": [ + 105, + 622, + 213, + 637 + ], + "score": 1.0, + "content": "6 RELATED WORK", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 33 + }, + { + "type": "text", + "bbox": [ + 107, + 649, + 503, + 671 + ], + "lines": [ + { + "bbox": [ + 106, + 648, + 504, + 662 + ], + "spans": [ + { + "bbox": [ + 106, + 648, + 504, + 662 + ], + "score": 1.0, + "content": "Several of the ideas in Ring Discrimination relate to existing work. Below, we explore these con-", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 659, + 429, + 672 + ], + "spans": [ + { + "bbox": [ + 105, + 659, + 429, + 672 + ], + "score": 1.0, + "content": "nections, and at the same time, place our work in a fast-paced and growing field.", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 34.5, + "bbox_fs": [ + 105, + 648, + 504, + 672 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 676, + 504, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 676, + 506, + 690 + ], + "spans": [ + { + "bbox": [ + 105, + 676, + 506, + 690 + ], + "score": 1.0, + "content": "Hard negative mining. While it has not been deeply explored in modern contrastive learning,", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 686, + 506, + 702 + ], + "spans": [ + { + "bbox": [ + 105, + 686, + 506, + 702 + ], + "score": 1.0, + "content": "negative mining has a rich line of research in the metric learning community. 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The authors show that weighting by this distribution samples more diverse neg-", + "type": "text", + "cross_page": true + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 147, + 506, + 162 + ], + "spans": [ + { + "bbox": [ + 106, + 148, + 200, + 162 + ], + "score": 1.0, + "content": "atives. Similarly, HDC", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 200, + 147, + 273, + 161 + ], + "score": 0.82, + "content": "( \\mathbf { Y } \\mathbf { u } \\mathbf { a } \\mathbf { n } \\operatorname { e t } \\mathbf { a } \\mathbf { l } . ) , [ 2 0 1 7 ]", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 274, + 148, + 506, + 162 + ], + "score": 1.0, + "content": "simulataneously optimizes a triplet loss using many levels", + "type": "text", + "cross_page": true + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 158, + 506, + 173 + ], + "spans": [ + { + "bbox": [ + 105, + 158, + 506, + 173 + ], + "score": 1.0, + "content": "of “hardness” in negatives, again improving the diversity. Although triplet objectives paved the way", + "type": "text", + "cross_page": true + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 168, + 506, + 184 + ], + "spans": [ + { + "bbox": [ + 105, + 168, + 506, + 184 + ], + "score": 1.0, + "content": "for modern NCE-based objectives, the focus on negative mining has largely been overlooked. Ring", + "type": "text", + "cross_page": true + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 179, + 506, + 195 + ], + "spans": [ + { + "bbox": [ + 105, + 179, + 506, + 195 + ], + "score": 1.0, + "content": "Discrimination, being inspired by the deep metric learning literature, reminds that negative sampling", + "type": "text", + "cross_page": true + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 191, + 506, + 205 + ], + "spans": [ + { + "bbox": [ + 105, + 191, + 506, + 205 + ], + "score": 1.0, + "content": "is still an effective way of learning stronger representations in the new NCE framework. As such,", + "type": "text", + "cross_page": true + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 204, + 505, + 215 + ], + "spans": [ + { + "bbox": [ + 106, + 204, + 505, + 215 + ], + "score": 1.0, + "content": "an important contribution was to do so while retaining the theoretical properties of NCE, namely in", + "type": "text", + "cross_page": true + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 214, + 506, + 227 + ], + "spans": [ + { + "bbox": [ + 105, + 214, + 506, + 227 + ], + "score": 1.0, + "content": "relation to mutual information. This, to the best of our knowledge, is novel as negative mining in", + "type": "text", + "cross_page": true + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 225, + 422, + 238 + ], + "spans": [ + { + "bbox": [ + 105, + 225, + 422, + 238 + ], + "score": 1.0, + "content": "metric learning literature was not characterized in terms of information theory.", + "type": "text", + "cross_page": true + } + ], + "index": 13 + } + ], + "index": 38, + "bbox_fs": [ + 105, + 676, + 506, + 734 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 82, + 505, + 236 + ], + "lines": [ + { + "bbox": [ + 106, + 82, + 505, + 95 + ], + "spans": [ + { + "bbox": [ + 106, + 82, + 249, + 95 + ], + "score": 1.0, + "content": "semi-hard negatives: Schroff et al.", + "type": "text" + }, + { + "bbox": [ + 250, + 82, + 277, + 95 + ], + "score": 0.68, + "content": "\\textcircled { | 2 0 1 5 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 277, + 82, + 366, + 95 + ], + "score": 1.0, + "content": "treats the furthest (in", + "type": "text" + }, + { + "bbox": [ + 366, + 83, + 379, + 93 + ], + "score": 0.83, + "content": "\\mathrm { L _ { 2 } }", + "type": "inline_equation" + }, + { + "bbox": [ + 379, + 82, + 505, + 95 + ], + "score": 1.0, + "content": "distance) example in the same", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 92, + 506, + 106 + ], + "spans": [ + { + "bbox": [ + 106, + 92, + 161, + 106 + ], + "score": 1.0, + "content": "minibatch as", + "type": "text" + }, + { + "bbox": [ + 161, + 95, + 172, + 105 + ], + "score": 0.84, + "content": "x _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 172, + 92, + 335, + 106 + ], + "score": 1.0, + "content": "as its negative, whereas Oh Song et al.", + "type": "text" + }, + { + "bbox": [ + 336, + 93, + 363, + 106 + ], + "score": 0.68, + "content": "\\underline { { \\left( \\bar { 2 } 0 1 6 \\right) } }", + "type": "inline_equation" + }, + { + "bbox": [ + 364, + 92, + 506, + 106 + ], + "score": 1.0, + "content": "weight each example in the mini-", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 104, + 505, + 118 + ], + "spans": [ + { + "bbox": [ + 105, + 104, + 203, + 118 + ], + "score": 1.0, + "content": "batch by its distance to", + "type": "text" + }, + { + "bbox": [ + 203, + 105, + 230, + 117 + ], + "score": 0.92, + "content": "g _ { \\boldsymbol { \\theta } } ( x _ { i } )", + "type": "inline_equation" + }, + { + "bbox": [ + 231, + 104, + 390, + 118 + ], + "score": 1.0, + "content": ", thereby being a continuous version of", + "type": "text" + }, + { + "bbox": [ + 390, + 104, + 475, + 117 + ], + "score": 0.25, + "content": "\\boxed { \\mathrm { S c h r o f f ~ e t ~ a l . } } \\textcircled { 2 0 1 5 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 475, + 104, + 478, + 118 + ], + "score": 1.0, + "content": ".", + "type": "text" + }, + { + "bbox": [ + 479, + 105, + 505, + 116 + ], + "score": 1.0, + "content": "More", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 115, + 506, + 128 + ], + "spans": [ + { + "bbox": [ + 105, + 115, + 361, + 128 + ], + "score": 1.0, + "content": "sophisticated negative sampling strategies developed over time.", + "type": "text" + }, + { + "bbox": [ + 362, + 115, + 436, + 127 + ], + "score": 0.85, + "content": "\\mathrm { I n } [ \\mathrm { W u \\ e t \\ a l . } ] \\mathbb { ( } 2 0 1 7 \\mathrm { ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 436, + 115, + 506, + 128 + ], + "score": 1.0, + "content": ", the authors pick", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 125, + 506, + 140 + ], + "spans": [ + { + "bbox": [ + 105, + 125, + 393, + 140 + ], + "score": 1.0, + "content": "negatives from a fixed normal distribution that is shown to approximate", + "type": "text" + }, + { + "bbox": [ + 394, + 127, + 406, + 137 + ], + "score": 0.83, + "content": "\\mathrm { L _ { 2 } }", + "type": "inline_equation" + }, + { + "bbox": [ + 406, + 125, + 506, + 140 + ], + "score": 1.0, + "content": "normalized embeddings", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 135, + 505, + 151 + ], + "spans": [ + { + "bbox": [ + 105, + 135, + 505, + 151 + ], + "score": 1.0, + "content": "in high dimensions. The authors show that weighting by this distribution samples more diverse neg-", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 147, + 506, + 162 + ], + "spans": [ + { + "bbox": [ + 106, + 148, + 200, + 162 + ], + "score": 1.0, + "content": "atives. Similarly, HDC", + "type": "text" + }, + { + "bbox": [ + 200, + 147, + 273, + 161 + ], + "score": 0.82, + "content": "( \\mathbf { Y } \\mathbf { u } \\mathbf { a } \\mathbf { n } \\operatorname { e t } \\mathbf { a } \\mathbf { l } . ) , [ 2 0 1 7 ]", + "type": "inline_equation" + }, + { + "bbox": [ + 274, + 148, + 506, + 162 + ], + "score": 1.0, + "content": "simulataneously optimizes a triplet loss using many levels", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 158, + 506, + 173 + ], + "spans": [ + { + "bbox": [ + 105, + 158, + 506, + 173 + ], + "score": 1.0, + "content": "of “hardness” in negatives, again improving the diversity. Although triplet objectives paved the way", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 168, + 506, + 184 + ], + "spans": [ + { + "bbox": [ + 105, + 168, + 506, + 184 + ], + "score": 1.0, + "content": "for modern NCE-based objectives, the focus on negative mining has largely been overlooked. Ring", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 179, + 506, + 195 + ], + "spans": [ + { + "bbox": [ + 105, + 179, + 506, + 195 + ], + "score": 1.0, + "content": "Discrimination, being inspired by the deep metric learning literature, reminds that negative sampling", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 191, + 506, + 205 + ], + "spans": [ + { + "bbox": [ + 105, + 191, + 506, + 205 + ], + "score": 1.0, + "content": "is still an effective way of learning stronger representations in the new NCE framework. As such,", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 204, + 505, + 215 + ], + "spans": [ + { + "bbox": [ + 106, + 204, + 505, + 215 + ], + "score": 1.0, + "content": "an important contribution was to do so while retaining the theoretical properties of NCE, namely in", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 214, + 506, + 227 + ], + "spans": [ + { + "bbox": [ + 105, + 214, + 506, + 227 + ], + "score": 1.0, + "content": "relation to mutual information. This, to the best of our knowledge, is novel as negative mining in", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 225, + 422, + 238 + ], + "spans": [ + { + "bbox": [ + 105, + 225, + 422, + 238 + ], + "score": 1.0, + "content": "metric learning literature was not characterized in terms of information theory.", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 6.5 + }, + { + "type": "text", + "bbox": [ + 106, + 242, + 505, + 417 + ], + "lines": [ + { + "bbox": [ + 105, + 241, + 506, + 254 + ], + "spans": [ + { + "bbox": [ + 105, + 241, + 479, + 254 + ], + "score": 1.0, + "content": "That being said, there are some cases of negative mining in contrastive literature. In CPC", + "type": "text" + }, + { + "bbox": [ + 480, + 242, + 506, + 254 + ], + "score": 1.0, + "content": "(Oord", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 253, + 506, + 265 + ], + "spans": [ + { + "bbox": [ + 105, + 253, + 506, + 265 + ], + "score": 1.0, + "content": "et al., 2018), the authors explore using negatives from the same speaker versus from mixed speakers", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 104, + 262, + 505, + 277 + ], + "spans": [ + { + "bbox": [ + 104, + 262, + 505, + 277 + ], + "score": 1.0, + "content": "in audio applications, the former of which can be interpreted as being more difficult. A recent paper,", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 274, + 506, + 287 + ], + "spans": [ + { + "bbox": [ + 106, + 274, + 506, + 287 + ], + "score": 1.0, + "content": "InterCLR (Xie et al., 2020), also finds that using “semi-hard negatives” is beneficial to contrastive", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 286, + 505, + 298 + ], + "spans": [ + { + "bbox": [ + 106, + 286, + 505, + 298 + ], + "score": 1.0, + "content": "learning whereas negatives that are too difficult or too easy produce worse representations. Where", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 296, + 505, + 308 + ], + "spans": [ + { + "bbox": [ + 106, + 296, + 505, + 308 + ], + "score": 1.0, + "content": "InterCLR uses a margin-based approach to sample negatives, we explore a wider family of negative", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 307, + 505, + 320 + ], + "spans": [ + { + "bbox": [ + 105, + 307, + 505, + 320 + ], + "score": 1.0, + "content": "distributions and show analysis that annealing offers a simple and easy solution to choosing between", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 318, + 506, + 331 + ], + "spans": [ + { + "bbox": [ + 106, + 318, + 506, + 331 + ], + "score": 1.0, + "content": "easy and hard negatives. Further, as InterCLR’s negative sampling procedure is a special case of", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 329, + 505, + 342 + ], + "spans": [ + { + "bbox": [ + 106, + 329, + 505, + 342 + ], + "score": 1.0, + "content": "CNCE, we provide theory grounding these approaches in information theory. Finally, a separate", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 340, + 506, + 352 + ], + "spans": [ + { + "bbox": [ + 106, + 340, + 506, + 352 + ], + "score": 1.0, + "content": "line of work in contrastive learning explores using neighboring examples (in embedding space) as", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 350, + 506, + 365 + ], + "spans": [ + { + "bbox": [ + 105, + 350, + 506, + 365 + ], + "score": 1.0, + "content": "“positive” views of the instance (Zhuang et al., 2019; Xie et al., 2020; Asano et al., 2019; Caron", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 104, + 360, + 506, + 376 + ], + "spans": [ + { + "bbox": [ + 104, + 360, + 130, + 376 + ], + "score": 1.0, + "content": "et al.,", + "type": "text" + }, + { + "bbox": [ + 131, + 362, + 156, + 375 + ], + "score": 0.57, + "content": "\\boxed { 2 0 2 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 156, + 360, + 313, + 376 + ], + "score": 1.0, + "content": "Li et al., 2020). That is, finding a set", + "type": "text" + }, + { + "bbox": [ + 313, + 362, + 334, + 375 + ], + "score": 0.91, + "content": "\\overline { { \\{ x _ { j } \\} } }", + "type": "inline_equation" + }, + { + "bbox": [ + 334, + 360, + 426, + 376 + ], + "score": 1.0, + "content": "such that we consider", + "type": "text" + }, + { + "bbox": [ + 427, + 363, + 474, + 375 + ], + "score": 0.9, + "content": "{ \\overline { { x _ { j } \\ } } } = { \\overline { { t ( x _ { i } ) } } }", + "type": "inline_equation" + }, + { + "bbox": [ + 474, + 360, + 506, + 376 + ], + "score": 1.0, + "content": "for the", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 374, + 506, + 385 + ], + "spans": [ + { + "bbox": [ + 105, + 374, + 173, + 385 + ], + "score": 1.0, + "content": "current instance", + "type": "text" + }, + { + "bbox": [ + 173, + 376, + 183, + 384 + ], + "score": 0.79, + "content": "x _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 184, + 374, + 506, + 385 + ], + "score": 1.0, + "content": ". While this does not deal with negatives explicitly, it shares similarities to our", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 384, + 505, + 397 + ], + "spans": [ + { + "bbox": [ + 105, + 384, + 505, + 397 + ], + "score": 1.0, + "content": "approach by employing other examples in the contrastive objective to learn better representations.", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 395, + 505, + 408 + ], + "spans": [ + { + "bbox": [ + 105, + 395, + 505, + 408 + ], + "score": 1.0, + "content": "In the Appendix, we discuss how one of these algorithms, LA (Zhuang et al., 2019), implicitly uses", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 406, + 380, + 419 + ], + "spans": [ + { + "bbox": [ + 105, + 406, + 380, + 419 + ], + "score": 1.0, + "content": "hard negatives and expand the Ring family with ideas inspired by it.", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 21.5 + }, + { + "type": "text", + "bbox": [ + 106, + 423, + 505, + 665 + ], + "lines": [ + { + "bbox": [ + 107, + 423, + 505, + 435 + ], + "spans": [ + { + "bbox": [ + 107, + 423, + 505, + 435 + ], + "score": 1.0, + "content": "Contrastive learning. We focused primarily on comparing Ring Discrimination to three recent", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 434, + 506, + 446 + ], + "spans": [ + { + "bbox": [ + 105, + 434, + 506, + 446 + ], + "score": 1.0, + "content": "and highly performing contrastive algorithms, but the field contains much more. The basic idea of", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 445, + 505, + 457 + ], + "spans": [ + { + "bbox": [ + 105, + 445, + 505, + 457 + ], + "score": 1.0, + "content": "learning representations to be invariant under a family of transformations is an old one, having been", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 455, + 507, + 469 + ], + "spans": [ + { + "bbox": [ + 105, + 455, + 249, + 469 + ], + "score": 1.0, + "content": "explored with self-organizing maps", + "type": "text" + }, + { + "bbox": [ + 250, + 455, + 350, + 469 + ], + "score": 0.8, + "content": "\\scriptstyle ( \\mathbf { B e c k e r } \\ \\& \\ \\mathbf { H i n t o n } , \\mathbf { \\left[ 1 9 9 2 \\right] } )", + "type": "inline_equation" + }, + { + "bbox": [ + 350, + 455, + 507, + 469 + ], + "score": 1.0, + "content": "and dimensionality reduction (Hadsell", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 104, + 466, + 505, + 479 + ], + "spans": [ + { + "bbox": [ + 104, + 466, + 505, + 479 + ], + "score": 1.0, + "content": "et al., 2006). Before IR, the idea of instance discrimination was studied (Dosovitskiy et al., 2014;", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 106, + 478, + 503, + 491 + ], + "spans": [ + { + "bbox": [ + 106, + 478, + 172, + 491 + ], + "score": 1.0, + "content": "Wang & Gupta,", + "type": "text" + }, + { + "bbox": [ + 172, + 478, + 198, + 491 + ], + "score": 0.56, + "content": "\\underline { { 2 0 1 5 } } )", + "type": "inline_equation" + }, + { + "bbox": [ + 198, + 478, + 503, + 491 + ], + "score": 1.0, + "content": "among many pretext objectives such as position prediction (Doersch et al.,", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 488, + 502, + 502 + ], + "spans": [ + { + "bbox": [ + 105, + 488, + 260, + 502 + ], + "score": 1.0, + "content": "2015), color prediction (Zhang et al.,", + "type": "text" + }, + { + "bbox": [ + 260, + 488, + 286, + 500 + ], + "score": 0.27, + "content": "\\boxed { 2 0 1 6 }", + "type": "inline_equation" + }, + { + "bbox": [ + 286, + 488, + 502, + 502 + ], + "score": 1.0, + "content": ", multi-task objectives (Doersch & Zisserman, 2017),", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 499, + 506, + 513 + ], + "spans": [ + { + "bbox": [ + 105, + 499, + 506, + 513 + ], + "score": 1.0, + "content": "rotation prediction (Gidaris et al., 2018; Chen et al., 2019), and many other “pretext” objectives", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 108, + 510, + 505, + 523 + ], + "spans": [ + { + "bbox": [ + 108, + 510, + 505, + 523 + ], + "score": 1.0, + "content": "(Pathak et al., 2017). As we have mentioned, one of the primary challenges to instance discrimi-", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 521, + 506, + 534 + ], + "spans": [ + { + "bbox": [ + 105, + 521, + 506, + 534 + ], + "score": 1.0, + "content": "nation is making such a large softmax objective tractable. Moving from a parametric (Dosovitskiy", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 104, + 532, + 506, + 546 + ], + "spans": [ + { + "bbox": [ + 104, + 532, + 130, + 546 + ], + "score": 1.0, + "content": "et al.,", + "type": "text" + }, + { + "bbox": [ + 130, + 532, + 156, + 545 + ], + "score": 0.68, + "content": "\\boxed { 2 0 1 4 }", + "type": "inline_equation" + }, + { + "bbox": [ + 156, + 532, + 506, + 546 + ], + "score": 1.0, + "content": "to a nonparametric softmax reduced issues with vanishing gradients, shifting the chal-", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 543, + 505, + 556 + ], + "spans": [ + { + "bbox": [ + 105, + 543, + 371, + 556 + ], + "score": 1.0, + "content": "lenge to efficient negative sampling. The memory bank approach", + "type": "text" + }, + { + "bbox": [ + 372, + 543, + 441, + 556 + ], + "score": 0.46, + "content": "\\mathbb { W } \\mathrm { u } \\mathrm { ~ e t ~ a l . } \\lVert \\mathbb { Z } \\mathrm { 0 . 8 } \\rVert", + "type": "inline_equation" + }, + { + "bbox": [ + 441, + 543, + 505, + 556 + ], + "score": 1.0, + "content": "is a simple and", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 106, + 555, + 504, + 567 + ], + "spans": [ + { + "bbox": [ + 106, + 555, + 504, + 567 + ], + "score": 1.0, + "content": "memory-efficient solution, quickly being adopted by the research community (Zhuang et al., 2019;", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 565, + 505, + 579 + ], + "spans": [ + { + "bbox": [ + 105, + 565, + 505, + 579 + ], + "score": 1.0, + "content": "Tian et al., 2019; He et al., 2019; Chen et al., 2020b; Misra & Maaten, 2020). With enough compu-", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 577, + 505, + 589 + ], + "spans": [ + { + "bbox": [ + 105, + 577, + 505, + 589 + ], + "score": 1.0, + "content": "tational resources, it is now also possible to reuse examples in a large minibatch and negatives of one", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 587, + 505, + 600 + ], + "spans": [ + { + "bbox": [ + 105, + 587, + 505, + 600 + ], + "score": 1.0, + "content": "another (Ye et al., 2019; Ji et al., 2019; Chen et al., 2020a). In our work, we focus on hard negative", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 106, + 599, + 505, + 611 + ], + "spans": [ + { + "bbox": [ + 106, + 599, + 505, + 611 + ], + "score": 1.0, + "content": "mining in the context of a memory bank or queue due to its computational efficiency. However,", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 608, + 505, + 622 + ], + "spans": [ + { + "bbox": [ + 105, + 608, + 505, + 622 + ], + "score": 1.0, + "content": "the same principles should be applicable to batch-based methods (e.g. SimCLR): assuming a large", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 105, + 619, + 505, + 633 + ], + "spans": [ + { + "bbox": [ + 105, + 619, + 505, + 633 + ], + "score": 1.0, + "content": "enough batch size, for each example, we only use a subset of the minibatch as negatives as in Ring.", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 104, + 630, + 505, + 645 + ], + "spans": [ + { + "bbox": [ + 104, + 630, + 505, + 645 + ], + "score": 1.0, + "content": "Finally, more recent work (Grill et al., 2020) removes negatives altogether, which is speculated to", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 105, + 641, + 506, + 655 + ], + "spans": [ + { + "bbox": [ + 105, + 641, + 506, + 655 + ], + "score": 1.0, + "content": "implicitly use negative samples via batch normalization (Ioffe & Szegedy, 2015); we leave a more", + "type": "text" + } + ], + "index": 50 + }, + { + "bbox": [ + 106, + 653, + 372, + 666 + ], + "spans": [ + { + "bbox": [ + 106, + 653, + 372, + 666 + ], + "score": 1.0, + "content": "thorough understanding of negatives in this setting to future work.", + "type": "text" + } + ], + "index": 51 + } + ], + "index": 40.5 + }, + { + "type": "title", + "bbox": [ + 108, + 683, + 190, + 695 + ], + "lines": [ + { + "bbox": [ + 104, + 681, + 192, + 699 + ], + "spans": [ + { + "bbox": [ + 104, + 681, + 192, + 699 + ], + "score": 1.0, + "content": "7 DISCUSSION", + "type": "text" + } + ], + "index": 52 + } + ], + "index": 52 + }, + { + "type": "text", + "bbox": [ + 106, + 709, + 504, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 709, + 506, + 722 + ], + "spans": [ + { + "bbox": [ + 106, + 709, + 506, + 722 + ], + "score": 1.0, + "content": "Computational cost of Ring. To measure the cost of CNCE, we compare the cost an epoch of", + "type": "text" + } + ], + "index": 53 + }, + { + "bbox": [ + 105, + 719, + 506, + 734 + ], + "spans": [ + { + "bbox": [ + 105, + 719, + 506, + 734 + ], + "score": 1.0, + "content": "training MoCo/IR versus MoCoRing/IRing on four image datasets. Table 5a reports the average", + "type": "text" + } + ], + "index": 54 + } + ], + "index": 53.5 + } + ], + "page_idx": 7, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 108, + 27, + 292, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 26, + 293, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 26, + 293, + 38 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2021", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 302, + 752, + 308, + 759 + ], + "lines": [ + { + "bbox": [ + 302, + 750, + 309, + 761 + ], + "spans": [ + { + "bbox": [ + 302, + 750, + 309, + 761 + ], + "score": 1.0, + "content": "8", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 82, + 505, + 236 + ], + "lines": [], + "index": 6.5, + "bbox_fs": [ + 105, + 82, + 506, + 238 + ], + "lines_deleted": true + }, + { + "type": "text", + "bbox": [ + 106, + 242, + 505, + 417 + ], + "lines": [ + { + "bbox": [ + 105, + 241, + 506, + 254 + ], + "spans": [ + { + "bbox": [ + 105, + 241, + 479, + 254 + ], + "score": 1.0, + "content": "That being said, there are some cases of negative mining in contrastive literature. In CPC", + "type": "text" + }, + { + "bbox": [ + 480, + 242, + 506, + 254 + ], + "score": 1.0, + "content": "(Oord", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 253, + 506, + 265 + ], + "spans": [ + { + "bbox": [ + 105, + 253, + 506, + 265 + ], + "score": 1.0, + "content": "et al., 2018), the authors explore using negatives from the same speaker versus from mixed speakers", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 104, + 262, + 505, + 277 + ], + "spans": [ + { + "bbox": [ + 104, + 262, + 505, + 277 + ], + "score": 1.0, + "content": "in audio applications, the former of which can be interpreted as being more difficult. A recent paper,", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 274, + 506, + 287 + ], + "spans": [ + { + "bbox": [ + 106, + 274, + 506, + 287 + ], + "score": 1.0, + "content": "InterCLR (Xie et al., 2020), also finds that using “semi-hard negatives” is beneficial to contrastive", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 286, + 505, + 298 + ], + "spans": [ + { + "bbox": [ + 106, + 286, + 505, + 298 + ], + "score": 1.0, + "content": "learning whereas negatives that are too difficult or too easy produce worse representations. Where", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 296, + 505, + 308 + ], + "spans": [ + { + "bbox": [ + 106, + 296, + 505, + 308 + ], + "score": 1.0, + "content": "InterCLR uses a margin-based approach to sample negatives, we explore a wider family of negative", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 307, + 505, + 320 + ], + "spans": [ + { + "bbox": [ + 105, + 307, + 505, + 320 + ], + "score": 1.0, + "content": "distributions and show analysis that annealing offers a simple and easy solution to choosing between", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 318, + 506, + 331 + ], + "spans": [ + { + "bbox": [ + 106, + 318, + 506, + 331 + ], + "score": 1.0, + "content": "easy and hard negatives. Further, as InterCLR’s negative sampling procedure is a special case of", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 329, + 505, + 342 + ], + "spans": [ + { + "bbox": [ + 106, + 329, + 505, + 342 + ], + "score": 1.0, + "content": "CNCE, we provide theory grounding these approaches in information theory. Finally, a separate", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 340, + 506, + 352 + ], + "spans": [ + { + "bbox": [ + 106, + 340, + 506, + 352 + ], + "score": 1.0, + "content": "line of work in contrastive learning explores using neighboring examples (in embedding space) as", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 350, + 506, + 365 + ], + "spans": [ + { + "bbox": [ + 105, + 350, + 506, + 365 + ], + "score": 1.0, + "content": "“positive” views of the instance (Zhuang et al., 2019; Xie et al., 2020; Asano et al., 2019; Caron", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 104, + 360, + 506, + 376 + ], + "spans": [ + { + "bbox": [ + 104, + 360, + 130, + 376 + ], + "score": 1.0, + "content": "et al.,", + "type": "text" + }, + { + "bbox": [ + 131, + 362, + 156, + 375 + ], + "score": 0.57, + "content": "\\boxed { 2 0 2 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 156, + 360, + 313, + 376 + ], + "score": 1.0, + "content": "Li et al., 2020). That is, finding a set", + "type": "text" + }, + { + "bbox": [ + 313, + 362, + 334, + 375 + ], + "score": 0.91, + "content": "\\overline { { \\{ x _ { j } \\} } }", + "type": "inline_equation" + }, + { + "bbox": [ + 334, + 360, + 426, + 376 + ], + "score": 1.0, + "content": "such that we consider", + "type": "text" + }, + { + "bbox": [ + 427, + 363, + 474, + 375 + ], + "score": 0.9, + "content": "{ \\overline { { x _ { j } \\ } } } = { \\overline { { t ( x _ { i } ) } } }", + "type": "inline_equation" + }, + { + "bbox": [ + 474, + 360, + 506, + 376 + ], + "score": 1.0, + "content": "for the", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 374, + 506, + 385 + ], + "spans": [ + { + "bbox": [ + 105, + 374, + 173, + 385 + ], + "score": 1.0, + "content": "current instance", + "type": "text" + }, + { + "bbox": [ + 173, + 376, + 183, + 384 + ], + "score": 0.79, + "content": "x _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 184, + 374, + 506, + 385 + ], + "score": 1.0, + "content": ". While this does not deal with negatives explicitly, it shares similarities to our", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 384, + 505, + 397 + ], + "spans": [ + { + "bbox": [ + 105, + 384, + 505, + 397 + ], + "score": 1.0, + "content": "approach by employing other examples in the contrastive objective to learn better representations.", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 395, + 505, + 408 + ], + "spans": [ + { + "bbox": [ + 105, + 395, + 505, + 408 + ], + "score": 1.0, + "content": "In the Appendix, we discuss how one of these algorithms, LA (Zhuang et al., 2019), implicitly uses", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 406, + 380, + 419 + ], + "spans": [ + { + "bbox": [ + 105, + 406, + 380, + 419 + ], + "score": 1.0, + "content": "hard negatives and expand the Ring family with ideas inspired by it.", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 21.5, + "bbox_fs": [ + 104, + 241, + 506, + 419 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 423, + 505, + 665 + ], + "lines": [ + { + "bbox": [ + 107, + 423, + 505, + 435 + ], + "spans": [ + { + "bbox": [ + 107, + 423, + 505, + 435 + ], + "score": 1.0, + "content": "Contrastive learning. We focused primarily on comparing Ring Discrimination to three recent", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 434, + 506, + 446 + ], + "spans": [ + { + "bbox": [ + 105, + 434, + 506, + 446 + ], + "score": 1.0, + "content": "and highly performing contrastive algorithms, but the field contains much more. The basic idea of", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 445, + 505, + 457 + ], + "spans": [ + { + "bbox": [ + 105, + 445, + 505, + 457 + ], + "score": 1.0, + "content": "learning representations to be invariant under a family of transformations is an old one, having been", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 455, + 507, + 469 + ], + "spans": [ + { + "bbox": [ + 105, + 455, + 249, + 469 + ], + "score": 1.0, + "content": "explored with self-organizing maps", + "type": "text" + }, + { + "bbox": [ + 250, + 455, + 350, + 469 + ], + "score": 0.8, + "content": "\\scriptstyle ( \\mathbf { B e c k e r } \\ \\& \\ \\mathbf { H i n t o n } , \\mathbf { \\left[ 1 9 9 2 \\right] } )", + "type": "inline_equation" + }, + { + "bbox": [ + 350, + 455, + 507, + 469 + ], + "score": 1.0, + "content": "and dimensionality reduction (Hadsell", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 104, + 466, + 505, + 479 + ], + "spans": [ + { + "bbox": [ + 104, + 466, + 505, + 479 + ], + "score": 1.0, + "content": "et al., 2006). Before IR, the idea of instance discrimination was studied (Dosovitskiy et al., 2014;", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 106, + 478, + 503, + 491 + ], + "spans": [ + { + "bbox": [ + 106, + 478, + 172, + 491 + ], + "score": 1.0, + "content": "Wang & Gupta,", + "type": "text" + }, + { + "bbox": [ + 172, + 478, + 198, + 491 + ], + "score": 0.56, + "content": "\\underline { { 2 0 1 5 } } )", + "type": "inline_equation" + }, + { + "bbox": [ + 198, + 478, + 503, + 491 + ], + "score": 1.0, + "content": "among many pretext objectives such as position prediction (Doersch et al.,", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 488, + 502, + 502 + ], + "spans": [ + { + "bbox": [ + 105, + 488, + 260, + 502 + ], + "score": 1.0, + "content": "2015), color prediction (Zhang et al.,", + "type": "text" + }, + { + "bbox": [ + 260, + 488, + 286, + 500 + ], + "score": 0.27, + "content": "\\boxed { 2 0 1 6 }", + "type": "inline_equation" + }, + { + "bbox": [ + 286, + 488, + 502, + 502 + ], + "score": 1.0, + "content": ", multi-task objectives (Doersch & Zisserman, 2017),", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 499, + 506, + 513 + ], + "spans": [ + { + "bbox": [ + 105, + 499, + 506, + 513 + ], + "score": 1.0, + "content": "rotation prediction (Gidaris et al., 2018; Chen et al., 2019), and many other “pretext” objectives", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 108, + 510, + 505, + 523 + ], + "spans": [ + { + "bbox": [ + 108, + 510, + 505, + 523 + ], + "score": 1.0, + "content": "(Pathak et al., 2017). As we have mentioned, one of the primary challenges to instance discrimi-", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 521, + 506, + 534 + ], + "spans": [ + { + "bbox": [ + 105, + 521, + 506, + 534 + ], + "score": 1.0, + "content": "nation is making such a large softmax objective tractable. Moving from a parametric (Dosovitskiy", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 104, + 532, + 506, + 546 + ], + "spans": [ + { + "bbox": [ + 104, + 532, + 130, + 546 + ], + "score": 1.0, + "content": "et al.,", + "type": "text" + }, + { + "bbox": [ + 130, + 532, + 156, + 545 + ], + "score": 0.68, + "content": "\\boxed { 2 0 1 4 }", + "type": "inline_equation" + }, + { + "bbox": [ + 156, + 532, + 506, + 546 + ], + "score": 1.0, + "content": "to a nonparametric softmax reduced issues with vanishing gradients, shifting the chal-", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 543, + 505, + 556 + ], + "spans": [ + { + "bbox": [ + 105, + 543, + 371, + 556 + ], + "score": 1.0, + "content": "lenge to efficient negative sampling. The memory bank approach", + "type": "text" + }, + { + "bbox": [ + 372, + 543, + 441, + 556 + ], + "score": 0.46, + "content": "\\mathbb { W } \\mathrm { u } \\mathrm { ~ e t ~ a l . } \\lVert \\mathbb { Z } \\mathrm { 0 . 8 } \\rVert", + "type": "inline_equation" + }, + { + "bbox": [ + 441, + 543, + 505, + 556 + ], + "score": 1.0, + "content": "is a simple and", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 106, + 555, + 504, + 567 + ], + "spans": [ + { + "bbox": [ + 106, + 555, + 504, + 567 + ], + "score": 1.0, + "content": "memory-efficient solution, quickly being adopted by the research community (Zhuang et al., 2019;", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 565, + 505, + 579 + ], + "spans": [ + { + "bbox": [ + 105, + 565, + 505, + 579 + ], + "score": 1.0, + "content": "Tian et al., 2019; He et al., 2019; Chen et al., 2020b; Misra & Maaten, 2020). With enough compu-", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 577, + 505, + 589 + ], + "spans": [ + { + "bbox": [ + 105, + 577, + 505, + 589 + ], + "score": 1.0, + "content": "tational resources, it is now also possible to reuse examples in a large minibatch and negatives of one", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 587, + 505, + 600 + ], + "spans": [ + { + "bbox": [ + 105, + 587, + 505, + 600 + ], + "score": 1.0, + "content": "another (Ye et al., 2019; Ji et al., 2019; Chen et al., 2020a). In our work, we focus on hard negative", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 106, + 599, + 505, + 611 + ], + "spans": [ + { + "bbox": [ + 106, + 599, + 505, + 611 + ], + "score": 1.0, + "content": "mining in the context of a memory bank or queue due to its computational efficiency. However,", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 608, + 505, + 622 + ], + "spans": [ + { + "bbox": [ + 105, + 608, + 505, + 622 + ], + "score": 1.0, + "content": "the same principles should be applicable to batch-based methods (e.g. SimCLR): assuming a large", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 105, + 619, + 505, + 633 + ], + "spans": [ + { + "bbox": [ + 105, + 619, + 505, + 633 + ], + "score": 1.0, + "content": "enough batch size, for each example, we only use a subset of the minibatch as negatives as in Ring.", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 104, + 630, + 505, + 645 + ], + "spans": [ + { + "bbox": [ + 104, + 630, + 505, + 645 + ], + "score": 1.0, + "content": "Finally, more recent work (Grill et al., 2020) removes negatives altogether, which is speculated to", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 105, + 641, + 506, + 655 + ], + "spans": [ + { + "bbox": [ + 105, + 641, + 506, + 655 + ], + "score": 1.0, + "content": "implicitly use negative samples via batch normalization (Ioffe & Szegedy, 2015); we leave a more", + "type": "text" + } + ], + "index": 50 + }, + { + "bbox": [ + 106, + 653, + 372, + 666 + ], + "spans": [ + { + "bbox": [ + 106, + 653, + 372, + 666 + ], + "score": 1.0, + "content": "thorough understanding of negatives in this setting to future work.", + "type": "text" + } + ], + "index": 51 + } + ], + "index": 40.5, + "bbox_fs": [ + 104, + 423, + 507, + 666 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 683, + 190, + 695 + ], + "lines": [ + { + "bbox": [ + 104, + 681, + 192, + 699 + ], + "spans": [ + { + "bbox": [ + 104, + 681, + 192, + 699 + ], + "score": 1.0, + "content": "7 DISCUSSION", + "type": "text" + } + ], + "index": 52 + } + ], + "index": 52 + }, + { + "type": "text", + "bbox": [ + 106, + 709, + 504, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 709, + 506, + 722 + ], + "spans": [ + { + "bbox": [ + 106, + 709, + 506, + 722 + ], + "score": 1.0, + "content": "Computational cost of Ring. To measure the cost of CNCE, we compare the cost an epoch of", + "type": "text" + } + ], + "index": 53 + }, + { + "bbox": [ + 105, + 719, + 506, + 734 + ], + "spans": [ + { + "bbox": [ + 105, + 719, + 506, + 734 + ], + "score": 1.0, + "content": "training MoCo/IR versus MoCoRing/IRing on four image datasets. Table 5a reports the average", + "type": "text" + } + ], + "index": 54 + }, + { + "bbox": [ + 105, + 267, + 505, + 279 + ], + "spans": [ + { + "bbox": [ + 105, + 267, + 505, + 279 + ], + "score": 1.0, + "content": "cost over 200 epochs. We observe that Ring models cost no more than 1.5 times the cost of standard", + "type": "text", + "cross_page": true + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 278, + 505, + 290 + ], + "spans": [ + { + "bbox": [ + 105, + 278, + 505, + 290 + ], + "score": 1.0, + "content": "contrastive algorithms, amounting to a difference of 3 to 7 minutes in ImageNet and 10 to 60 seconds", + "type": "text", + "cross_page": true + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 289, + 505, + 301 + ], + "spans": [ + { + "bbox": [ + 105, + 289, + 505, + 301 + ], + "score": 1.0, + "content": "in three other datasets per epoch. In the context of deep learning, we do not find the cost increases", + "type": "text", + "cross_page": true + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 300, + 505, + 312 + ], + "spans": [ + { + "bbox": [ + 105, + 300, + 505, + 312 + ], + "score": 1.0, + "content": "to be substantial. In particular, since (1) the memory structure in IR and MoCo allow us to store and", + "type": "text", + "cross_page": true + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 310, + 506, + 324 + ], + "spans": [ + { + "bbox": [ + 105, + 310, + 506, + 324 + ], + "score": 1.0, + "content": "reuse embeddings and (2) gradients are not propagated through the memory structure, the additional", + "type": "text", + "cross_page": true + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 321, + 506, + 334 + ], + "spans": [ + { + "bbox": [ + 105, + 321, + 506, + 334 + ], + "score": 1.0, + "content": "compute of Ring amounts to one matrix multiplication, which is cheap on modern hardware. We", + "type": "text", + "cross_page": true + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 331, + 479, + 345 + ], + "spans": [ + { + "bbox": [ + 106, + 331, + 479, + 345 + ], + "score": 1.0, + "content": "used a single Titan X GPU with 8 CPU workers, and PyTorch Lightning (Falcon et al., 2019).", + "type": "text", + "cross_page": true + } + ], + "index": 24 + } + ], + "index": 53.5, + "bbox_fs": [ + 105, + 709, + 506, + 734 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "table", + "bbox": [ + 139, + 80, + 295, + 133 + ], + "blocks": [ + { + "type": "table_body", + "bbox": [ + 139, + 80, + 295, + 133 + ], + "group_id": 3, + "lines": [ + { + "bbox": [ + 139, + 80, + 295, + 133 + ], + "spans": [ + { + "bbox": [ + 139, + 80, + 295, + 133 + ], + "score": 0.553, + "html": "
ModelCIFAR10 (sec.)ImageNet (min.)
IR136.0 ±443.9 ±1
IRing141.1 ± 5 (1.1x)51.0±1 (1.2x)
MoCo318.4±1661.1 ±1
MoCoRing383.4 ± 12 (1.2x)64.9 ±1 (1.1x)
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DatasetArch.MoCo-v2MoCoRing-v2
CIFAR10ResNet-1890.191.9(+1.8)
CIFAR10ResNet-5092.494.1(+1.6)
CIFAR100ResNet-1865.167.3(+2.2)
STL10ResNet-1874.876.7(+1.9)
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Transfer TaskMoCoMoCoRing
LibriSpeech Spk.ID (Panayotov et al.201595.596.6(+1.1)
AudioMNISTBecker et al. 2018)87.491.3(+3.9)
Google CommandsWarden 1201838.541.4(+2.9)
Fluent Actions Lugosch et al.] 201936.536.8(+0.3)
Fluent ObjectsLugosch et al. 1201941.944.1(+2.2)
Fluent Locations Lugoschet al. 1201960.963.9(+3.0)
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DatasetSimCLRSimCLRing
CIFAR1088.989.3(+0.4)
CIFAR10063.564.1(+0.6)
STL1071.272.1(+0.9)
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We observe that Ring models cost no more than 1.5 times the cost of standard", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 278, + 505, + 290 + ], + "spans": [ + { + "bbox": [ + 105, + 278, + 505, + 290 + ], + "score": 1.0, + "content": "contrastive algorithms, amounting to a difference of 3 to 7 minutes in ImageNet and 10 to 60 seconds", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 289, + 505, + 301 + ], + "spans": [ + { + "bbox": [ + 105, + 289, + 505, + 301 + ], + "score": 1.0, + "content": "in three other datasets per epoch. In the context of deep learning, we do not find the cost increases", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 300, + 505, + 312 + ], + "spans": [ + { + "bbox": [ + 105, + 300, + 505, + 312 + ], + "score": 1.0, + "content": "to be substantial. In particular, since (1) the memory structure in IR and MoCo allow us to store and", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 310, + 506, + 324 + ], + "spans": [ + { + "bbox": [ + 105, + 310, + 506, + 324 + ], + "score": 1.0, + "content": "reuse embeddings and (2) gradients are not propagated through the memory structure, the additional", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 321, + 506, + 334 + ], + "spans": [ + { + "bbox": [ + 105, + 321, + 506, + 334 + ], + "score": 1.0, + "content": "compute of Ring amounts to one matrix multiplication, which is cheap on modern hardware. We", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 331, + 479, + 345 + ], + "spans": [ + { + "bbox": [ + 106, + 331, + 479, + 345 + ], + "score": 1.0, + "content": "used a single Titan X GPU with 8 CPU workers, and PyTorch Lightning (Falcon et al., 2019).", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 21 + }, + { + "type": "text", + "bbox": [ + 107, + 349, + 505, + 405 + ], + "lines": [ + { + "bbox": [ + 106, + 349, + 505, + 362 + ], + "spans": [ + { + "bbox": [ + 106, + 349, + 505, + 362 + ], + "score": 1.0, + "content": "Comparison with the state-of-the-art. Unlike the experiments in Sec. 5, we now choose the op-", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 360, + 505, + 373 + ], + "spans": [ + { + "bbox": [ + 106, + 360, + 505, + 373 + ], + "score": 1.0, + "content": "timal hyperparameters for MoCo-v2 (Chen et al., 2020b) separately for CIFAR10, CIFAR100, and", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 370, + 504, + 384 + ], + "spans": [ + { + "bbox": [ + 105, + 370, + 163, + 384 + ], + "score": 1.0, + "content": "STL10. Table", + "type": "text" + }, + { + "bbox": [ + 163, + 371, + 176, + 384 + ], + "score": 0.42, + "content": "5 \\mathsf { b }", + "type": "inline_equation" + }, + { + "bbox": [ + 177, + 370, + 504, + 384 + ], + "score": 1.0, + "content": "compares MoCo-v2 and its CNCE equivalent, MoCoRing-v2 using linear evalua-", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 381, + 505, + 396 + ], + "spans": [ + { + "bbox": [ + 105, + 381, + 505, + 396 + ], + "score": 1.0, + "content": "tion. We observe comparable improvements as found in Table 1 even with optimal hyperparameters.", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 393, + 497, + 406 + ], + "spans": [ + { + "bbox": [ + 105, + 393, + 497, + 406 + ], + "score": 1.0, + "content": "Notably, the gains generalize to ResNet-50 encoders. Refer to Sec. F for hyperparameter choices.", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 27 + }, + { + "type": "text", + "bbox": [ + 107, + 410, + 505, + 477 + ], + "lines": [ + { + "bbox": [ + 105, + 407, + 506, + 424 + ], + "spans": [ + { + "bbox": [ + 105, + 407, + 506, + 424 + ], + "score": 1.0, + "content": "Generalization to other modalities. Thus far, we have focused on visual representation learning,", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 420, + 505, + 434 + ], + "spans": [ + { + "bbox": [ + 105, + 420, + 505, + 434 + ], + "score": 1.0, + "content": "although the same ideas apply to other domains. To exemplify the generality of CNCE, we apply", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 431, + 506, + 444 + ], + "spans": [ + { + "bbox": [ + 105, + 431, + 326, + 444 + ], + "score": 1.0, + "content": "MoCoRing to learning speech representations. Table", + "type": "text" + }, + { + "bbox": [ + 327, + 432, + 339, + 444 + ], + "score": 0.68, + "content": "5 \\mathrm { { c } }", + "type": "inline_equation" + }, + { + "bbox": [ + 339, + 431, + 506, + 444 + ], + "score": 1.0, + "content": "reports linear evaluation on six transfer", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 442, + 506, + 456 + ], + "spans": [ + { + "bbox": [ + 105, + 442, + 506, + 456 + ], + "score": 1.0, + "content": "datasets, ranging from predicting speaker identity to speech recognition to intent prediction. We", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 454, + 506, + 466 + ], + "spans": [ + { + "bbox": [ + 105, + 454, + 506, + 466 + ], + "score": 1.0, + "content": "find significant gains of 1 to 4 percent over 4 datasets and 6 transfer tasks with an average of 2.2", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 464, + 362, + 478 + ], + "spans": [ + { + "bbox": [ + 105, + 464, + 252, + 477 + ], + "score": 1.0, + "content": "absolute percentage points. See Sec.", + "type": "text" + }, + { + "bbox": [ + 252, + 464, + 263, + 478 + ], + "score": 0.33, + "content": "\\mathrm { ~ E ~ }", + "type": "inline_equation" + }, + { + "bbox": [ + 264, + 464, + 362, + 477 + ], + "score": 1.0, + "content": "for experimental details.", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 32.5 + }, + { + "type": "text", + "bbox": [ + 106, + 482, + 505, + 614 + ], + "lines": [ + { + "bbox": [ + 105, + 480, + 506, + 495 + ], + "spans": [ + { + "bbox": [ + 105, + 480, + 506, + 495 + ], + "score": 1.0, + "content": "Batch-based negative sampling. In Ring, we assumed to have a memory structure that stores", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 106, + 493, + 505, + 505 + ], + "spans": [ + { + "bbox": [ + 106, + 493, + 505, + 505 + ], + "score": 1.0, + "content": "embeddings, which led to an efficient procedure of mining semi-hard negatives. However, an-", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 504, + 505, + 516 + ], + "spans": [ + { + "bbox": [ + 105, + 504, + 505, + 516 + ], + "score": 1.0, + "content": "other flavor of contrastive algorithms removes the memory structure entirely, using the examples", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 104, + 513, + 505, + 528 + ], + "spans": [ + { + "bbox": [ + 104, + 513, + 505, + 528 + ], + "score": 1.0, + "content": "in the minibatch as negatives of one another. Here, we motivate a possible extension of Ring", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 525, + 505, + 538 + ], + "spans": [ + { + "bbox": [ + 105, + 525, + 505, + 538 + ], + "score": 1.0, + "content": "to SimCLR, and leave more careful study to future work. 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For every example", + "type": "text" + }, + { + "bbox": [ + 241, + 549, + 252, + 558 + ], + "score": 0.84, + "content": "x _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 252, + 548, + 505, + 559 + ], + "score": 1.0, + "content": "in the minibatch, only consider the subset of the minibatch", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 107, + 558, + 505, + 572 + ], + "spans": [ + { + "bbox": [ + 107, + 558, + 274, + 571 + ], + "score": 0.9, + "content": "\\{ x : x \\subseteq M , \\exp \\{ g _ { \\theta } ( t ( x _ { i } ) ) ^ { \\cdot } g _ { \\theta } ( t ^ { \\prime } ( x ) ) \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 275, + 558, + 300, + 572 + ], + "score": 1.0, + "content": "in the", + "type": "text" + }, + { + "bbox": [ + 300, + 559, + 306, + 569 + ], + "score": 0.72, + "content": "\\ell \\cdot", + "type": "inline_equation" + }, + { + "bbox": [ + 306, + 558, + 334, + 572 + ], + "score": 1.0, + "content": "-th and", + "type": "text" + }, + { + "bbox": [ + 335, + 560, + 342, + 569 + ], + "score": 0.76, + "content": "u", + "type": "inline_equation" + }, + { + "bbox": [ + 342, + 558, + 410, + 572 + ], + "score": 1.0, + "content": "-th percentiles in", + "type": "text" + }, + { + "bbox": [ + 410, + 559, + 427, + 571 + ], + "score": 0.9, + "content": "M \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 427, + 558, + 505, + 572 + ], + "score": 1.0, + "content": "as negative exam-", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 106, + 570, + 504, + 582 + ], + "spans": [ + { + "bbox": [ + 106, + 570, + 139, + 582 + ], + "score": 1.0, + "content": "ples for", + "type": "text" + }, + { + "bbox": [ + 139, + 572, + 149, + 581 + ], + "score": 0.85, + "content": "x _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 150, + 570, + 504, + 582 + ], + "score": 1.0, + "content": ". This can be efficiently implemented as a matrix operation using an element-wise mask.", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 580, + 505, + 593 + ], + "spans": [ + { + "bbox": [ + 105, + 580, + 377, + 593 + ], + "score": 1.0, + "content": "Thus, we ignore gradient signal for examples too far or too close to", + "type": "text" + }, + { + "bbox": [ + 377, + 582, + 387, + 592 + ], + "score": 0.86, + "content": "x _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 388, + 580, + 505, + 593 + ], + "score": 1.0, + "content": "in representation. As before,", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 106, + 591, + 506, + 605 + ], + "spans": [ + { + "bbox": [ + 106, + 591, + 151, + 605 + ], + "score": 1.0, + "content": "we anneal", + "type": "text" + }, + { + "bbox": [ + 151, + 593, + 159, + 601 + ], + "score": 0.76, + "content": "u", + "type": "inline_equation" + }, + { + "bbox": [ + 159, + 591, + 259, + 605 + ], + "score": 1.0, + "content": "from 100 to 10 and set", + "type": "text" + }, + { + "bbox": [ + 259, + 592, + 286, + 602 + ], + "score": 0.89, + "content": "\\ell = 1", + "type": "inline_equation" + }, + { + "bbox": [ + 287, + 591, + 318, + 605 + ], + "score": 1.0, + "content": ". Table", + "type": "text" + }, + { + "bbox": [ + 318, + 591, + 331, + 604 + ], + "score": 0.77, + "content": "\\textcircled { 5 } \\textcircled { \\scriptsize { \\mathrm { d } } }", + "type": "inline_equation" + }, + { + "bbox": [ + 332, + 591, + 506, + 605 + ], + "score": 1.0, + "content": "report consistent but moderate gains over", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 106, + 603, + 400, + 615 + ], + "spans": [ + { + "bbox": [ + 106, + 603, + 400, + 615 + ], + "score": 1.0, + "content": "SimCLR, showing promise but room for improvement in future research.", + "type": "text" + } + ], + "index": 47 + } + ], + "index": 41.5 + }, + { + "type": "title", + "bbox": [ + 108, + 630, + 252, + 642 + ], + "lines": [ + { + "bbox": [ + 105, + 628, + 253, + 645 + ], + "spans": [ + { + "bbox": [ + 105, + 628, + 253, + 645 + ], + "score": 1.0, + "content": "8 CONCLUDING REMARKS", + "type": "text" + } + ], + "index": 48 + } + ], + "index": 48 + }, + { + "type": "text", + "bbox": [ + 107, + 654, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 655, + 505, + 667 + ], + "spans": [ + { + "bbox": [ + 106, + 655, + 505, + 667 + ], + "score": 1.0, + "content": "To conclude, we presented a family of mutual information estimators that approximate the partition", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 105, + 666, + 505, + 677 + ], + "spans": [ + { + "bbox": [ + 105, + 666, + 505, + 677 + ], + "score": 1.0, + "content": "function using samples from a class of conditional distributions. We proved several theoretical", + "type": "text" + } + ], + "index": 50 + }, + { + "bbox": [ + 106, + 677, + 505, + 689 + ], + "spans": [ + { + "bbox": [ + 106, + 677, + 505, + 689 + ], + "score": 1.0, + "content": "statements about this family, showing a bound on mutual information and a tradeoff between bias", + "type": "text" + } + ], + "index": 51 + }, + { + "bbox": [ + 105, + 686, + 506, + 702 + ], + "spans": [ + { + "bbox": [ + 105, + 686, + 506, + 702 + ], + "score": 1.0, + "content": "and variance. Then, we applied these estimators as objectives in contrastive representation learning.", + "type": "text" + } + ], + "index": 52 + }, + { + "bbox": [ + 105, + 698, + 506, + 712 + ], + "spans": [ + { + "bbox": [ + 105, + 698, + 506, + 712 + ], + "score": 1.0, + "content": "In doing so, we found that our representations outperform existing approaches consistently across a", + "type": "text" + } + ], + "index": 53 + }, + { + "bbox": [ + 105, + 710, + 506, + 722 + ], + "spans": [ + { + "bbox": [ + 105, + 710, + 506, + 722 + ], + "score": 1.0, + "content": "spectrum of contrastive objectives, data distributions, and transfer tasks. Overall, we hope our work", + "type": "text" + } + ], + "index": 54 + }, + { + "bbox": [ + 105, + 720, + 493, + 734 + ], + "spans": [ + { + "bbox": [ + 105, + 720, + 493, + 734 + ], + "score": 1.0, + "content": "to encourage more exploration of negative sampling in the recent growth of contrastive learning.", + "type": "text" + } + ], + "index": 55 + } + ], + "index": 52 + } + ], + "page_idx": 8, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 108, + 27, + 292, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 26, + 293, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 26, + 293, + 38 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2021", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 302, + 751, + 308, + 759 + ], + "lines": [ + { + "bbox": [ + 302, + 751, + 309, + 762 + ], + "spans": [ + { + "bbox": [ + 302, + 751, + 309, + 762 + ], + "score": 1.0, + "content": "9", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "table", + "bbox": [ + 139, + 80, + 295, + 133 + ], + "blocks": [ + { + "type": "table_body", + "bbox": [ + 139, + 80, + 295, + 133 + ], + "group_id": 3, + "lines": [ + { + "bbox": [ + 139, + 80, + 295, + 133 + ], + "spans": [ + { + "bbox": [ + 139, + 80, + 295, + 133 + ], + "score": 0.553, + "html": "
ModelCIFAR10 (sec.)ImageNet (min.)
IR136.0 ±443.9 ±1
IRing141.1 ± 5 (1.1x)51.0±1 (1.2x)
MoCo318.4±1661.1 ±1
MoCoRing383.4 ± 12 (1.2x)64.9 ±1 (1.1x)
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DatasetArch.MoCo-v2MoCoRing-v2
CIFAR10ResNet-1890.191.9(+1.8)
CIFAR10ResNet-5092.494.1(+1.6)
CIFAR100ResNet-1865.167.3(+2.2)
STL10ResNet-1874.876.7(+1.9)
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Transfer TaskMoCoMoCoRing
LibriSpeech Spk.ID (Panayotov et al.201595.596.6(+1.1)
AudioMNISTBecker et al. 2018)87.491.3(+3.9)
Google CommandsWarden 1201838.541.4(+2.9)
Fluent Actions Lugosch et al.] 201936.536.8(+0.3)
Fluent ObjectsLugosch et al. 1201941.944.1(+2.2)
Fluent Locations Lugoschet al. 1201960.963.9(+3.0)
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DatasetSimCLRSimCLRing
CIFAR1088.989.3(+0.4)
CIFAR10063.564.1(+0.6)
STL1071.272.1(+0.9)
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Unlike the experiments in Sec. 5, we now choose the op-", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 360, + 505, + 373 + ], + "spans": [ + { + "bbox": [ + 106, + 360, + 505, + 373 + ], + "score": 1.0, + "content": "timal hyperparameters for MoCo-v2 (Chen et al., 2020b) separately for CIFAR10, CIFAR100, and", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 370, + 504, + 384 + ], + "spans": [ + { + "bbox": [ + 105, + 370, + 163, + 384 + ], + "score": 1.0, + "content": "STL10. 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Table", + "type": "text" + }, + { + "bbox": [ + 327, + 432, + 339, + 444 + ], + "score": 0.68, + "content": "5 \\mathrm { { c } }", + "type": "inline_equation" + }, + { + "bbox": [ + 339, + 431, + 506, + 444 + ], + "score": 1.0, + "content": "reports linear evaluation on six transfer", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 442, + 506, + 456 + ], + "spans": [ + { + "bbox": [ + 105, + 442, + 506, + 456 + ], + "score": 1.0, + "content": "datasets, ranging from predicting speaker identity to speech recognition to intent prediction. We", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 454, + 506, + 466 + ], + "spans": [ + { + "bbox": [ + 105, + 454, + 506, + 466 + ], + "score": 1.0, + "content": "find significant gains of 1 to 4 percent over 4 datasets and 6 transfer tasks with an average of 2.2", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 464, + 362, + 478 + ], + "spans": [ + { + "bbox": [ + 105, + 464, + 252, + 477 + ], + "score": 1.0, + "content": "absolute percentage points. See Sec.", + "type": "text" + }, + { + "bbox": [ + 252, + 464, + 263, + 478 + ], + "score": 0.33, + "content": "\\mathrm { ~ E ~ }", + "type": "inline_equation" + }, + { + "bbox": [ + 264, + 464, + 362, + 477 + ], + "score": 1.0, + "content": "for experimental details.", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 32.5, + "bbox_fs": [ + 105, + 407, + 506, + 478 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 482, + 505, + 614 + ], + "lines": [ + { + "bbox": [ + 105, + 480, + 506, + 495 + ], + "spans": [ + { + "bbox": [ + 105, + 480, + 506, + 495 + ], + "score": 1.0, + "content": "Batch-based negative sampling. In Ring, we assumed to have a memory structure that stores", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 106, + 493, + 505, + 505 + ], + "spans": [ + { + "bbox": [ + 106, + 493, + 505, + 505 + ], + "score": 1.0, + "content": "embeddings, which led to an efficient procedure of mining semi-hard negatives. However, an-", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 504, + 505, + 516 + ], + "spans": [ + { + "bbox": [ + 105, + 504, + 505, + 516 + ], + "score": 1.0, + "content": "other flavor of contrastive algorithms removes the memory structure entirely, using the examples", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 104, + 513, + 505, + 528 + ], + "spans": [ + { + "bbox": [ + 104, + 513, + 505, + 528 + ], + "score": 1.0, + "content": "in the minibatch as negatives of one another. Here, we motivate a possible extension of Ring", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 525, + 505, + 538 + ], + "spans": [ + { + "bbox": [ + 105, + 525, + 505, + 538 + ], + "score": 1.0, + "content": "to SimCLR, and leave more careful study to future work. 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ModelAircraftCUBirdsDTDFungiMNISTFashionMNISTTrafficSignVGGFlowerMSCOCO
IR40.917.939.22.796.991.797.168.152.4
IRing40.6 (-0.3)17.9 (+0.0)39.5 (+0.3)3.4(+0.7)97.8 (+0.9)91.6 (+0.1)98.8 (+1.7)68.5 (+0.4)52.5 (+0.1)
MoCo41.518.039.73.196.990.997.364.552.0
MoCoRing41.6(+0.1)18.6 (+0.6)39.5 (-0.2)3.6(+0.5)97.9 (+1.0)91.3 (+0.4)99.3 (+2.0)69.1 (+4.6)52.6 (+0.6)
CMC40.115.838.34.397.591.594.667.151.4
CMCRing40.8 (+0.7)16.8 (+1.0)40.6(+2.3)4.2 (-0.1)97.9 (+0.4)92.1 (+0.6)97.1( (+2.5)69.1 (+2.0)52.1 (+0.7)
LA41.317.839.02.397.292.398.266.952.3
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COCO: Object DetectionCOCO: Inst. SegmentationCOCO: Keypoint DetectionVOC: Object Detection
Arch.Mask R-CNN,R18-FPN,1x scheduleR-CNN,R18-FPNFaster R-CNN, R18-C4
ModelApbbAPAPApmkAPAPApkpAP5APApbbAPAP
IR8.619.06.68.517.47.434.663.032.95.514.53.3
IRing10.922.98.711.020.99.637.266.135.77.620.34.4
MoCo6.014.34.010.821.49.737.666.536.97.317.94.1
MoCoRing9.420.37.612.022.910.838.767.737.98.022.14.8
LA10.222.08.110.020.39.036.365.335.17.620.04.3
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DatasetArch.MoCo-v2MoCoRing-v2
CIFAR10ResNet-1890.191.9(+1.8)
CIFAR10ResNet-5092.494.1(+1.6)
CIFAR100ResNet-1865.167.3(+2.2)
STL10ResNet-1874.876.7(+1.9)
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DatasetSimCLRSimCLRing
CIFAR1088.989.3(+0.4)
CIFAR10063.564.1(+0.6)
STL1071.272.1(+0.9)
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ModelCIFAR10 (sec.)ImageNet (min.)
IR136.0 ±443.9 ±1
IRing141.1 ± 5 (1.1x)51.0±1 (1.2x)
MoCo318.4±1661.1 ±1
MoCoRing383.4 ± 12 (1.2x)64.9 ±1 (1.1x)
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Transfer TaskMoCoMoCoRing
LibriSpeech Spk.ID (Panayotov et al.201595.596.6(+1.1)
AudioMNISTBecker et al. 2018)87.491.3(+3.9)
Google CommandsWarden 1201838.541.4(+2.9)
Fluent Actions Lugosch et al.] 201936.536.8(+0.3)
Fluent ObjectsLugosch et al. 1201941.944.1(+2.2)
Fluent Locations Lugoschet al. 1201960.963.9(+3.0)
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